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H ANDBOOK OF THE G EOMETRY OF BANACH S PACES

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H ANDBOOK OF THE G EOMETRY OF BANACH S PACES Volume 2 Edited by

W.B. JOHNSON Texas A&M University, College Station, Texas, USA

J. LINDENSTRAUSS The Hebrew University of Jerusalem, Jerusalem, Israel

2003 ELSEVIER Amsterdam • Boston • London • New York • Oxford • Paris San Diego • San Francisco • Singapore • Sydney • Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands © 2003 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-proﬁt educational classroom use. Permissions may be sought directly from Elsevier Science & Technology Rights Department in Oxford, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier Science homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and ‘Obtaining Permissions’. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier’s Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made. First edition 2003 Library of Congress Cataloging-in-Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data Handbook of the geometry of Banach spaces Vol. 2 1. Banach spaces I. Johnson, William B., 1944- II. Lindenstrauss, J. 515.7’32 ISBN 0444513051

ISBN: 0-444-51305-1 ∞ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).

Printed in The Netherlands.

Preface 1. Introduction The aim of this Handbook is to present an overview of the main research directions and results in Banach space theory obtained during the last half century. The scope of the theory, having widened considerably over the years, now has deep and close ties with many areas of mathematics, including harmonic analysis, complex analysis, partial differential equations, classical convexity, probability theory, combinatorics, logic, approximation theory, geometric measure theory, operator theory, and others. In choosing a topic for an article in the Handbook we considered both the interest the topic would have for non specialists as well as the importance of the topic for the core of Banach space theory, which is the study of the geometry of inﬁnite dimensional Banach spaces and n-dimensional normed spaces with n ﬁnite but large (local theory). Many of the leading experts on the various aspects of Banach space theory have written an exposition of the main results, problems, and methods in areas of their expertise. The enthusiastic response we received from the community was gratifying, and we are deeply appreciative of the considerable time and effort our contributors devoted to the preparation of their articles. Our expectation is that this Handbook will be very useful as a source of information and inspiration to graduate students and young research workers who are entering the subject. The material included will be of special interest to researchers in Banach space theory who may not be aware of many of the beautiful and far reaching facets of the theory. We ourselves were surprised by the new light thrown by the Handbook on directions with which we were already basically familiar. We hope that the Handbook is also valuable for mathematicians in related ﬁelds who are interested in learning the new directions, problems, and methods in Banach space theory for the purpose of transferring ideas between Banach space theory and other areas. Our introductory article, “Basic concepts in the geometry of Banach spaces”, is intended to make the Handbook accessible to a wide audience of researchers and students. In this chapter those concepts and results which appear in most aspects of the theory and which go beyond material covered in most textbooks on functional and real analysis are presented and explained. Some of the results are given with an outline of proof; virtually all are proved in the books on Banach space theory referenced in the article. In principle, the basic concepts article contains all the background needed for reading any other chapter in the Handbook. Each article past the basic concepts one is devoted to one speciﬁc direction of Banach space theory or its applications. Each article contains a motivated introduction as well as v

vi

Preface

an exposition of the main results, methods, and open problems in its speciﬁc direction. Most have an extensive bibliography. Many articles, even the basic concepts one, contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers. The format of the chapters is as varied as the personal scientiﬁc styles and tastes of the contributors. In our view this makes the Handbook more lively and attractive. The chapters in Volume 1 were ordered alphabetically according to the ﬁrst author of the chapter. The chapters in this second volume are ordered by the same principle. Chapters which should have been included in Volume 1 according to this principle but were not ready by the deadline we set for Volume 1 appear instead in the present volume. At the end of this volume there appear a few addenda and corrigenda to articles in Volume 1. Even though this Handbook is quite voluminous it was inevitable that a few aspects of Banach space theory are not covered in it, at least in the depth they deserve. Some of these omissions stem from our planning and others from the fact that a few of the researchers who intended to contribute to this Handbook regretfully were not able to participate. Examples of such aspects are (i) Geometric non-linear functional analysis. (ii) Questions related to parameters (like various approximation numbers and widths) as well as the general area of applications to approximation theory. (iii) The connection between Banach space theory and axiomatic set theory. (iv) Probabilistic inequalities and majorizing measures. Nevertheless, we believe that the Handbook of the Geometry of Banach Spaces presents a reasonably comprehensive and accessible view of the present state of the subject. As for the history of the subject, a short account of the history of local theory is contained in an article by B. Maurey in this volume. A comprehensive history of Banach space theory is now being prepared by A. Pietsch. This volume ends with an author index and a subject index which is common to both volumes.

William B. Johnson and Joram Lindenstrauss

List of Contributors Argyros, S.A., University of Athens, Athens (Ch. 23) Godefroy, G., Université Paris VI, Paris (Ch. 23) Gowers, W.T., Centre for Mathematical Sciences, Cambridge (Ch. 24) Kalton, N., University of Missouri, Columbia, MO (Chs. 25, 26) Ledoux, M., Université Paul-Sabatier, Toulouse (Ch. 27) Mankiewicz, P., Institute of Mathematics, PAN, Warsaw (Ch. 28) Maurey, B., Université Marne la Vallée, Marne la Vallée (Chs. 29, 30) Montgomery-Smith, S., University of Missouri, Columbia, MO (Ch. 26) Odell, E., The University of Texas, Austin, TX (Ch. 31) Pełczy´nski, A., Institute of Mathematics, PAN, Warsaw (Ch. 32) Pisier, G., Université Paris VI, Paris and Texas A&M University, College Station, TX (Chs. 33, 34) Preiss, D., University College London, London (Ch. 35) Rosenthal, H.P., The University of Texas at Austin, Austin, TX (Chs. 23, 36) Schechtman, G., Weizmann Institute of Science, Rehovot (Ch. 37) Schlumprecht, Th., Texas A&M University, College Station, TX (Ch. 31) Tomczak-Jaegermann, N., University of Alberta, Edmonton (Ch. 28) Tzafriri, L., The Hebrew University of Jerusalem, Jerusalem (Ch. 38) Wojciechowski, M., Institute of Mathematics, PAN, Warsaw (Ch. 32) Wojtaszczyk, P., Warsaw University, Warsaw (Ch. 39) Xu, Q., Université de Franche-Comté, Besançon (Ch. 34) Zinn, J., Texas A&M University, College Station, TX (Ch. 27) Zippin, M., The Hebrew University of Jerusalem, Jerusalem (Ch. 40) Zizler, V., University of Alberta, Edmonton (Ch. 41)

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Contents of Volume 1 Preface List of Contributors

v vii

1. Basic concepts in the geometry of Banach spaces W.B. Johnson and J. Lindenstrauss 2. Positive operators Y.A. Abramovitch and C.D. Aliprantis 3. Lp spaces D. Alspach and E. Odell 4. Convex geometry and functional analysis K. Ball 5. ΛP -sets in analysis: Results, problems and related aspects J. Bourgain 6. Martingales and singular integrals in Banach spaces D.L. Burkholder 7. Approximation properties P.G. Casazza 8. Local operator theory, random matrices and Banach spaces K.R. Davidson and S.J. Szarek 9. Applications to mathematical ﬁnance F. Delbaen and W. Schachermayer 10. Perturbed minimization principles and applications R. Deville and N. Ghoussoub 11. Operator ideals J. Diestel, H. Jarchow and A. Pietsch 12. Special Banach lattices and their applications S.J. Dilworth 13. Some aspects of the invariant subspace problem P. Enﬂo and V. Lomonosov 14. Special bases in function spaces T. Figiel and P. Wojtaszczyk 15. Inﬁnite dimensional convexity V.P. Fonf, J. Lindenstrauss and R.R. Phelps 16. Uniform algebras as Banach spaces T.W. Gamelin and S.V. Kislyakov ix

1 85 123 161 195 233 271 317 367 393 437 497 533 561 599 671

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17. Euclidean structure in ﬁnite dimensional normed spaces A.A. Giannopoulos and V.D. Milman 18. Renormings of Banach spaces G. Godefroy 19. Finite dimensional subspaces of Lp W.B. Johnson and G. Schechtman 20. Banach spaces and classical harmonic analysis S.V. Kislyakov 21. Aspects of the isometric theory of Banach spaces A. Koldobsky and H. König 22. Eigenvalues of operators and applications H. König

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Author Index Subject Index

975 993

781 837 871 899 941

Contents of Volume 2 Preface List of Contributors Contents of Volume 1

v vii ix

23. Descriptive set theory and Banach spaces S.A. Argyros, G. Godefroy and H.P. Rosenthal 24. Ramsey methods in Banach spaces W.T. Gowers 25. Quasi-Banach spaces N. Kalton 26. Interpolation of Banach spaces N. Kalton and S. Montgomery-Smith 27. Probabilistic limit theorems in the setting of Banach spaces M. Ledoux and J. Zinn 28. Quotients of ﬁnite-dimensional Banach spaces; random phenomena P. Mankiewicz and N. Tomczak-Jaegermann 29. Banach spaces with few operators B. Maurey 30. Type, cotype and K-convexity B. Maurey 31. Distortion and asymptotic structure E. Odell and Th. Schlumprecht 32. Sobolev spaces A. Pełczy´nski and M. Wojciechowski 33. Operator spaces G. Pisier 34. Non-commutative Lp -spaces G. Pisier and Q. Xu 35. Geometric measure theory in Banach spaces D. Preiss 36. The Banach spaces C(K) H.P. Rosenthal 37. Concentration, results and applications G. Schechtman xi

1007 1071 1099 1131 1177 1201 1247 1299 1333 1361 1425 1459 1519 1547 1603

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38. Uniqueness of structure in Banach spaces L. Tzafriri 39. Spaces of analytic functions with integral norm P. Wojtaszczyk 40. Extension of bounded linear operators M. Zippin 41. Nonseparable Banach spaces V. Zizler Addenda and corrigenda to Chapter 7, Approximation properties by Peter G. Casazza Addenda and corrigenda to Chapter 8, Local operator theory, random matrices and Banach spaces by K.R. Davidson and S.J. Szarek Addenda and corrigenda to Chapter 11, Operator ideals by J. Diestel, H. Jarchow and A. Pietsch Addenda and corrigenda to Chapter 15, Inﬁnite dimensional convexity by V.P. Fonf, J. Lindenstrauss and R.R. Phelps

1635 1671 1703 1743

1817 1819 1821 1823

Author Index

1825

Subject Index

1849

CHAPTER 23

Descriptive Set Theory and Banach Spaces Spiros A. Argyros Department of Mathematics, University of Athens, 15780, Athens, Greece E-mail: [email protected]

Gilles Godefroy Equipe d’Analyse, Université Paris VI, 4 place Jussieu, F-75252, Paris Cedex 05, France E-mail: [email protected]

Haskell P. Rosenthal∗ Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082, USA E-mail: [email protected]

Contents I. I.1. I.2. I.3. I.4. I.5. I.6. II. II.1. II.2. II.3. II.4. II.5. II.6. III. III.1. III.2.

Basic concepts in descriptive set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trees and analytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Set derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compact sets of ﬁrst Baire class functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representable Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Descriptive set theoretic complexity of families of Banach spaces . . . . . . . . . . . . . . . . Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The c0 -theorem and Banach space invariants associated with Baire-1 functions . . . . . . . . Connections between Baire-1 functions, D-functions, and Banach spaces containing c0 or 1 The c0 -theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spreading models associated with two special classes of Baire-1 functions . . . . . . . . . . . Transﬁnite analogs and the Index Theorem for spaces not containing c0 . . . . . . . . . . . . Some open universality problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weakly null sequences and asymptotic p spaces . . . . . . . . . . . . . . . . . . . . . . . . . Compact families of ﬁnite subsets of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schreier families and the repeated averages hierarchy . . . . . . . . . . . . . . . . . . . . . . Schreier families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Research partially supported by NSF Grant DMS-0070547.

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2 Edited by William B. Johnson and Joram Lindenstrauss © 2003 Elsevier Science B.V. All rights reserved 1007

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The repeated averages hierarchy (RA-hierarchy) . . . . . . . . . . . . . III.3. Restricted unconditionality and dichotomies for weakly null sequences III.4. Asymptotic p spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tsirelson and mixed Tsirelson norms . . . . . . . . . . . . . . . . . . . III.5. Notes and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract The goal of this chapter is to reveal the power of descriptive set theory in penetrating the structure of Banach spaces. The chapter is divided into three subchapters, each with its own introduction. Subchapters one, two and three were mostly written by the second, third, and ﬁrst authors, respectively. Space limitations forced us to leave out many fundamental results and on-going research in this exciting interface. We do hope, however, that our article gives a strong ﬂavor of this aspect of Banach space theory.

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I. Basic concepts in descriptive set theory In this chapter, we introduce the basic concepts in descriptive set theory, and display some of their applications to Banach space theory. Section 1 shows the classical theory of analytic subsets of Polish spaces, the rank theorem for trees, and the countable height of closed well-founded trees in Polish spaces. Set derivations are introduced in Section 2, and ﬁrst Baire class functions in Section 3, which is devoted to the proof of a dichotomy result on compact sets of ﬁrst Baire class functions; application is given to Rosenthal’s 1 -theorem. Representable Banach spaces are deﬁned and studied in Section 4, where a classiﬁcation theorem is provided. Section 5 is devoted to the topological complexity of families of Banach spaces, in relation with universality problems and index theory. N OTATION . If E is a set, we denote by E N (resp. E [N] ) the set of all all sequences (resp. ﬁnite sequences) in E. If s ∈ E [N] and n ∈ E. s n stands for the concatenation of s and n, and |s| is the length of s. We denote by ω1 the ﬁrst uncountable ordinal or equivalently the set of all countable ordinals. The letter α will usually denote a countable ordinal, hence α ∈ ω1 . We let π1 = A × B → A be the ﬁrst projection: π1 ((x, y)) = x. If M is a topological space, T(M) denotes the collection of all closed subsets of M. I.1. Trees and analytic sets A topological space S is called a Polish space if it is homeomorphic to a separable complete metric space. The space NN consisting of all sequences of natural numbers, equipped with the product of the discrete topologies, is an important example of a Polish space, which is homeomorphic (through continued fractions) to the set [0, 1] \ Q equipped with the topology induced by the real line. Let N[N] be the set of ﬁnite sequences of integers. If s ∈ N[N] , we denote by |s| the length of s. If σ ∈ NN , we write s σ if σ “starts with s”, that is, s(i) = σ (i) for all 1 i |s|. Set V s = σ ∈ NN : s σ . Then for any σ ∈ NN , the family {Vs : s σ } is a basis of clopen neighborhoods of σ . A subset T of N[N] is called a tree if t ∈ T and s < t implies that s ∈ T . We denote by [T ] the “boundary” of T , namely: [T ] = σ ∈ NN : s ∈ T for all s σ . It is easily seen that [T ] is a closed subset of NN , and that any closed subset of NN can be obtained in this way. A tree T is well founded if [T ] = ∅. We denote by S the set of all trees of integers (hence S is contained in the power set of N[N] ) and by WF the subset of S consisting of well-founded trees. P ROPOSITION I.1.1. Let S be a Polish space, and A ⊂ S be a subset of S. The following assertions are equivalent:

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(i) There is a continuous map ϕ : NN → S such that A = ϕ(NN ). (ii) There is a Polish space P and a continuous map ψ : P → S such that ψ(P ) = A. (iii) There is a closed subset F of S × NN such that A = π1 (F ). A subset A of S satisfying (i), (ii), (iii) is called an analytic subset of S. Indeed, (ii) implies (i) since any Polish space is a continuous image of NN ; (i) implies (iii) by considering the graph of ϕ, and (iii) implies (i) since a closed subset of a Polish space is Polish. Using the ﬂexibility of the space NN allows us to show through condition (iii) that many sets are analytic. L EMMA I.1.2. Let (An ) be a sequence of analytic subsets of a Polish space S. Then the sets n∈N

An ,

An

n∈N

are analytic. Indeed write An = π1 (Fn ), with Fn closed in S × NN . The set F = (x, σ, n) ∈ S × NN × N: (x, σ ) ∈ Fn is a closed subset of S × NN × N, and π1 (F ) = n∈N An . The result follows since NN × N N N . A similar argument works for the intersection, using this time that (NN )N NN . Since closed sets are trivially analytic, Lemma I.1.2 shows that all sets in the σ -ﬁeld generated by closed sets (i.e., all Borel sets) are analytic. Note that Proposition I.1.1 shows that continuous functions map analytic sets to analytic sets. As shown by Souslin, it is not so for Borel sets. We now display the link with trees. Let A = π1 (F ) be analytic, with F closed in S × NN . For all x ∈ S, let B(x, δ) = {y ∈ S: d(x, y) < δ}. We set

T (x) = s ∈ N[N] : B x, |s|−1 × Vs ∩ F = ∅ . It is clear that T (x) is a tree, and moreover that x ∈ A ⇔ T (x) ∈ / WF. Let us denote by ω1 = {α; α < ω1 } the set of all countable ordinals. We deﬁne the height h(T ) of a tree T as follows. Given T ∈ S, we “trim it” and deﬁne T = s ∈ N[N] ; ∃ n ∈ N such that s n ∈ T .

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We proceed by transﬁnite induction to deﬁne

T (α+1) = T (α) and if β is a limit ordinal, T (β) =

T (α) .

α<β

It is easily seen that T is well-founded if and only if there is a (countable) ordinal α < ω1 such that T (α) = ∅. We deﬁne h(T ) ∈ ω1 to be the least of such ordinals. If T ∈ / WF, we let h(T ) = ω1 . Returning to A = π1 (F ) an analytic subset of S, we may now deﬁne r : S → ω1 ∪ {ω1 } by r(x) = h(T (x)). Then r(x) = ω1 if and only if x ∈ A, and C =S \A=

Cα

α<ω1

with Cα = r −1 ({λ; λ < α}). Such a set C, whose complement A = S \ C is analytic, is called a coanalytic set. By the above, we can associate with C a natural map r : C → ω1 , called the Lusin–Sierpinski index. Roughly speaking, the larger r(x) ∈ ω1 is, the harder it is to check that x ∈ C, or equivalently the harder it is to check that T (x) ∈ WF. We need a natural topology on the set S of all trees. For this, we recall that S ⊆ [N] 2(N ) =def K. It is clear that S is closed in K equipped with the product topology, and it follows that S equipped with the induced topology is homeomorphic to the Cantor set. The subset WF of S is coanalytic in S. Now we may state (without proof) the rank theorem for trees. T HEOREM I.1.3. For any countable ordinal α ∈ ω1 , the set Bα = T ∈ S: h(T ) < α is a Borel subset of S. Moreover, if A is any analytic subset of WF ⊆ S then sup h(T ) < ω1 .

T ∈A

This result implies the “separation theorem”: if A and A are disjoint analytic subsets of S, and r : S \ A = C → ω1 is deﬁned as above by r(x) = h(T (x)), then r(A ) is analytic in WF and thus by Theorem I.1.3 there is an α ∈ ω1 such that r(A ) ⊆ Bα . Hence B = r −1 (Bα ) is a Borel subset of S such that A ⊆ B and A ∩ B = ∅, and we have shown that

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disjoint analytic sets can be separated by a Borel set. In particular, a subset of S is Borel if and only if it is analytic and coanalytic. An easy transﬁnite induction shows that for any α ∈ ω1 , there is a T ∈ WF such that h(T ) = α. Hence the “rank Theorem” I.1.3 shows that WF is a coanalytic non-Borel subset of S. Moreover, Theorem I.1.3 shows through the Lusin–Sierpinski index that any coanalytic set is the union of ω1 Borel sets Cα = r −1 (Bα ), sometimes called its constituents. So far, we considered only trees on integers, i.e., subsets of N[N] . A tree on a Polish space S is more generally, a subset T of S [N] such that t ∈ T and s < t implies s ∈ T . The space S [N] is the direct sum of the Polish spaces {S k ; k 1} and as such it is an analytic metric space. Well-founded trees on S and their height are deﬁned as before. Then the following holds: T HEOREM I.1.4. Let S be a Polish space, and T ⊆ S [N] an analytic (in particular, a closed) well-founded tree. Then its height is countable, i.e., h(T ) < ω1 .

I.2. Set derivations Let K be a compact metric space. A derivation is a map d from the set F(K) of closed subsets of K to itself which satisﬁes the following properties: (a) F ⊆ G ⇒ d(F ) ⊆ d(G). (b) d(F ) ⊆ F . The set F(K) is compact when equipped with the Hausdorff distance dH . A derivation will be called Borel if it is a Borel map from (F(K), dH ) to itself. We can iterate derivations in the obvious way, letting for ξ ω1 d ξ (F ) =

d d α (F ) . α<ξ

It follows easily from the fact that the topology of K has a countable basis that for any F ∈ F(K), there is an α < ω1 such that d α (F ) = d ω1 (F ). We let rd (F ) = min α < ω1 : d α (F ) = d ω1 (F ) . With this notation, one has T HEOREM I.2.1. Let d : F(K) → F(K) be a Borel derivation. The set C = F ∈ F(K): d ω1 (F ) = ∅ is a coanalytic subset of F(K). If A ⊆ C is analytic, then sup rd (F ): F ∈ A < ω1 .

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In particular, if sup rd (F ): F ∈ C = ω1 then C is coanalytic non-Borel. (For a proof, see Theorem 34.10, p. 272 of [72].) E XAMPLE I.2.2. (a) Let d(F ) = F be the set of all accumulation points of F . Then d is a Borel derivation. Hence if K is not scattered, the set of all scattered closed subsets of K is coanalytic non Borel in F(K). (b) Similarly, let X be a separable Banach space, and K = (BX∗ , ω∗ ). Pick any ε > 0 and deﬁne V : V ω∗ -open in F, · - diam(V ) < ε . dε (F ) = F \ Then dε is a Borel derivation, which will be used later for deﬁning the Szlenk index. In the above, ω∗ -open sets can be replaced by ω∗ -slices, to deﬁne a “slower” derivation.

I.3. Compact sets of ﬁrst Baire class functions We ﬁrst recall Baire’s fundamental theorem. T HEOREM I.3.1. Let E be a complete metric space, let X be a Banach space, and f : E → X. The following are equivalent: (i) For every non-empty closed subset F ⊆ E, f |F has a point of continuity. (ii) The map f is the pointwise limit of a sequence (fn ) of continuous mappings from E into X. Functions which satisfy (i)–(ii) are called ﬁrst Baire class functions. E XAMPLE I.3.2. Let Y be a separable Banach space, and f = Id : (BY ∗ , ω∗ ) → (BY ∗ , · ). Then f is a ﬁrst Baire class function if and only if Y ∗ is separable, if and only if any ω∗ -closed F = ∅ in BY ∗ has ω∗ -open subsets of arbitrarily small · -diameter. The natural link between ﬁrst Baire class functions and derivations is provided by the following observation: let f : K → X and ε > 0 be given, and deﬁne dfε (F ) = F \

V open: diam f (V ) < ε .

Then f is a ﬁrst Baire class function if and only if (dfε )(α) (K) = ∅ for some α < ω1 . The least such α, denoted β(f, ε), provides an index which measures “how far” f is from being continuous. Also, we set β(f ) = sup{β(f, ε): ε > 0} and call this the Baire-1 index of f .

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Now let S be a Polish space. We denote by B1 (S) the space of ﬁrst Baire class functions from S to R, equipped with the topology τρ (S) of pointwise convergence on S. We use derivations for proving T HEOREM I.3.3. Let {fn } be a sequence of continuous functions from S to R. Let K be the closure of {fn } in Rs equipped with the product topology τρ (S). The following are equivalent: (i) Every f ∈ K \ {fn ; n 1} is the pointwise limit of a subsequence of {fn }. (ii) Every f ∈ K is a Borel function. (iii) K ⊆ B1 (S). P ROOF. Only (ii) ⇒ (i) requires a proof. Pick f ∈ K \ {fn : n 1}, and ﬁx a free ultraﬁlter U on N such that f (x) = lim fn (x) n→U

for all x ∈ S. For (u, v) ∈ Q2 with u < v, we deﬁne a derivation Du,v : for A ⊆ S, with A closed, deﬁne Du,v (A) by: x ∈ Du,v (A) if for any neighborhood V of x, one has

n 1: inf fn < u < v < sup fn ∈ U. V ∩A

V ∩A

Assume that F = ∅ is such that Du,v (F ) = F . For any ε = ε(k) ∈ {0, 1}[N] , we construct by induction non-empty open sets Vε in F and a sequence {kn : n −1} of integers such that for all s and t, (i) diam(Vs ) < 2−|s| ; (ii) V s ⊂ Vt if s extends t; (iii) if |s| = n, Vs0 ⊂ {fkn < u} and Vs1 ⊂ {fkn > v}. Indeed we may assume that diam(F ) 1. We start with Vφ = F and k−1 = 1. If the construction is done up to |s| = n and fkn−1 , we let As = n 1; inf fn < u < v < sup fn . Vs ∩F

Vs ∩F

Since As ∈ U for all |s| = n, one has

As ∩ {k kn−1 } = ∅.

|s|=n

We pick kn in this set and then construct the open sets Vs (|s| = n + 1) using fkn . For any σ = σ (k) ∈ {0, 1}N , we deﬁne

h(σ ) = {Vs : σ extends s}.

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Clearly, h = {0, 1}N → T(S) is continuous. If V is any free ultraﬁlter on N and g = limn→V (fkn ◦ h), then g(σ ) >

u+v ⇔ σ ∈ V. 2

Since a non-trivial ultraﬁlter is a non-Borel subset of {0, 1}N , we have a proof that (ii) implies Du,v (F ) = F for any F = ∅ and any u < v. (α) (S) for all α ∈ ω1 . By the above, there is ξ(u, v) = ξ < ω1 We now deﬁne Fα = Du,v such that Fξ = ∅, hence S=

{Fα \ Fα+1 ; α < ξ }.

Let {V : 1} be a basis of the topology of S. For any (u, v) ∈ Q2 , there is an α < ξ and an 1 such that at least one of the two sets Au,v ,α = n 1: inf fn u V ∩Fα

and u,v B,α = n 1: sup fn v V ∩Fα

belong to U . We denote by D the countable collection of all subsets of N of the form Au,v ,α u,v or B,α that belong to U , where (u, v) ∈ Q2 , 1 and α < ξ(u, v) are arbitrary. The set D is the basis of a ﬁlter H contained in U , and it is easily checked that for all x ∈ S, f (x) = lim fn (x). n→H

It is now easy, using the fact that H has a countable basis, to ﬁnd a subsequence of {fn } which τρ (S)-converges to f . This shows (i). R EMARK I.3.4. The above proof shows that if {fn } has no pointwise convergent subsequence, there are u < v, a copy of K = h({0, 1}N) of the Cantor set and a subsequence {fkn } such that (fkn ◦ h)(ε) v if ε(n) = 1. In other words, the subsequence {fkn } is equivalent to the sequence of the coordinate functions on {0, 1}N . Note in particular that {fkn } has no pointwise convergent subsequence, and no Borel pointwise cluster point. It is easily seen that the sequence of coordinate functions on {0, 1}N is equivalent in the Banach space C({0, 1}N ) to the natural basis of 1 (N). Hence Theorem I.3.3 and its proof lead to the 1 -theorem ([112]).

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T HEOREM I.3.5. Let (xn )n1 be a bounded sequence in a Banach space X. Then exactly one of the following properties holds true: (a) (xn ) has a subsequence (xkn ) equivalent to the 1 -basis. (b) Every subsequence of (xn ) has a weak Cauchy subsequence. The argument consists in applying Theorem I.3.3 to S = (BX∗ , ω∗ ), and to fn = xn . Note that condition (i) implies the theorem of Odell and Rosenthal [102]: when a separable Banach space X does not contain 1 (N), every x ∗∗ ∈ X∗∗ is the ω∗ -limit of a weak Cauchy sequence in X. (For another exposition of the 1 -theorem, see [62], Theorem 3.3 of Chapter 24 of this Handbook.)

I.4. Representable Banach spaces When a Banach space X is not separable, it is natural to look for a subset of X which witnesses the non-separability, such as an uncountable biorthogonal system. This cannot be done in full generality. However, we show in this section that it can indeed be done when the Banach space is sufﬁciently regular. We need the following D EFINITION I.4.1. A Banach space X is representable if it is isomorphic to a subspace Y of ∞ (N), such that Y is analytic in RN – equivalently, analytic in (∞ (N), ω∗ ). It is plain that separable spaces are representable. If Y is separable, Y ∗ is representable as a Kσ in RN . If K is a separable compact subset of (B1 (S), τρ ), where S is a Polish space (called a Rosenthal compact in [56]), then C(K) is representable. The following result motivates Deﬁnition I.4.1. T HEOREM I.4.2. Let X be a non-separable representable Banach space. Then X has a biorthogonal system of cardinality of the continuum. That is, there exists a bounded subset {(xσ , xσ∗ ): σ ∈ {0, 1}N} of X × X∗ such that xt∗ (xs ) = δst . P ROOF. We still denote by ω∗ the restriction to X of the ω∗ topology of ∞ (N). Let φ : NN → (X, ω∗ ) be a continuous onto map. Fix ε > 0. An easy transﬁnite induction provides {xα : α < ω1 } ⊆ X and {xα∗ : α < ω2 } ⊆ X∗ such that (i) xα 1. xα∗ < 1 + ε for all α. (ii) If β < α, xα∗ (xβ ) = 0 and xα∗ (xα ) = 1. Pick σα ∈ NN such that φ(σα ) = xα . Since NN is a Polish space, there is an α0 < ω1 such that for all α > α0 , every neighborhood of σα contains uncountably many of {σγ : γ < ω2 }. Let δ0 be some complete metric on NN . For n 1, we construct balls {Bsn : s ∈ {0, 1}n } in NN of δ0 -radius less than 1/n, such that n+1 ⊆ Bsn for s ∈ {0, 1}n and i ∈ {0, 1}; (a) Bsi (b) there exist fsn ∈ 1 such that fsn < 1 + ε, and fsn > 1 − ε on φ(Bsn ), |fsn | 1/n on φ(Bsn ) if s = s.

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For doing so, we ﬁrst pick α1 > α0 . By ω∗ -approximation of xα∗1 from 1 , there is an f11 ∈ 1 with f11 < 1 + ε and f11 [φ(σα1 )] = 1. Since φ is continuous, we ﬁnd B11 containing σα1 of radius less than 1 such that f11 [φ(σ )] > 1 − ε for all σ ∈ B11 . Pick now σ2 > β1 > α1 such that σβ1 ∈ B11 and σα2 ∈ B11 . One has

xα∗2 φ(σβ1 ) = 0;

xα∗2 φ(σα2 ) = 1

and again, we ﬁnd by ω∗ -approximation f12 ∈ 1 with f12 < 1 + ε and

f12 φ(σβ1 ) = 0;

f12 φ(σα2 ) = 1.

Since (f12 ◦ φ) is continuous on NN and every neighborhood of σβ1 contains uncountably many σγ ’s, there is an α3 > α2 such that σα3 ∈ B11 and f12 [φ(σα3 )] < 1/4. We have now

xα∗3 φ(σα3 ) = 1;

xα∗3 φ(σα1 ) = 0.

Approximating xα∗3 , we ﬁnd f22 ∈ 1 with f22 < 1 + ε and

f22 φ(σα3 ) = 1;

f22 φ(σα1 ) = 0.

We may now ﬁnd B12 containing σα2 and B22 containing σα3 such that (a) and (b) are satisﬁed. It should now be clear how to proceed to complete the construction. To ﬁnish the proof, we deﬁne A : {0, 1}N → NN by A(σ ) =

Bσn|n

n1

and we let xσ = φ(A(σ )]. We pick xσ∗ ∈ X∗ a ω∗ -cluster point in X∗ of the bounded sequence [r(fσn|n )], where r : 1 → X∗ is the canonical map of restriction to X. Conditions (a) and (b) show that the set (xσ , xσ∗ ) works. This result opens the way to a classiﬁcation theorem, which we state without proof; (i) follows from Theorem I.4.2. T HEOREM I.4.3. Let X be a representable Banach space. Then (i) X is separable if and only if X contains no uncountable biorthogonal system. (ii) X does not contain 1 ({0, 1}N) if and only if X contains no uncountable Markushevich basis, if and only if (BX∗ , ω∗ ) is an angelic compact space. (iii) X contains 1 ({0, 1}N ) if and only if X containsa total uncountable biorthogonal system, i.e., (xt , xt∗ ) such that xs∗ (xt ) = δs,t and t ker(xt∗ ) = {0}.

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I.5. Descriptive set theoretic complexity of families of Banach spaces Let S be a Polish space; recall that F(S) denotes the set of all closed subsets of S. Let S be any metrizable compactiﬁcation of S. The map F → F from F(S) to F( S) is one-to-one and maps F(S) to F0 ( S) = {F ; F ∩ P = F }. We recall some basic properties (see [36]): S) is a Gδ -subset of F( S), hence a Polish space for the Hausdorff metric of F( S). 1. F0 ( 2. The induced Borel structure (called the Effros–Borel structure) makes F(S) a standard Borel space. 3. This Borel structure is generated by the sets {F ∈ F(P ); F ∩ Vn = ∅}, where (Vn )n is a basis of the topology of S. Hence it does not depend upon the compactiﬁcation. It is a classical theorem of Banach that every separable Banach space is isometric to a subspace of C({0, 1}N). The subset V of F(C({0, 1}N)) consisting of all vector subspaces is Borel in the Effros–Borel structure, and it is therefore a standard Borel space (i.e., it is Borel-isomorphic to R). This frame allows us to speak about Borel, resp. analytic, resp. coanalytic families of Banach spaces since all these notions can be deﬁned in a standard Borel space. (Cf. also pp. 262–266 of [72] for a treatment of coanalytic families of Banach spaces.) We denote by E ⊆ V 2 the graph of the equivalence relation of linear isomorphism. T HEOREM I.5.1. The set E is analytic non-Borel in V 2 . The equivalence relation of linear isomorphism has no analytic section. In fact, there is a separable space U such that its equivalence class U is not Borel. Also, it follows from the results in [26] that Lp is not Borel for all p = 2 with 1 < p < ∞. This space U is the universal space constructed in [107]. It is not known whether 2 (N) is the only equivalence class which is Borel. Theorem I.5.1 says that the quotient (V/ ) is not a standard Borel space in any natural structure. T HEOREM I.5.2. Let E ⊆ V be an analytic set of separable Banach spaces, stable under linear isomorphism. If E contains all separable reﬂexive spaces, then there is X ∈ E such that X is universal, i.e., X contains an isomorphic copy of every separable Banach space. Theorem I.5.2 yields the result of Bourgain that a separable Banach space is universal provided every separable reﬂexive space embeds into it [23]. In turn, the proof of I.5.2 is founded on Bourgain’s arguments. See Section II.1.6 and also Theorem III.4.7 and the paragraph immediately following for further discussion of Bourgain’s work in this connection. Of course (I.5.2) (and in fact the work in [22]) yields that the class SD of Banach spaces with separable dual, is coanalytic non-Borel. The Szlenk index is then a natural rank deﬁned on this coanalytic class. G. Lancien has exhibited a “universal control function” deﬁned on the countable ordinals, showing that the “dentability index” of any space X in SD may be controlled by its Szlenk index [82] (see also p. 805 of [58]). Thus it follows that the slice derivation is not much slower than the Szlenk derivation given in I.2.2(b). A similar theory can be developed for basic sequences, where linear isomorphism is replaced by equivalence between bases. In this frame, the analogous result to Theorem I.5.1

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holds true; in particular, the set of basic sequences up to equivalence is not a standard Borel space in any natural way.

I.6. Notes and remarks 1. The classical theory of analytic sets goes back to Souslin [126] and Lusin [87]. Classical references include the books [79] (§33–39), [36], and [98]. We refer to [73] and references therein for applications of descriptive set theory to harmonic analysis. Transﬁnite ranks belong to the classical theory [88]. The derivation of Theorem I.1.4 from Theorem I.1.3 is given in ([74], pp. 146–147). Both are special cases of the Kunen–Martin uniform boundedness theorem ([98], p. 101), which involves arbitrary cardinals. An application of Theorem I.1.4 is given in [26]: there exist uncountably many pairwise non-isomorphic nonHilbertian complemented subspaces of Lp , 1 < p < ∞, p = 2. (Such are all Lp spaces; see [2].) It remains an open question if the cardinality of isomorphism types equals the continuum. 2. Derivations were ﬁrst considered by Cantor in his 1870 solution of the uniqueness problem for trigonometric series; it is well-known that this work led him to the creation of set theory. Cantor’s derivation is Example 2.2(a). Theorem I.2.1 is in particular an extension of Hurewicz’s theorem [67] asserting that the set of countable closed subsets of [0, 1] is coanalytic non-Borel in the Hausdorff topology. It is a special case of a theorem due to Moschovakis (see [98,133]). A link between Theorem I.2.1 and Theorem I.1.4 is obtained via the tree T ⊆ F(K)[N] consisting of all ﬁnite sequences (F0 , F1 , . . . , Fn ) such that d(Fi ) = Fi−1 for 1 i n. Finally, we note the following result due to Ghoussoub and Maurey [55]: Every Banach space with the Radon Nikodym Property (or more generally, the Point of Continuity Property) has a subspace isometric to a dual Banach space. The proof of this result involves a transﬁnite inductive argument and concepts in our chapter. For an exposition, see [117] and [52]; also see [52] for discussion of the RNP and PCP. 3. Theorem I.3.1 is Baire’s main theorem, whose proof goes back to [13] in the case of realvalued functions of a real variable. This is the ﬁrst time where transﬁnite arguments were used in studying sequences of functions. Easy modiﬁcations of the original proof provide the general result. Finite and transﬁnite indices which measure the complexity of ﬁrst Baire class functions have recently been intensively studied ([74,33,32,34,75,76,84,83,119]), in particular in the context of Banach space theory ([123,66,1,42–46,11,77]). They are an operative tool in the proof of the c0 -theorem [118] (given as Theorem II.2.2 below). The study of pointwise compact subsets of the ﬁrst Baire class originated in [114]; these sets show up in the proof of Rosenthal’s 1 -theorem [112] and Odell–Rosenthal’s characterizations of Banach spaces not containing 1 [102]; see also [116]. Theorem 4.3 is Rosenthal’s dichotomy [115]. Most of the problems left open in [114] were solved in [25]. We refer to [130] for recent deep classiﬁcation results. An ordinal ranking of these compact sets has been deﬁned in [91], using some results from [56]. Theorem I.3.3 is from [38], where it was used for an effective version of the results; see also [109].

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4. Representable Banach spaces are deﬁned in [61], where it is shown how to adjust Stegall’s proof [127] in order to obtain Theorem I.4.2. The more complete Theorem I.4.3 [50] follows from Theorem I.4.2 and results from [25,37] and [125]. Under an appropriate determinacy axiom, such results extend to subspaces of ∞ (N) which belong to a projective class in the w∗ -topology [60]. 5. This section presents the frame in which the descriptive complexity of families of Banach spaces can be deﬁned and studied, as shown in [17] (see [20]). Theorems I.5.1 and I.5.2 are established in [20]. These results have important antecedents in Bourgain’s work. Bourgain was the ﬁrst discoverer of the connection between descriptive set theory and universality results. In [22], Bourgain showed in just a few lines, that a separable Banach space X is universal provided C(K) embeds in X for every countable compact metric space K, thus also giving an extension of Szlenk’s theorem [128]. His elegant argument simply exploits the classical fact that the family of closed countable subsets of [0, 1] is coanalytic non-Borel in F([0, 1]]). (See [116] for an exposition of these results of Bourgain.) A primary motivation for [22] is Bourgain’s ordinal-proof, via the Szlenk index for operators and the Kunen–Martin boundedness theorem, of H.P. Rosenthal’s result that a separable Banach space X is universal provided there is a C(K)-space and an operator T : C(K) → X such that T ∗ X∗ is non-separable. See [120] for an exposition of this further application of descriptive set-theoretic methods. We refer to [18,19,21,78,70,71,82,81] and references therein for related work. The note [57] surveys recent applications of the Szlenk index. The book [72] provides updated information on descriptive set theory.

II. The c0 -theorem and Banach space invariants associated with Baire-1 functions Let X be a separable inﬁnite-dimensional Banach space, and let K denote the unit ball of X∗∗ in its weak* topology; let x ∗∗ ∈ X∗∗ . The main purpose of this subchapter is to discuss the connection between the Baire-properties of x ∗∗ |K and the Banach space structure of X. Thus, x ∗∗ is called a Baire-1 element of X∗∗ if x ∗∗ |K is Baire-1, and a D-element if x ∗∗ |K is a difference of bounded semi-continuous functions. Section II.1 gives a fairly self-contained proof of the result that the Baire-1 elements of X∗∗ \ X correspond to nontrivial weak-Cauchy sequences, while the D elements correspond to non-trivial weakly unconditionally summing series (Theorem II.1.2). Then X contains an isomorph of c0 iff X∗∗ \ X has a D-element, while it contains an isomorph of 1 iff X∗∗ has a non-Baire-1 element (Theorem II.1.3). Section II.1 also introduces the classes of (s) and (ss) sequences, fundamental for discussing the c0 -theorem in Section II.2. Some applications of the c0 theorem are reviewed, such as: If X is non-reﬂexive and every subspace of X has weakly sequentially complete dual, then c0 embeds in X (Corollary II.2.5). The proof of the c0 -theorem involves a fundamental intrinsic characterization of the D-functions on a compact metric space K, through the transﬁnite oscillations. These are deﬁned, and then a proof is sketched of the c0 -theorem itself, with particular attention to the “real variables – descriptive set-theoretic” part given in Theorem II.2.17. Section II.3 is devoted to the subclasses of Baire-1 functions called B1/2 (K) and B1/4 (K). The fundamental connections here: sequences generating an 1 -spreading model are associated

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with Baire-1 elements of X∗∗ which are not in B1/2 (K), while sequences generating a summing basis spreading model are associated with the ones in B1/4 (K) (Theorems II.3.5 and II.3.6). Section II.4 treats transﬁnite spreading models and certain transﬁnite subclasses of B1/4 -functions, and the quite recent Index Theorem for spaces not containing c0 . The subchapter concludes with some open questions in II.5 concerning the possible universality of certain kinds of Banach spaces in terms of their descriptive set-theoretic structure.

II.1. Connections between Baire-1 functions, D-functions, and Banach spaces containing c0 or 1 Let K be a compact metric space. We change our notation slightly from Chapter I and let B1 (K) denote the class of all bounded (real or complex) valued functions on K of the ﬁrst Baire class. It follows easily that B1 (K) is a Banach algebra in the supremum norm. D EFINITION II.1.1. f : K → C is called a (complex) difference of bounded semicontinuous functions (a D-function) if there exist continuous functions ϕ1 , ϕ2 , . . . on K so that ϕj (k) < ∞ and ϕj converges to f pointwise. (II.1.1) sup k∈K

We then set ϕj (k): (ϕj )is a sequence in C(K) f D = inf sup k∈K

satisfying (II.1.1) .

(II.1.2)

We also let DBSC(K) denote the set of all D-functions on K, and sometimes abbreviate DBSC(K) by D(K). Recall that an extended real valued function f : K → [−∞, ∞] is called upper semicontinuous if f (x) = limy→x f (y) for all x ∈ K; f is called lower semi-continuous if f (x) = limy→x f (y) for all x ∈ K; f is semi-continuous if it is either upper or lower semicontinuous. (Following Bourbaki, we use non-exclusive lim sups and lim infs; thus, e.g., limy→x f (y) = limε↓0 sup{f (y): d(y, x) < ε}, d the metric on K). It then follows from a result of Baire that f ∈ D(K) if and only if there are bounded lower semi-continuous functions u1 , . . . , u4 on K so that f = (u1 − u2 ) + i(u3 − u4 ). D(K) is also a Banach algebra under the D-norm, and in general · D is not equivalent to the sup norm · ∞ on D(K), that is, since obviously D(K) ⊂ B1 (K), in general D(K) is not closed in B1 (K). Now let X be a separable Banach space and let K equal BX∗ in its weak* topology. Let ∗∗ denote the set of all x ∗∗ ∈ X ∗∗ with x ∗∗ |K ∈ B (K), and let X ∗∗ denote the set of all XB 1 D 1 ∗∗ (resp. X ∗∗ ) as the Baire-1 x ∗∗ ∈ X∗∗ with x ∗∗ |K ∈ D(K). We refer to the members of XB D 1 (resp. D-elements) of X∗∗ . The Baire-1 elements of X∗∗ were ﬁrst introduced in [64].

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The following result shows the fundamental connection between classes of Baire-1 functions and the Banach space structure of X. T HEOREM II.1.2. Let X and K be as above, and let x ∗∗ ∈ X∗∗ . ∗∗ if and only if there is a weak-Cauchy sequence (x ) in X with x → (a) x ∗∗ ∈ XB j j 1 ∗∗ ∗ x ω . Moreover one can then choose (xj ) with xj x ∗∗ for all j . ∗∗ if and only if there exists a sequence (x ) in X so that (b) x ∗∗ ∈ XD x j j is weakly ∗ unconditionally summing (i.e., |x (xj )| < ∞ for all x ∗ ∈ X∗ ) and nj=1 xj → / X, given ε > 0, one can choose (xj ) so that x ∗∗ ω∗ . Moreover if x ∗∗ ∈ (xj ) is equivalent to the c0 -basis

(II.1.3)

and ∞ ∗ x (xj ) < x ∗∗ |K + ε. sup D

x ∗ ∈BX∗ j =1

(II.1.4)

Before giving the proof, we note an immediate consequence of II.1.2 and a result of Odell and Rosenthal [102]. T HEOREM II.1.3. Let X be a separable Banach space. ∗∗ \ X = ∅. (a) c0 → X if and only if XD ∗∗ = ∅. 1 (b) → X if and only if X∗∗ \ XB 1 ((a) follows immediately from II.1.2(b); (b) from [102]; see also the remark following Theorem I.3.5 above.) To prove II.1.2, we give some preliminary notations, also used in the sequel. Given sequences (bj ) and (ej ) in a linear space, (ej ) is called the difference sequence of (bj ) if e1 = b1 and ej = bj − bj −1 for all j . The summing basis refers to the unit vectors basis for SER, the space of all converging series of scalars, i.e., all sequences (cj ) with cj convergent, under the norm (cj )SER = supn | nj=1 cj |. It is easily seen that SER is isomorphic to c0 ; in fact if (ej ) is the usual c0 -basis and bn = ni=1 ei for all n, then n n j =1 cj bj = supk | j =k cj | for all n, so (bj ) is 2-equivalent to the summing basis, and of course (bj ) is also a basis for c0 . D EFINITION II.1.4. Let (bj ) be a given sequence in a Banach space. (a) (bj ) is called non-trivial weak-Cauchy if (bj ) is weak-Cauchy and non-weakly convergent. if (bj ) is a weak-Cauchy basic sequence so that cj (b) (bj ) is called an (s)-sequence converges whenever cj bj converges. In the above, “(s)” stands for “summing”. It follows easily that an (s) sequence is nontrivial weak-Cauchy, hence its closed linear span cannot be weakly sequentially complete. The following result yields universality of (s)-sequences in non-weakly sequentially complete Banach spaces. For a proof, see Proposition 2.2 of [117].

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P ROPOSITION II.1.5. Let (xj ) be a non-trivial weak-Cauchy sequence in a Banach space. Then (xj ) has an (s)-subsequence. We shall also need to use the difference sequences of (s)-sequences, which are (somewhat surprisingly) characterized as follows. D EFINITION II.1.6. A sequence (ej ) in a Banach space is called a (c)-sequence provided it is a semi-normalized basic sequence so that ( nj=1 ej )∞ n=1 is weak-Cauchy. P ROPOSITION II.1.7. Let (bn ) and (ej ) be sequences in a Banach space with (ej ) the difference sequence of (bj ). Then (bj ) is (s) if and only if (ej ) is (c). We sketch the proof; for details, see Proposition 2.1 of [117]. (Throughout, if (xj ) is a sequence in a Banach space, [xj ] denotes its closed linear span. If (xj ) is basic, (xj∗ ) denotes its sequence of biorthogonal functionals in [xj ]∗ ; xj∗ (xi ) = δij for all i and j .) Suppose ﬁrst (bj ) is an (s)-sequence, and let (Pj ) be its sequence of basis projections: j Pj x = i=1 ci bi if x = ∞ i=1 ci bi . It follows that the ej ’s are linearly independent; an elementary argument yields that if (Qk ) is the sequence of basis projections for (ej ) (just deﬁned on their linear span), then deﬁning en∗ en∗ =

∞

bj∗

(the series converging ω∗ ),

(II.1.5)

j =n

(en∗ ) is biorthogonal to (en ) and Qn = Pn−1 + en∗ ⊗ bn

for all n

(II.1.6)

(where (en∗ ⊗ bn )(α) = en∗ (x)bn for all x ∈ X). Since (bj ) is (s), the sequence (en∗ ) is uniformly bounded, and (II.1.6) then also yields (Qn ) is uniformly bounded, hence (ej ) is basic and so a (c)-sequence. But if (ej ) is a (c)-sequence, (II.1.6) yields that conversely (Pn ) is uniformly bounded, whence (bj ) is (c). We need one last natural Banach space concept. D EFINITION II.1.8. Let (xj ) and (fj ) be sequences in a Banach space. (xj ) is called WUC (Weakly Unconditionally Cauchy) if ∞ ∗ ∗ x (xj ): x ∈ BX∗ < ∞. (xj ) = sup WUC j =1

(fj ) is called DUC (Difference (weakly) Unconditionally Cauchy) if (fj − fj −1 )∞ j =1 is WUC (where f0 = 0); we then set (fj )DUC = (fj − fj −1 )WUC . application of the uniform boundedness principle yields that (xj ) is WUC if A routine |x ∗ (xj )| < ∞ for all x ∗ ∈ X∗ (WUC sequences are also called weakly unconditionally

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summing in the literature). Since the sequence of partial sums of a WUC sequence is weakCauchy, DUC sequences are weak-Cauchy. We need a natural permanence property of DUC sequences. Given (xj ) and (yj ) sequences in a Banach space, (yj ) is called a convex block basis of (xj ) if there exist 0 = n0 < n1 < n2 < · · · and (λj ) non-negative scalars so that for all j ,

λi = 1

and yj =

nj−1

λi xi .

nj−1
(We don’t require that (xj ) be basic, although this will usually be the case.) The following natural permanence property is established in Proposition 3.2 of [117]. P ROPOSITION II.1.9. Let (yj ) be a convex block basis of a DUC sequence (xj ). Then (yj ) is also DUC and (yj )DUC (xj )DUC . Part (b) of Theorem II.1.2 follows easily from the next result and our preliminary comments. P ROPOSITION II.1.10. (a) A sequence in a Banach space is equivalent to the summing basis if and only if it is an (s)-sequence which is DUC. (b) Let (yj ) and (xj ) be sequences in a Banach space so that (i) yj − xj → 0 weakly as j → ∞, (ii) (yj ) is DUC and non-weakly convergent. Then some convex block basis of (xj ) is equivalent to the summing basis. P ROOF. (a) Since the summing basis is DUC and (s), any sequence equivalent to it has these properties also. Suppose conversely that (bj ) is a DUC-(s) sequence with difference sequence (ej ). Since (ej ) is semi-normalized basic by Proposition II.1.7, (ej∗ ) is uniformly bounded, hence letting λ = supj ej∗ we have, given n and scalars c1 , . . . , cn , that n max |cj | λ cj ej . 1j n

(II.1.7)

j =1

On the other hand, (ej ) is WUC; thus choosing x ∗ ∈ BX∗ with ∗ |x ( cj ej )|, we have n n

cj ej |cj | x ∗ (ej ) max |cj | (ej )WUC . j =1

n

j =1 cj ej

=

(II.1.8)

j =1

Of course (II.1.7) and (II.1.8) yield that (ej ) is equivalent to the c0 basis, whence (bj ) is equivalent to the summing basis. (b) (yj ) is non-trivial weak-Cauchy and hence has an (s)-subsequence (yj ). (yj ) is again DUC by Proposition II.1.9, and hence (yj ) is equivalent to the summing basis by

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Proposition II.1.10(a). Since yj − yj → 0 weakly, also yj − xj → 0 weakly, so “far-out” convex combinations of (xj ) converge in norm. It follows that there exist convex block bases (uj ) of (xj ) and (vj ) of (yj ) respectively so that uj − vj < ∞. Now (vj ) is also equivalent to the summing basis (indeed (vj ) is clearly also (s) and it is DUC by Proposition II.1.9). Thus (uj ) is equivalent to the summing basis, by Section I.3. We also note that if ε > 0 is given in advance, we may insure that uj − vj < ε, which yields by Proposition II.1.9 that (uj ) < (yj )DUC + ε. DUC

The next result, due to Choquet [35], (see also [108]) is our last needed ingredient for the proof of Theorem II.1.2. For K a compact metric space, we identify C(K)∗ with M(K), the space of all scalar-valued Borel measures on K. L EMMA II.1.11. Let K be a compact metric space and Y = C(K). Let E = BY ∗ in the ω∗ -topology, and regard K as canonically embedded in E. Let F ∈ X∗∗ with F |E ∈ B1 (E). Then F (μ) =

f (k) dμ(k)

(II.1.9)

K

for all μ ∈ Y ∗ = M(K). P ROOF. We follow the argument in [102]. For S a closed subset of K, let P(S) denote the set of all probability measures μ ∈ M(K) with μ(K \ S) = 0. Let Pa (S) denote the set of purely atomic members of P(S), and for μ ∈ P(K), let Pμ denote the set of λ in P(K) with λ absolutely continuous with respect to μ. Also, let supp μ = {k ∈ K: μ(U ) > 0 for every open neighborhood U of μ}. Of course supp μ is a closed subset of K. Then Pμ and Pa (S) are both ω∗ -dense in P(S) where S = supp μ.

(II.1.10)

Indeed, if W = Pμ or Pa (S), W is convex and f ∞ = supν∈W | f dν| for all f ∈ C(S), so (II.1.10) follows by the Hahn–Banach theorem. Now deﬁne G on M(K) by G(μ) =

F (k) dμ(k) for all μ ∈ M(K).

(II.1.11)

K

It follows from the bounded convergence theorem that G|E ∈ B1 (E), hence also H |E ∈ B1 (E), where H = F − G. Suppose the assertion of the lemma is false. Thus H = 0. Since the linear span of P(K) equals M(K), it follows that there is a ν ∈ P(K) with H (ν) = 0; by multiplying H by −1 if necessary, we may assume that Re H (ν) > 0. Let Z denote the space of λ ∈ M(K) with λ absolutely continuous with respect to ν. Applying the Radon–Nikodym and Riesz

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representation theorems, it follows that there is a bounded Borel-measurable function ϕ on K so that ϕ dλ for all λ ∈ Z. (II.1.12) H (λ) = K

Since then Re H (ν) = K Re ϕ dλ > 0, it follows that (Re ϕ)+ dν > 0, where (Re ϕ)+ denotes the positive part of Re ϕ. Thus we may choose a c > 0 so that ν(L) > 0, where L = {k ∈ K: ϕ(k) c}. Thus if λ ∈ P(K) is such that λ(K \ L) = 0, then (II.1.13) Re ϕ dλ = ϕ dλ c. L

Finally, let μ ∈ P(K) be deﬁned by μ(B) =

ν(B ∩ L) ν(L)

for all Borel sets B

and let S = supp μ. Now it follows from (II.1.12) and (II.1.13) that Re H (λ) c

for all λ ∈ Pμ .

(II.1.14)

On the other hand, if k ∈ K and δk denotes the point-mass probability at k, then G(δk ) = F (δk ) and thus H (λ) = 0

for all λ ∈ Pa (S).

(II.1.15)

Now P(S) is a weak*-closed subset of E, yet (II.1.10), (II.1.14), and (II.1.15) yield that H |P(S) has no point of continuity in P(S). This contradicts the fact that H |E ∈ B1 (E), by the Baire characterization theorem (Theorem I.3.1 above). We are ﬁnally prepared for the P ROOF OF T HEOREM II.1.2. Let X, K and X∗∗ be as in the statement of Theorem I.1.2. Let Y = C(K); we may regard X ⊂ Y by the Hahn–Banach theorem. (a) Let f = x ∗∗ |K. If there is a weak Cauchy sequence (xj ) converging w∗ to x ∗∗ , then trivially f ∈ B1 (K) since the xj ’s may be regarded as continuous functions on K, and of course xj |K → f pointwise. Suppose conversely that f ∈ B1 (K). Choose (fj ) a sequence of continuous functions on K with fj → f pointwise, where f = x ∗∗ |K, and assume without loss of generality that x ∗∗ = 1. Let τ : C → C be the continuous retraction onto the disc given by τ (z) = z if |z| 1, τ (z) = z/|z| otherwise; then replacing fj by τ ◦ fj for all j , we may assume that fj 1 for all j . Now we claim that ﬁxing n, then

dist X, conv {fj : j n} = 0.

(II.1.16)

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Indeed, if this were false, by the Hahn–Banach separation theorem, we could choose y ∗ ∈ Y ∗ and an r > 0 with y ∗ (x) = 0 for all x ∈ X

and

Re y ∗ (fj ) r

for all j.

(II.1.17)

By the Riesz representation theorem, there is a complex valued Borel measure μ on K representing y ∗ ; we thus have that x(k) dμ(k) = 0 for all x ∈ X and K (II.1.18) Re fj (k) dμ(k) r for all j. But identifying X∗∗ with X⊥⊥ in Y ∗∗ , we also have that then x ∗∗ (μ) = 0, since (II.1.18) yields that μ ∈ X⊥ . Now in fact, if E = BY ∗ , then x ∗∗ |E ∈ B1 (E). Indeed, letting T : X → Y denote the canonical isometric injection, we have that T ∗ (E) = K, and this (fn ◦ (T ∗ |E)) is a sequence of continuous functions on E converging pointwise to x ∗∗ |E. Thus by Lemma I.1.11, ∗∗ x ∗∗ (k) dμ(k) = 0, (II.1.19) x (μ) = K

but

x ∗∗ (k) dμ(k) r > 0

Re

(II.1.20)

K

by (II.1.18) and the bounded convergence theorem. This contradiction yields (II.1.16). But then it follows that there exists a convex block basis (uj ) of (vj ) and a sequence (xj ) in X so that uj − xj → 0

and xj 1

for all j.

(II.1.21)

But it’s clear that still uj → f pointwise, hence also xj → f pointwise on K. Again by the Riesz representation theorem, we obtain that (xj ) is weak-Cauchy with xj → x ∗∗ w∗ . (b) If (xj ) is as in (a), (xj ) is WUC, which implies immediately that f = x ∗∗ |K ∈ ∗∗ . Given ε > 0, we may choose (f ) a seDBSC(K). Suppose conversely, that x ∗∗ ∈ XD j quence in C(K) (with f0 = 0) so that fj → f ∞

pointwise and

(fj − fj −1 )(k) < f D + ε

(II.1.22) for all k ∈ K.

j =1

Again invoking the Riesz representation theorem, we have that (fj ) < f D + ε, DUC

(II.1.23)

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and moreover (fj ) is weak-Cauchy with fj → x ∗∗ ω∗ . By part (a), there exists also a weakCauchy sequence, (xj ) in X with xj → x ∗∗ ω∗ . But then fj − xj → 0 weakly and (fj ) is / X). The conclusion of II.1.10(b) then yields a non-weakly convergent (assuming x ∗∗ ∈ convex block basis (xj ) of (xj ), so that (xj ) is equivalent to the summing basis, and the proof of II.1.10(b) yields that if ε > 0 is given, (xj ) may be chosen so that (II.1.4) holds. We recall ﬁnally the following concept. D EFINITION II.1.12 ([107]). A Banach space X has property (u) provided for any weakCauchy sequence (xj ) in X, there exists a DUC sequence (yj ) in X with xj − yj → 0 weakly. The following result is now an immediate consequence of Theorem II.1.2 and Proposition II.1.10. C OROLLARY II.1.13. Let X be a given Banach space. The following are equivalent. (1) X has property (u). (2) Every non-trivial weak-Cauchy sequence in X has a convex block basis equivalent to the summing basis. ∗∗ = X ∗∗ (if X is separable). (3) XB D 1 II.2. The c0 -theorem We ﬁrst recall the following class of basic sequences introduced in [116]. D EFINITION II.2.1. A sequence (bj ) in a Banach space is called strongly summing (s.s.) weak-Cauchy basicsequence so that whenever (cj ) is a sequence of scalars if (bj ) is a with supn nj=1 cj bj < ∞, cj converges. The following result yields a general subsequence principle characterizing Banach spaces containing c0 , analogous to the 1974 result in [112] characterizing spaces containing 1 . T HEOREM II.2.2 (The c0 -theorem). Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing subsequence or a convex block basis equivalent to the summing basis. The alternatives of this result are mutually exclusive (we indicate why this is so below). We ﬁrst draw some immediate consequences (throughout, let X denote an inﬁnitedimensional real or complex Banach space). C OROLLARY II.2.3. The following are equivalent. (1) No subspace of X is isomorphic to c0 . (2) Every non-trivial weak-Cauchy sequence in X has an (s.s.) subsequence.

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P ROOF. (1) ⇒ (2) The second alternative of II.1.2 is excluded, since a sequence equivalent to the summing basis spans a space isomorphic to c0 . (2) ⇒ (1) The summing basis is obviously not (s.s.), hence no sequence equivalent to it is (s.s.). Since X has no sequence equivalent to the summing basis, c0 does not embed in X. This corollary in turn yields a dual characterization of spaces containing 1 . C OROLLARY II.2.4. The following are equivalent. (1) No subspace of X is isomorphic to 1 . (2) For every linear subspace Y of X, every non-trivial weak-Cauchy sequence in Y ∗ has an (s.s.) subsequence. P ROOF. (2) ⇒ (1) is trivial, for if Y is isomorphic to 1 , Y ⊂ X, then c0 is isomorphic to a subspace of Y ∗ , hence Y ∗ has a sequence equivalent to the summing basis. (1) ⇒ (2) Suppose to the contrary that (2) failed. By Corollary II.2.3, Y ∗ contains a subspace isomorphic to c0 , for some Y ⊂ X, hence Y contains a subspace isomorphic to 1 by a result of Bessaga and Pełczy´nski [16]. The next result gives one of the many motivations for the c0 -theorem. C OROLLARY II.2.5. If X is non-reﬂexive and Y ∗ is weakly sequentially complete for all subspaces Y of X, then c0 embeds in X; moreover X has property (u). To show this, we ﬁrst note the following fundamental permanence property of (s.s.) sequences. L EMMA II.2.6. Let (bj ) be an (s.s.) sequence in X. Then ( weak-Cauchy sequence.

n

∗ ∞ j =1 bj )n=1

is a non-trivial

P ROOF. Let F ∈ [bj ]∗∗ . Since (bj ) is basic, it follows that n ∗ F bj bj < ∞. sup n

(II.2.1)

j =1

n ∗ ∗ ∞ Hence since (bj ) is (s.s.), ∞ j =1 F (bj ) converges, proving ( j =1 bj )n=1 is weak-Cauchy. ∗ ∗ But of course (bj ) is a semi-normalized basic sequence, hence (bj ) is a (c)-sequence, so ( nj=1 bj∗ )∞ n=1 is an (s)-sequence and thus non-weakly convergent. R EMARK . A stronger permanence property is given in Proposition II.2.10. We pass now to the P ROOF OF C OROLLARY II.2.5. The hypotheses imply that 1 does not embed in X, since c0 embeds in (1 )∗ and c0 is not weakly sequentially complete. Since X is non-reﬂexive,

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we may choose a bounded sequence in X with no weakly convergent subsequence. But this sequence in turn has a weak-Cauchy subsequence (xj ) by the 1 -theorem. (xj ) has no (s.s.) subsequence by Lemma II.2.6, so since it is non-trivial weak-Cauchy, it has a convex block basis (fj ) equivalent to the summing basis, whence [fj ] is isomorphic to c0 , so c0 embeds in X. Moreover the same argument applies to any non-trivial weak-Cauchy sequence (xj ) in X; letting (fj ) be as above, (xj − fj ) is weakly null and (fj ) is DUC, so X has property (u). A reﬁnement of this argument yields the following equivalence (see Corollary 1.5 of [118]). C OROLLARY II.2.7. The following are equivalent. (1) Y ∗ is weakly sequentially complete for all subspaces Y of X. (2) X has property (u) and 1 does not embed in X. R EMARK . It is known that setting K = ωω + 1, then C(K) fails property (u) (cf. Proposition 5.3 of [66]), hence Corollary II.2.5 yields the existence of a subspace Y of C(K) with Y ∗ non-weakly sequentially complete. It was apparently unknown before the work in [118] if C(K) contained such a subspace Y , for any K countable compact metric. Actually, a particular (s.s.) sequence was discovered in C(ωω + 1) prior to the formulation and proof of Theorem II.1.2, and the study of this example led eventually to the above general results. We may combine the results given in Section II.1 and the above corollaries to obtain the ∗∗ = X ∗∗ \ X ∗∗ ). following result (where we let XND D C OROLLARY II.2.8. The following are equivalent. (1) Neither c0 nor 1 embeds in X. ∗∗ ∩ X ∗∗ = X ∗∗ \ X. (2) XB ND 1 (3) For all non-reﬂexive subspaces Y of X, there exists a subspace Z of Y so that neither Z nor Z ∗ is weakly sequentially complete. The proof of the c0 -theorem involves the following natural companion notion for (s.s.) sequences. D EFINITION II.2.9. A basic sequence (ej ) in a Banach space is called coefﬁcient converging (c.c.) if (i) ( nj=1 ej )∞ n=1 is a weak-Cauchy sequence and (ii) whenever (cj ) is a sequence of scalars with supn nj=1 cj ej < ∞, the sequence (cj ) converges. Note that (s.s.) sequences are (s) and (c.c.) sequences are (c) (since the conditions (i) and (ii) force (ej ) to be semi-normalized). The following gives some rather satisfying permanence properties relating (s.s.) and (c.c.) sequences (and also implies Lemma II.2.6).

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P ROPOSITION II.2.10. Let (bj ) be a sequence in a Banach space with difference sequence (ej ). The following are equivalent. (1) (bj ) is (s.s.). (2) (ej ) is (c.c.). (3) (bj ) is basic and (bj∗ ) is (c.c.). (4) (ej ) is basic and (ej∗ ) is (s.s.). Moreover if (bj ) is (s.s.), every convex block basis of (bj ) is also (s.s.). (For the proof, see Section 2 of [118].) We next deal with the proof of the c0 -theorem. We ﬁrst note a “real-variables” formulation. T HEOREM II.2.11. Let K be a compact metric space, f : K → C a bounded discontinuous function, and (fn ) be a uniformly bounded sequence of continuous functions on K with fn → f pointwise. (a) (fn ) has a convex block basis equivalent to the summing basis (in C(K)) if and only if f ∈ D(K). / D(K). (b) (fn ) has an (s.s.) subsequence if and only if f ∈ Let us note that part (a) follows from the results of Section II.1. Indeed, if f ∈ D(K), then by Proposition II.1.10 and Theorem II.1.2, (fn ) has a convex block basis equivalent to the summing basis. But if (gn ) is a convex block basis of (fn ) with (gn ) equivalent to the summing basis, then also gn → f pointwise and of course supk∈K |(gn − gn−1 )(k)| < ∞, so evidently f ∈ D(K). Finally, note that if (fn ) is (s.s.), f ∈ / D(K). Indeed, otherwise (fn ) would have a convex block basis (gn ) which is equivalent to the summing basis; but also (gn ) would be (s.s.) by the last statement of Proposition II.2.10. Thus the summing basis would be (s.s.), a contradiction. If, e.g., we let K equal BX∗ in its weak* topology, we also immediately obtain that the two alternatives of Theorem II.2.2 are mutually exclusive. The hard part of the c0 -theorem is thus the “if” assertion in II.2.11(b). An important ingredient in the proof of this is an intrinsic characterization of functions in DBSC(K) (for K an arbitrary separable metric space), involving the transﬁnite oscillations of a given scalar valued functions on K. First, if g is an extended real valued function on K, we denote the upper semicontinuous envelope of g by Ug, that is, Ug(x) = limy→x g(y) for all x ∈ X. D EFINITION II.2.12. Given f : K → C and α a countable ordinal, the α-th oscillation of f , oscα f , is deﬁned as follows: osc0 f ≡ 0. If β = α + 1, deﬁne

(II.2.2) osc β f (x) = lim f (y) − f (x) + oscα f (y) for all x ∈ K. y→x

If β is a limit ordinal, set osc β f (x) = sup oscα f (x) for all x ∈ K. α<β

Finally, set oscβ f = U osc βf .

(II.2.3)

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The transﬁnite oscillations (oscα f )α<ω1 thus form an increasing transﬁnite sequence of [0, ∞]-valued upper semi-continuous functions, and it is easily seen that osc1 f osc f 2 osc1 f , where osc f (x) = limy,z→x |f (y) − f (z)| (the usual deﬁnition of the oscillation of f ). We may deﬁne D(K) = DBSC(K) exactly as in the case of K compact (Deﬁnition II.1.1) and again obtain that D(K) is a Banach algebra. Then the elements of D(K) are characterized as those functions on K whose transﬁnite oscillations are bounded for all α. T HEOREM II.2.13. Let K be a separable metric space and f : K → C a bounded function. There exists a (least) countable ordinal α so that oscα f = oscβ f for all β > α. Then f ∈ D(K) if and only if oscα f is bounded. When f is real valued and this occurs, then f D = |f | + oscα f ∞ .

(II.2.4)

Moreover, setting λ = |f | + oscα f ∞ , u=

λ − oscα f + f 2

and v =

λ − oscα f − f , 2

thus u and v are non-negative lower semi-continuous functions with f = u − v and f D = u + v∞ . The c0 -theorem is established by exploiting the fact that if f ∈ / D(K), then the transﬁnite oscillations must blow up at some countable α. It turns out to be more convenient to work with the “positive oscillations”, introduced in earlier work of Kechris and Louveau [74] (with different terminology). These are denoted by pα (f ), and are deﬁned in exactly the same way as the transﬁnite oscillations oscα (f ), except that one deletes the absolute value signs in Deﬁnition II.2.12. We then have the following simple relationship between the transﬁnite oscillations and positive oscillations. P ROPOSITION II.2.14. For any f : K → R and countable ordinal α pα (f ) oscα f pα (f ) + pα (−f ).

(II.2.5)

Evidently then f ∈ / D(K) precisely when the pα (f )’s blow up. We now sketch the ﬂow of the proof of the c0 -theorem. D EFINITION II.2.15. A (c)-sequence (ej ) in X is called an ε-(c.c.) sequence if whenever (cj ) is a sequence of scalars with cj = 0 for inﬁnitely many j and nj=1 cj ej 1 for all n, then limj →∞ |cj | ε. Now suppose that (fn ) is a uniformly bounded sequence of continuous functions on K with fn → f pointwise, where f ∈ / D(K) (K compact metric). We seek to ﬁnd an (s.s.)

Descriptive set theory and Banach spaces

1033

subsequence of (fn ). Of course by Proposition II.1.5, we may assume that (fn ) is (s). It is ﬁrst shown that it is enough to construct subsequences of (fn ) whose difference sequences are ε-(c.c.) (given ε > 0). L EMMA II.2.16. (fj ) has an (s.s.) subsequence provided, for every ε > 0 and subsequence (gj ) of (fj ), there is a subsequence (bj ) of (gj ) whose difference sequence (ej ) is ε-(c.c.). For the sake of simplicity, let us suppose that we deal only with real scalars; so f is then real-valued. The following real variables result then yields the c0 -theorem. T HEOREM II.2.17. Let α be a countable ordinal and x ∈ K be given with 0 < pα (f )(x) =def λ < ∞. Let U be an open neighborhood of x and 0 < η < 1. There exists (bj ) a subsequence of (fj ) so that letting (ej ) be the difference sequence of (bj ), then given 1 m1 < m2 < · · · an inﬁnite sequence of indices, there exists a t in U and a k so that k

em2j (t) > (1 − η)λ;

(II.2.6)

j =1

em2j (t) > 0 for all 1 j k;

(II.2.7)

ei (t) < ηλ.

(II.2.8)

i ∈{m / 1 ,m2 ,...} im1

We now sketch the proof of the c0 -theorem: let β be the least countable ordinal with oscβ f unbounded (which is well deﬁned by Theorem II.2.13). Then by Proposition II.2.14, either pβ (f ) or pβ (−f ) must be unbounded, so by replacing f by −f and fn by −fn for all n if necessary, we may assume pβ (f ) is unbounded. Now β must be a limit ordinal, since f is bounded, and (II.2.1) easily yields that then oscα+1 f is bounded if oscα f is. But of course then p˜β (f ) is also unbounded, and we have that pα (f ) is bounded for all α < β and sup pα (f )∞ = ∞.

(II.2.9)

α<β

Now by replacing (fn ) by a given subsequence (fn ), it is really enough to show that given ε > 0, (fn ) has a subsequence (bn ) whose difference sequence is ε-(c.c.). By (II.2.9), we may choose α < β and x ∈ K so that 2 def λ = pα f (x) > . ε

(II.2.10)

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We now claim that if η is sufﬁciently small and (bi ) satisﬁes the conclusion of Theorem II.2.17, its difference sequence (ej ) is ε-c.c. Were this false, we could choose a sequence of scalars (cj ) so that n cj ej 1 for all n

(II.2.11)

j =1

and so that there are 1 = m1 < m2 < · · · with cm2j−1 = 0

and cm2j > ε

for all j.

(II.2.12)

(It is obvious that one may always assume c1 = 0.) Now it is also the case that there is a τ < ∞ so that the basis constant of any difference subsequence of a subsequence of (fj ) is at most τ . Now by Theorem II.2.17, choose t in K and k satisfying (II.2.6)–(II.2.8). Thus we obtain m m 2k 2k ci ei ci ei (t) i=1

i=1

ε

k j =1

em2j (t) −

|ci | ei (t)

(by (II.2.7))

i ∈{m / 1 ,m2 ,...}

ε(1 − η)λ − 2τ ηλ

(II.2.13)

(by (II.2.6), (II.2.8), and the basis constant estimate τ ). Of course if η is chosen small m 2k enough, (II.2.10) and (II.2.13) yield that i=1 ci ei > 2, contradicting (II.2.11). We note that the hypothesis of a given open neighborhood U of x is not used in the above deduction; this is used in a crucial way in the proof of Theorem II.2.17, which is achieved by transﬁnite induction. A remarkable feature of the argument is that only the statement itself at step α, is needed to obtain the step at α + 1 (the main part of this argument).

II.3. Spreading models associated with two special classes of Baire-1 functions We ﬁrst recall the following two special classes introduced in [66]. Let K be a ﬁxed compact metric space. D EFINITION II.3.1. (a) B1/2(K) denotes the set of all f : K → C so that there is a sequence (fn ) in DBSC(K) with fn → f uniformly. (b) B1/4(K) denotes the set of all f : K → C so that there is a sequence (fn ) in DBSC(K) with fn → f uniformly and supn fn D < ∞.

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Of course B1/2(K) is a Banach algebra under the sup-norm and B1/2 (K) ⊂ B1 (K). B1/4(K) is also a Banach algebra under the norm f 1/4 = inf λ > 0: there exists (fn ) in D(K) with fn → f uniformly and sup fn D λ .

(II.3.1)

n

We next give some intrinsic criteria. (A function f is called a simple D-function if f ∈ D(K) and has only ﬁnitely many values.) P ROPOSITION II.3.2. Let f : K → C be given. The following are equivalent. (1) f ∈ B1/2 (K). (2) f is a uniform limit of a sequence of simple D-functions on K. (3) β(f ) ω; i.e., βn (f, ε) < ∞ for all ε > 0. R EMARK . β(f, ε) and β(f ) denote the Baire-1 oscillation indices deﬁned following Example I.3.2. T HEOREM II.3.3. Let f : K → C be given. The following are equivalent. (1) f ∈ B1/4 (K). (2) There exists a sequence (fn ) of simple D-functions with fn → f uniformly and supn fn D < ∞. (3) oscω f ∞ < ∞. R EMARK . The following remarkable identity is obtained by Farmaki in [45]: for f real valued in B1/4(K), f B1/4 = |f | + osc ω f ∞ . We next recall the basic concepts concerning spreading models in Banach spaces. D EFINITION II.3.4. Let (ej ) be a basis for a Banach space E and let (xj ) be a seminormalized basic sequence in a Banach space X. (xj ) is said to generate (ej ) as spreading model if for all ε > 0 and all k, there is an N so that for any N n1 < n2 < · · · < nk , (xn1 , . . . , xnk ) is (1 + ε)-equivalent to (e1 , . . . , ek ).

(II.3.2)

(ej ) is called a spreading model for (xj ) if some subsequence of (xj ) generates (ej ), and ﬁnally (ej ) is called a spreading model for X if it is a spreading model for some basic sequence in X. It is easily seen that if (ej ) is a spreading model for X, (ej ) is 1-spreading; that is, (ej ) is isometrically equivalent to its subsequences. Remarkable results of Brunel and Sucheston (based on Ramsey theory), yield that if (xj ) is any semi-normalized basic sequence in a

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Banach space, then some subsequence of (xj ) generates a spreading model (ej ), which is moreover unconditional if (xj ) is weakly null (see [27,28]; also see Theorem 2.2 of [62] for a nice exposition of the ﬁrst assertion). It then follows from results in Section II.1 above that if (ej ) is a conditional spreading model for (xj ), (xj ) has a non-trivial weak-Cauchy subsequence, and hence an (s)-subsequence. In turn, it follows that then (ej ) is also an (s)-sequence. Moreover if (xj ) is already an (s)-sequence and (ej ) is a spreading model for (xj ), either (ej ) is conditional, or (ej ) is equivalent to the 1 -basis. Now ﬁx X a separable Banach space, and assume K is ω∗ compact with Ext(BX∗ ) ⊂ K ⊂ BX∗ .

(II.3.3)

The following results yield fundamental equivalences connecting spreading models and the Baire-1 classes given in Deﬁnition II.3.1. T HEOREM II.3.5. Let (xn ) be a non-trivial weak-Cauchy sequence in X, and let f = x ∗∗ |K where (xn ) converges ω∗ to x ∗∗ . (1) If f ∈ / B1/2(K), then some subsequence of (xn ) generates a spreading model equivalent to the 1 -basis. (2) If every convex block basis of (xn ) has a spreading model equivalent to the 1 -basis, f∈ / B1/2(K). T HEOREM II.3.6. Let (xn ) and f be as in Theorem II.3.5. (1) If f ∈ B1/4 (K), then some convex block basis of (xn ) generates a spreading model equivalent to the summing basis. (2) If (xn ) generates a spreading model equivalent to the summing basis, f ∈ B1/4(K). Let us note that one can ﬁnd examples of the above result for X = C(K) with K a countable compact metric space. In fact, if K = ωω + 1, then B1 (K) \ B1/2 (K) = ∅; also then the classical Schreier sequence (xj ) is an example of an (s)-sequence with spreading 2 model equivalent to the 1 -basis. If K = ωω + 1, then B1/4(K) \ DBSC(K) = ∅ (see [66]). It follows using the c0 -theorem and Theorem II.3.6 that there exists an (s.s.) sequence (xn ) in C(K) which has a spreading model equivalent to the summing basis. Actually, a Banach space X is constructed in [66] such that c0 does not embed in X, yet X∗ is separable and for some x ∗∗ ∈ X∗∗ \ X, f ∈ B1/4(K) of K = (BX∗ , ω∗ ) and x ∗∗ |K. We conclude this section with a brief discussion of the class of functions of ﬁnite Baire index. It follows from the results in Section II.2 that if K is a compact metric space and f : K → C is such that β(f ) < ω, then f ∈ DBSC(K). Actually, f is then a “strong D-function”; that is, there exists a sequence (fn ) of simple D-functions on K with fn − f D → 0 [33]. Of course it follows via Section II.1 that if (fn ) in C(K) is such that (fn ) is uniformly bounded and fn → f pointwise with f discontinuous of ﬁnite index, then (fn ) has a convex block basis equivalent to the summing basis.

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II.4. Transﬁnite analogs and the Index Theorem for spaces not containing c0 Let (bj ) be a ﬁxed normalized basis for a Banach space B. We modify the terminology of Chapter I slightly and say that a ﬁnite sequence (x1 , . . . , xn ) in a Banach space X is λ-equivalent to (b1 , . . . , bn ) if n n n 1 cj x j cj bj λ cj x j λ j =1

j =1

for all scalars c1 , . . . , cn . (II.4.1)

j =1

Now suppose X is a separable Banach space containing no subspace isomorphic to B. Given λ 1, set

Tλ = T X, (bj ), λ = (x1 , . . . , xn ): xi ∈ X for all i and

(x1 , . . . , xn ) is λ-equivalent to (b1 , . . . , bn ), n = 1, 2, . . . .

(II.4.2)

It follows that this is a well founded closed tree in X, hence by Theorem I.1.4, it has a height h(Tλ ) < ω1 . Thus also

def h X, (bj ) = sup h(Tλ ) < ω1 .

(II.4.3)

λ1

Now suppose in addition that (bj ) is 1-spreading, and that (xn ) is an inﬁnite seminormalized basic sequence in X so that (bj ) is equivalent to a spreading model for (xn ). Now given λ 1, let

F (xj ), (bj ), λ = F (xj ), λ = (n1 , . . . , nk ): k 1, n1 < n2 < · · · < nk ,

and (xnj )kj =1 is λ-equivalent to (b1 , . . . , bk ) .

(II.4.4)

Then it follows that F ((xj ), λ) is a well founded tree (regarded as a subset of N[N] ) and we have h(F ((xj ), λ)) h(Tλ ). Thus we obtain,

def

ω + 1 h F (xj ) = sup h F (xj ), λ h X, (bj ) < ω1 .

(II.4.5)

1λ

Of course one may replace the inﬁnitely branching tree N[N] by {F ⊂ N: #F < ∞}, identifying ﬁnite subsets of N with their increasing enumerations as in (II.4.3). The well founded tree F = (F (xj ), λ) has the additional property that it is hereditary and pointwise closed. That is, given F ∈ F and G ⊂ F , G ∈ F , and ﬁnally given (Fn ) in F with (1Fn ) converging pointwise, then there is an F ∈ F with 1Fn → 1F pointwise.

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A bootstrapping method for (rapidly) deﬁning subtrees of the inﬁnitely branching tree, of arbitrarily large height, was formulated by Alspach and Argyros as follows [1]. (We use the convention: if F, G are ﬁnite non-empty subsets of N, F < G means max F < min G; also n F means n min F .) D EFINITION II.4.1. For every limit ordinal ξ , ﬁx (ξn ) a sequence of successor ordinals strictly increasing to ξ . We recursively deﬁne for ξ < ω1 , a family Sξ of ﬁnite subsets of N, as follows: S0 = {n}: n ∈ N ∪ {∅}. If Sξ has been deﬁned, Sξ +1 =

n

Fi : n 1, n F1 < · · · < Fn , and Fi ∈ Fξ for 1 i n ∪ {∅}.

i=1

If ξ is a limit ordinal and Sα has been deﬁned for all α < ξ , Sξ = {F : for some n 1, F ∈ Sξn and n F } ∪ {∅}. Sξ is termed the Schreier family of order ξ . Note that S1 = {F ⊂ N: #F min F if F = ∅}.

(II.4.6)

It is established in [1] that for all ξ < ω1 , Sξ is hereditary, pointwise closed, and a wellfounded tree with h(Sξ ) = ωξ + 1. For further properties, see Section II.3.2. It is easily seen that a Banach space X has a spreading model equivalent to (bj ) if and only if X has a basic sequence (xj ) so that for some λ 1,

F (xj ), λ ⊃ S1 .

(II.4.7)

This forms the basis for the following concept. D EFINITION II.4.2. Let (ej ) be a 1-spreading basis for a Banach space E, let (xj ) be a semi-normalized basic sequence in a Banach space X, and let 1 ξ < ω1 . (xj ) is said to ξ -generate a spreading model equivalent to (ej ) if there is a λ 1 so that Sξ ⊂ F ⊂ N: F is ﬁnite and (xj )j ∈F is λ-equivalent to (ej )j ∈F .

(II.4.8)

See Chapter III for several results concerning the special cases where (ej ) is the usual basis for c0 or 1 ; in particular, see Theorem III.3.12 for a duality equivalence. Also see [47] and [48] for interesting dichotomies concerning sequences which ξ -generate spreading models equivalent to the c0 and 1 bases, respectively. We next formulate a certain transﬁnite analogue of part of Theorem II.3.5.

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T HEOREM II.4.3. Let X be a separable Banach space, let K satisfy (II.3.3), and let (xn ), and f be as in Theorem II.3.5; let 1 ξ < ω1 . If the Baire index β(f ) > ωξ , then some subsequence of (xj ) ξ -generates a spreading model equivalent to the 1 basis. This yields an important boundedness result due to Bourgain [24]. C OROLLARY II.4.4. Let X be a separable Banach space which contains no subspace isomorphic to 1 , and let K satisfy (II.3.3). There exists a countable ordinal α so that

β x ∗∗ |K α

for all x ∗∗ ∈ X∗∗ .

(II.4.9)

N OTE . β(f ) denotes the Baire-1 index of f , deﬁned following Example I.3.2. P ROOF. Let (bj ) be the usual basis for 1 . By the boundedness result cited from Section II.1, we have that there is a countable ordinal α so that

h T X, (bj ), λ α

for all λ 1.

(II.4.10)

Suppose now that β(f ) > ωξ . Then by the results of Section II.1 and Theorem II.4.3, there is a weak-Cauchy sequence (xj ) in X with xj → x ∗∗ ω∗ so that (xj ) ξ -generates a spreading model equivalent to the 1 -basis. Thus by (II.4.5), and the basic properties of Sξ ,

ωξ h F (xj ) α.

(II.4.11)

It follows that β(f ) α · ω.

(II.4.12)

It has recently been discovered jointly by the ﬁrst author of the present paper and V. Kanellopoulos, that the analogue of Bourgain’s theorem holds in the case of c0 , thus afﬁrming a conjecture of the third author [10]. To formulate this precisely, suppose K is a compact metric space and f : K → C is a bounded function with f ∈ / D(K). It follows from Theorem II.2.13 that there is a least ordinal α < ω1 with oscα f ∞ = ∞. We denote this α by rND (f ), the “non-D index of f ”. T HEOREM II.4.5 (The c0 -index theorem). Let X be a separable Banach space not containing a subspace isomorphic to c0 , and let K = (BX∗ , ω∗ ). Then there exists a countable ordinal α so that rND (x ∗∗ |K) α for all x ∗∗ ∈ X∗∗ . We indicate only some ideas of the intricate proof of this result. Let (bj ) denote the summing basis for SER (the space of converging series). Let K be as above, and suppose f ∈ B1 (K) is discontinuous and satisﬁes oscξ f ∞ < ∞ for some countable ordinal ξ . Suppose also that (fn ) is a uniformly bounded sequence in C(K) converging pointwise

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to f . Then the “end-goal” of the work in [10], is the proof that there exists a convex block basis (xj ) of (fn ) and a λ > 1 such that

h F (xj ), (bj ), λ ξ. (II.4.13) ∗∗ , then Of course it then follows that assuming f = x ∗∗ |K with x ∗∗ ∈ XB 1

h X, (bj ) ξ.

(II.4.14)

But then were the conclusion of Theorem II.4.5 false, we would obtain that h(X, bj )) = ω1 , which implies that X has an inﬁnite basic sequence equivalent to the summing basis by Theorem I.1.4, a contradiction. To achieve this end-goal, the authors in [10] introduce the concept of the ξ -th variation of a sequence of functions f¯ = (fn ), denoted νξ (f¯ ). First, this requires the novel introduction of a transﬁnite family of ﬁnite subsets of doubleton’s in N, deﬁned as follows, via the notations: given p and q doubleton subsets of N, write p < q if max p < min q; given F a ﬁnite non-empty set of such doubletons and n ∈ N , write n F if n min p for all p ∈ F . Now set P0 = {∅}; P1 = {{(n, m)}: n, m ∈ N, n = m}. If Pξ is deﬁned, let Pξ +1 = {p} ∪ F : p < F and F ∈ Pξ ∪ Pξ . If ξ is a countable limit ordinal, choose (ξn ) a strictly increasing sequence with ξ = supn ξn and set Pξ = {F : ∃ n ∈ N with n F and F ∈ Pξn } ∪ {∅}. Now given a sequence of complex-valued functions f¯ = (fn ) deﬁned on a set K and given V ⊂ K, deﬁne fi (t) − fj (t). (II.4.15) v˜ξ (f¯, V ) = sup sup F ∈Pξ t ∈V (i,j )∈F

Then deﬁne

vξ (f¯, V ) = inf v˜ξ (fm+j )∞ j =1 . m

(II.4.16)

Finally, set vξ (f¯ ) = Vξ (f¯, K). The following basic permanence property for the transﬁnite variations is then established in [10]. (See Deﬁnition II.2.12 for the deﬁnition of the transﬁnite oscillation oscξ f .) T HEOREM II.4.6. Let K, f¯ = (fn ), and f be as above. (a) Given ξ < ω1 and V an open subset of K, sup oscξ f (t) vξ (f¯, V ). t ∈V

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(b) There exists a convex block basis g¯ of f¯ such that vξ (g¯ ) = oscξ f ∞

for all ξ < ω1 .

The end-goal is ﬁnally achieved in [10] by showing that given a countable ordinal ξ , if (xn ) is a uniformly bounded point-wise converging basic sequence in C(K), which dominates the summing basis and satisﬁes v˜ωξ (xn ) < ∞, then for some 1 < λ < ∞, (II.4.13) holds. The proof of II.4.6(a) is achieved following the ideas in the proof of the c0 theorem; in particular, of Theorem II.2.17. The convex block basis in II.4.6(b), termed in [10] the “optimal sequence associated to f ”, is obtained through II.4.6(a) and the following concept and theorem. First, let K and f be as above, and also let g be a bounded positive real valued function on K. Deﬁne the relative oscillation of f with respect to g, osc g f , by osc g f (t) = lim f (y) − f (t) + g(y) y→t

for all t ∈ K.

Note that osc α+1 f = osc g f where g = oscα f . The following result in [10], called there the optimal sequences theorem, is then the remaining basic ingredient needed for the proof of Theorem II.4.6, and hence of II.4.5. (The term “optimal sequences” really refers to “optimal convex block basis” in the terminology of this subchapter.) T HEOREM II.4.7. Let K, (fn ), and g be as above, with g upper semi-continuous. Then given ε > 0, there exists a convex block basis (gn ) of (fn ) so that for all n and m, |gn − gm | + g < osc g f ∞ + ε. ∞ We next consider some transﬁnite Banach algebras of ﬁrst Baire class functions, and a conjecture, stronger than Theorem II.4.5. Fix K a compact metric space, and for each countable inﬁnite ordinal α let Dα denote the family of all scalar valued bounded functions f with oscα f bounded. It can be seen that Dα is a Banach algebra under the norm f Dα = |f | + oscα f ∞ . One also has that for all 1 ξ and ωξ α < ωξ +1 , Dα = Dωξ ; in particular, this shows K (e.g., K = [0, 1]), it is that rND (f ) = ωξ for some ξ , for any f . For uncountable also known that Dωξ+1 = Dωξ . By Theorem II.2.13, ωα<ω1 Dα = D(K). Now following [43], deﬁne also the following transﬁnite analogues of B1/4(K), namely a transﬁnite descending family of Banach spaces (Vξ (K))ξ <ω1 , as follows. V1 (K) = B1/4 (K) and · 1 = · B1/4 (K) . If Vξ (K) has been deﬁned, let Vξ +1 (K) denote the family of all functions f on K so that there is a sequence (fn ) in D(K) with f − fn ξ → 0 and

sup fn D < ∞, n

(II.4.17)

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under the norm f ξ +1 = inf{supn fn D : (fn ) satisﬁes (II.4.17)}. Finally, for ξ a limit ordinal, Vξ (K) denotes the set of all f in α<ξ Vα (K) with f ξ =def supα<ξ f α < ∞. One then has that Vξ (K) ⊂ Dωξ

for all ξ 1

(II.4.18)

(and V1 (K) = Dω by Theorem II.3.6). We may now formulate a transﬁnite analogue of Theorem II.3.6, due to Farmaki ([43] and [44]). T HEOREM II.4.8. Let X, (xn ), and f be as in Theorem II.3.5, let K satisfy (II.3.3), and let 1 ξ be a countable ordinal. (1) If f ∈ Vξ (K), there exists a convex block basis (yn ) of (xn ) which ξ -generates a spreading model equivalent to the summing basis. (2) If (xn ) ξ -generates a spreading model equivalent to the summing basis, then f ∈ Dωα (K). The next result is established in exactly the same way as Corollary II.4.4. C OROLLARY II.4.9. Let X and K be as above, and suppose c0 does not embed in X. Then there is a countable ordinal α so that if x ∗∗ ∈ X∗∗ ∼ X, then x ∗∗ |K ∈ / Vα (K). C ONJECTURE . Vα (K) = Dωα (K) for all compact metric spaces K. Evidently this conjecture, if true, yields the c0 -index theorem, in virtue of Corollary II.4.9 and (II.4.18). Of course Theorem II.4.5 may be regarded as additional evidence for the conjecture. Also, the third author of this article has proved the conjecture for all ordinals α with α < ω (presently unpublished). We conclude this section with the earlier result of Bossard [19], which establishes Theorem II.4.5 in the special case when 1 does not embed in X. Let us denote the set of weak-Cauchy sequences in X by WC(X). Now XN is a Polish space in the Tychonoff topology; it is easily seen that ∞ (X) is a Borel set in this topology, and moreover, WC(X) is a coanalytic subset. Now for any compact metric space K and f = (fn ) in WC(C(K)), deﬁne β(f) = β(F ) and rND (f) = rND (F ),

where fn → F pointwise.

(II.4.19)

(If F ∈ DBSC(K), we simply set rND (f) = ω1 .) The special case of Theorem II.4.5 is deduced from the following “pure descriptive set theory” result. T HEOREM II.4.10. Let K be a compact metric space. There exist coanalytic ranks β¯ and r¯ND on certain coanalytic subsets of ∞ (X) (in the Tychonoff topology), so that ¯ = β(f) and r˜ND (f) = rND (f) β(f)

for all f ∈ WC C(K) .

(II.4.20)

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P ROOF OF T HEOREM II.4.5, ASSUMING 1 DOES NOT EMBED IN X. We may regard X as a subspace of C(K) (K as in the statement of II.4.8). Since c0 does not embed in X,

rND (xn ) < ω1

for all (xn ) ∈ WC(X).

(II.4.21)

By Bourgain’s boundedness result (Corollary II.4.4), there exists a countable ordinal η so that

β (xn ) η

for all (xn ) ∈ WC(X).

(II.4.22)

Since β˜ is a coanalytic rank, by the coanalytic rank theorem,

¯ η def f ∈ ∞ C(K) : β(f) = Bη

is a Borel set.

(II.4.23)

But then ∞ (X) ∩ Bη is also a Borel set, and thus again by the coanalytic rank theorem and (II.4.21), there is a countable ordinal γ so that r¯ND (f) γ

for all f ∈ ∞ (X) ∩ Bη .

(II.4.24)

Finally, let (xn ) ∈ WC(X). Then (xn ) ∈ ∞ (X) ∩ Bη by (II.4.22) and (II.4.19), hence by (II.4.24) rND (xn ) = r¯ND (xn ) γ (where the equality in (II.4.25) holds by (II.4.19)).

(II.4.25)

II.5. Some open universality problems Fix X a separable Banach space, and let K = (BX∗ , ω∗ ). Note that X is universal (cf. Theorem I.5.2) provided C(K) embeds in X where K = [0, 1] or Δ, the Cantor set. The classical (L∞ -normalized) Haar basis (hj )∞ j =1 is in turn a monotone basis for a Banach subspace of L∞ ([0, 1]), isometric to C(Δ), and thus we have by Theorem I.1.4 that X is universal if and only if h(X, (hj )) = ω1 . This observation may be useful in attacking the following problems, whose formulation reﬂects the third author’s belief (or prejudice) that the answers are all afﬁrmative. ∗∗ , with r ∗∗ P ROBLEM 1. Assume for all ordinals α < ω1 , there exists x ∗∗ in XB ND (x |K) > 1 α. Is X universal?

R EMARK . Of course Theorem II.4.5 asserts that then c0 embeds in X. It is not evident that the hypotheses of Problem 1 imply that X∗ is non-separable. ∗∗ with β(x ∗∗ |K) > α. P ROBLEM 2. Assume for all ordinals α < ω1 , there exist x ∗∗ in XB 1 Does L1 embed in X?

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Note that the hypotheses in Problem 2 imply that 1 embeds in X, by Bourgain’s boundedness theorem (Corollary II.4.4 above). Bossard has raised the question in [19] as to whether these hypotheses imply that X is universal. The third named author suggests this is false, and that in fact the following query may have an afﬁrmative answer. P ROBLEM 3. Does there exist a separable X satisfying the hypotheses of Problem 2 such that X is cotype 2? For the next problem, we deﬁne (following [64]), the arbitrary intrinsic Baire-classes of the double dual of X, as follows. Set X0∗∗ = X. Suppose β > 0 is a countable ordinal and Xα∗∗ is deﬁned for all α < β. Let Xβ = {x ∗∗ ∈ X∗∗ : there exists a sequence (xn∗∗ ) of elements of α<β Xα∗∗ with xn∗∗ → ∗∗ . x ∗∗ ω∗ }. Note that Theorem II.1.2(a) yields that X1∗∗ = XB 1

∗∗ = X ∗∗ for all α < ω . Is X universal? P ROBLEM 4. Assume that Xα+1 1 α

Of course the theorem in [102] yields that X2∗∗ = X1∗∗ implies 1 embeds in X. Also, ∗∗ for all α, it follows from the hypothesis of Problem 4 that X∗∗ = since Xα∗∗ ⊂ Xα+1 def ω1 ∗∗ X is non-separable. α<ω1 α The converse to Problem 4 holds. This is not at all immediate, but does follow from known results. First, if S is an uncountable compact metric space, then it follows easily from the bounded convergence theorem that (C(S))∗∗ α may be identiﬁed with Bα (S), the α-th Baire class of bounded functions on S, via the identiﬁcation xf∗∗ (μ) =

f (s) dμ(s) for all μ ∈ M(S),

(II.5.1)

where f ∈ Bα (S) and M(S) is identiﬁed with C(S)∗ . Thus since a classical result in descriptive set theory asserts that Bα (S) = Bα+1 (S),

C(S)

∗∗ α+1

∗∗ = C(S) α

for all countable ordinals α.

(II.5.2)

Secondly, if Y is a complemented subspace of a separable Banach space X, then it follows easily that Yα∗∗ = Xα∗∗ ∩ Y ∗∗

for all α < ω1 ,

(II.5.3)

where Y ∗∗ is identiﬁed with Y ⊥⊥ . Indeed, if P : X → Y is a bounded linear projection, it follows by induction that

P ∗∗ Xα∗∗ = Yα∗∗

for all α,

(II.5.4)

which then easily yields (II.5.3) by induction. Now assuming X is universal, then a result of Pełczy´nski yields that X contains a complemented subspace Y isometric to C([0, 1])

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∗∗ = Y ∗∗ and so X ∗∗ = (see [120] for a discussion and references). Hence by (II.5.2), Yα+1 α α+1 ∗∗ Xα for all α, by (II.5.3). We offer no belief concerning the answer to the next and last problem raised here.

P ROBLEM 5. Does there exist a separable Banach space X with X3∗∗ = X2∗∗ = X1∗∗ ? More generally, given α a countable ordinal with α > 1, does there exist a separable Banach ∗∗ = X ∗∗ = X ∗∗ for all β < α? space X with Xα+1 α β McWilliams constructed a separable Banach space X such that X2∗∗ is not norm closed [94]. It is natural to consider the following question (which we have not studied): does McWilliams space X satisfy X3∗∗ = X4∗∗ ? The space is also built out of spaces which could conceivably give a positive answer to the ﬁrst question in Problem 5.

II.6. Notes and remarks 1. Theorem II.1.2(a) is due to Odell and Rosenthal [102], with II.1.3(b) forming the main motivation for its formulation. Theorem II.1.2(b) may be deduced from results of Bessaga and Pełczy´nski [16]. The idea of an (s)-sequence appears in [66]; the crystallization and proof of II.1.5 appears in [118], as do the concepts of (c)-sequences and the proof of Proposition II.1.7. The proofs of Propositions II.1.9 and II.1.10 involve reﬁnements of arguments in [16]. The basic concept of property (u) is introduced by Pełczy´nski in [106]; the terminology “DUC-sequences” was introduced in [117], just to focus attention on the structure of non-trivial weak-Cauchy sequences, as primary objects. Suppose X and K are as in Theorem II.1.2, and x ∗∗ ∈ X∗∗ \ X is such that x ∗∗ |K ∈ DSC(K); that is, it is a difference of (possibly unbounded) semi-continuous functions on K. It is still the case that c0 embeds in X – cf. Theorem 4.1 of [66]. For a study of the class DSC(K), see [34]. 2. All of the material in this section is taken from [118]. Lemma II.2.6 and the c0 -theorem itself give the main motivation for the introduction of (s.s.) sequences. A theorem analogous to the ﬁrst part of II.2.13 is obtained by Kechris and Louveau [74], using the “positiveoscillations” instead. The identity (II.2.4) (obtained in [118]) and its consequence (the “moreover” part of II.2.13) yields the surprising consequence that the inf in the deﬁnition of the D-norm, (II.1.2), is actually attained for real valued f . A reﬁnement of Theorem II.2.17 is obtained by Farmaki ([44], Theorem 8), which mainly keeps track of the rank of the “stopping time” implicit in the statement of II.2.17. In fact, it is shown in [44] that if α = ωξ , then given m1 < m2 < · · · as in the statement of II.2.17, k may be chosen so that (m1 , . . . , mk ) ∈ Sξ (the transﬁnite Schreier family deﬁned in Section II.4). In turn, this result is used in [44] in the proof of Theorem II.4.8 above. Let us say that a Banach space has (wcd) (weakly complete dual) if its dual is weakly sequentially complete. We note the various properties (V ) and (V ∗ ) introduced by Pełczy´nski in [106]; these are discussed in [120]. It is proved in [106] that (u) ⇒ (V ) ⇒ (V ∗ ) ⇒ (wcd).

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Thus it follows immediately from Corollary II.2.7 that a Banach space has any of these properties hereditarily if and only if it satisﬁes condition (2) of Corollary II.2.7. An interesting situation where this occurs is that of M-embedded Banach spaces, that is Banach spaces X which are M-ideals in their double duals X∗∗ , i.e., such that x ∗ + x ⊥ = x ∗ + x ⊥ for all x ∗ ∈ X∗ and x ⊥ ∈ X⊥ (where X, X∗ are regarded as subspaces of their double duals; thus X∗∗∗ = X∗ ⊕ X⊥ ). It is not hard to see that this property is hereditary and implies the weak sequential completeness of X∗ , whence such a space X has property (u). Historically, this was one of the last permanence properties discovered; see [59]; also see [65] for a nice survey on M-embedded Banach spaces. A remarkable application of the c0 -theorem is obtained by Lopez in [85]. Recall that a basic sequence (ej ) is shrinking if [ej∗ ] = [ej ]∗ . Lopez’s result (which also uses a deep result of Elton) goes as follows: let X be a non-reﬂexive Banach space with a shrinking semi-normalized basis (ej ) such that x ∗∗ ∈ X∗∗ \ X implies x ∗∗ (ej∗ ) → 0 as j → ∞. If X has no inﬁnite-dimensional reﬂexive subspace, (ej ) is equivalent to the c0 -basis. If c0 does not embed in X, then every non-reﬂexive subspace of X contains an order-1 quasi-reﬂexive subspace. Evidently Corollary II.2.8 yields a criterion for c0 or 1 to embed in a given Banach space X. Another such criterion has recently been obtained by Lopez, Martin and Payá [86]: assume X is real with an inﬁnite set A ⊂ SX such that |x ∗ (a)| = 1 for all a ∈ A and extreme points x ∗ of BaX∗ . Then c0 or 1 embeds in X. The proof is achieved using the 1 -theorem and Fonf’s theorem that c0 embeds in X if ex(BaX∗ ) is countable [51]. 3. Proposition II.3.2 and the spreading model characterization of B1/2(K) are obtained by Haydon, Odell, and Rosenthal in [66], where part 1 of Theorem II.3.6 is also proved. Part 2 of II.3.6 is due to V. Farmaki (answering an open question raised in [66]). This also follows rather easily from the equivalences (1) ⇔ (3) in Theorem II.3.3, discovered independently (with different proofs) in [43] and [119]. Functions of ﬁnite Baire index are discussed in [119] as well as in [33], using transﬁnite (actually just the ﬁnite) oscillations. 4. Theorem II.4.3 is due jointly to Kiriakouli and Negrepontis (see Section 3 in [96]). It is given as Corollary 3.5 in [76]. The index-boundedness principle for 1 , Corollary II.4.4, is due to Bourgain [24]. Actually, several basic notions in Chapter I ﬁrst appear in [24]. The remarkable c0 -index theorem (Theorem II.4.5) was discovered by Argyros and Kanellopoulos “in between” drafts of the present paper. The power in Deﬁnition II.4.2 lies in the (descriptive set-theoretic) theorem from II.1 that if a separable Banach space X contains sequences ξ -generating a spreading model equivalent to a given 1-spreading basis (ej ) for every countable ordinal ξ , then E embeds in X, where E is the Banach space with basis (ej ). Now the Bourgain 1 -index theorem investigates sequences ξ -generating the 1 -basis, while the c0 -index theorem studies sequences ξ -generating the summing basis. Farmaki [47] has done recent important work investigating sequences ξ -generating the c0 -basis , via a dichotomy with sequences semiboundedly complete of order ξ ; this generalizes fundamental seminal work of Elton in the case ξ = ω [40]. See also her recent work [48], yielding a dichotomy for sequences ξ -generating the 1 -basis .

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5. Let X and K be as in II.5, and let α be a countable ordinal. Call Xα∗∗ the intrinsic α-th ∗∗ , the α-th Baire class of X ∗∗ , by Baire class of X∗∗ . Regard X ⊂ C(K) and deﬁne XB α ∗∗ XB = x ∗∗ ∈ X⊥⊥ : x ∗∗ |K ∈ Bα (K) . α

(II.6.1)

Similarly, let B(K) denote the bounded Borel measurable functions on K, and let the set of Borel elements of X∗∗ be deﬁned as ∗∗ XB = x ∗∗ ∈ X⊥⊥ : x ∗∗ |K ∈ B(K) .

(II.6.2)

∗∗ if and only if x ∗∗ |K ∈ B(K) and x ∗∗ |K It can be seen that for x ∗∗ ∈ X∗∗ , x ∗∗ ∈ XB satisﬁes the barycentric calculus; i.e., k dμ(k) = x ∗∗ (k) dμ(k) (II.6.3) x ∗∗ K

K

for all Borel probability measures μ on K, where K k dμ(k) denotes the “ω∗ -integral.” Now it follows easily, by transﬁnite induction and the bounded convergence theorem, that ∗∗ Xα∗∗ ⊂ XB α

for all α < ω1 .

∗∗ is norm-closed, since B (K) is uniformly closed. We also set X ∗∗ = Moreover XB α ω1 α ∗∗ X and term its elements the intrinsic Borel elements of X∗∗ . Of course Xω∗∗1 α<ω1 α ∗∗ . Now if X is a L -space, we have that is norm-closed and Xω∗∗1 ⊂ XB ∞ ∗∗ Xα∗∗ = XB α

for all α < ω1

∗∗ and also Xω∗∗1 = XB .

(II.6.4)

Indeed, the argument for the converse to Problem 4 yields the stronger result that if X is a subspace of a Banach space Y , then Xα∗∗ = Yα∗∗

X⊥⊥ provided X⊥ is complemented in Y ∗ .

(II.6.5)

(For if Q : Y ∗ → X⊥ is a projection and P = I − Q, then thanks to weak*-continuity, one still has that P ∗ (Yα∗∗ ) = Xα∗∗ for all α.) Thus if X is a L∞ space, X⊥ is “automatically” complemented in Y ∗ (see [120]), and so (II.6.5) holds, and one then obtains (II.6.4). In particular, if S is a compact metric space and X = C(S), then (II.6.4) holds and moreover for all α, Xα∗∗ may be identiﬁed with Bα (S) via (II.5.1). ∗∗ (and moreover the Now Theorem II.1.2(a) shows that for all separable X, X1∗∗ = XB 1 ∗∗ coincides with the one given in Section 1; this is really a consequence deﬁnition of XB 1 of Choquet’s more general result that Baire-1 afﬁne functions on a compact convex set satisfy the barycentric calculus [35]). However (II.6.4) badly fails for general spaces X and α > 1, as witnessed by the following remarkable example of Talagrand [129]. There ∗∗ \ X = ∅. Thus exists a separable Banach space X with the Schur property such that XB 2 ∗∗ ∗∗ ⊂ XB . X = Xω∗∗1 = XB 2

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The example of McWilliams given at the end of Section 5 also yields a space with ∗∗ since X ∗∗ is not norm-closed. These counterexamples nevertheless suggest X2∗∗ = XB 2 2 the problem of identifying natural classes of spaces X which satisfy (II.6.4). (For analogous issues in the context of compact convex sets, see [111] and [29].) In particular, if X is a separable C ∗ -algebra, does (II.6.4) hold? The importance of the “barycentric calculus” property implicit in (II.6.1) is illustrated by the following simple example due to Choquet [35] (see also [102]). Let X = C([0, 1]). Deﬁne x ∗∗ ∈ X∗∗ by x ∗∗ (μ) = dμ/dm, the absolutely continuous part of μ with respect to Lebesgue measure m. Then x ∗∗ |K ∈ B2 (K) ∗∗ . We note also the following remarkable result of Christensen [36], showing but x ∗∗ ∈ / XB 2 that (II.6.4) holds for X = 1 : X = x ∗∗ ∈ X∗∗ : x ∗∗ |K ∈ B(K) . Finally, we raise the following problem, related to Problem 4. Suppose X has property (u). Is X1∗∗ = X2∗∗ ? The answer is known to be yes, provided X is isomorphic to a subspace of an order continuous Banach lattice. III. Weakly null sequences and asymptotic p spaces This subchapter is devoted to the structure and the complexity, measured by ordinal scales, of weakly null sequences, as well as to the presentation of results related to the modern class of asymptotic p spaces. It contains four sections. The ﬁrst three concern weakly null sequences. The results contained here are mainly divided in two parts. The ﬁrst part deals with restricted forms of unconditionality, such as near, convex, and Schreier unconditionality, and the second with dichotomies for subsequences or convex block subsequences of weakly null sequences. One of the central consequences of the results presented here is the complete effective description, for a general weakly null sequence, of convex combinations tending in norm to zero. The proofs in most of the cases are combinatorial. We present (or sketch) some of them to explain the nature of the techniques, and also to introduce the reader to the subject. The key ingredients are the classical inﬁnite Ramsey theorem, used almost everywhere, and also the more recent Schreier families and Repeated Averages Hierarchy. With a mix of them we accomplish new combinatorial results, like the “large families lemma”, which have their own interest. In the fourth section, we deal with “non-trivial” (not containing p ) asymptotic p spaces. These spaces are related to the famous Tsirelson space. The main intention is to present methods of producing Tsirelson and mixed Tsirelson norms. These are examples, or the frame for more advanced constructions, concerning the solutions of many important problems of Banach space theory. We also list properties of these spaces and we make a brief presentation of certain results for asymptotic p and Hereditarily Indecomposable spaces. III.1. Compact families of ﬁnite subsets of N Throughout this section, M, N will denote families of ﬁnite subsets of N, which, in most of the cases, will be additionally assumed to be compact in the topology of pointwise con-

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vergence. Such compact families are closely connected to weakly null sequences in Banach spaces. To indicate this fact let us consider such a family M containing all singletons. We deﬁne a norm on the space c00 (N) of all eventually zero sequences in the following manner: x(n): F ∈ M . xM = sup n∈F

It is not difﬁcult to see that the natural basis (en )n∈N of (c00 (N), · M ) is a normalized weakly null sequence (in fact, (en )n∈N is a shrinking basis for the completion of (c00 (N), · M )). Conversely, given a weakly null sequence (xn )n∈N in some Banach space X and δ > 0, the family Fδ = {G ⊂ N: ∃ x ∗ ∈ BX∗ with |x ∗ (xn )| > δ for n ∈ G} is a compact family of ﬁnite subsets of N with the property that for every convex combination a x with x > δ, there exists a G ∈ F such that x= ∞ n n δ/2 n=1 n∈G an > δ/2. These observations make clear the necessity of studying and understanding the structure of such families. We start with some N OTATION . (i) We denote by [N] (resp. [N]<∞ ) the set of all inﬁnite (resp. ﬁnite) subsets of N. For M ∈ [N] we denote by [M] the set of inﬁnite subsets of M. (ii) Let M be a family of ﬁnite subsets of N and let L ∈ [N], L = (j )j ∈N . We set: M(L) = {j }j ∈G : G ∈ M and M[L] = {F = G ∩ L: G ∈ M}. Observe that if M is a compact family then the same holds for the families M(L), M[L]. The family M is called hereditary provided for all F , G in [N]<∞ with F ⊂ G and G ∈ M we also have that F ∈ M. It is called spreading if for every F = {mi }i=1 , G = {ni }i=1 with mi ni and F ∈ M, then also G ∈ M. Let us notice that if M is hereditary and spreading then the families M(L), M[L] are subfamilies of M and each one of them is also hereditary and spreading. Finally the norm deﬁned by a compact hereditary family M, as it was explained above, makes the basis (en )n∈N a weakly null unconditional basis for the space XM = (c00 (N), · M ). Most of the results of this section rely essentially on an important principle of inﬁnite combinatorics, namely the inﬁnite Ramsey theorem. To recall its statement we let [N] be endowed with the topology of pointwise convergence. T HEOREM III.1.1 (cf. [39,124]). Let A be an analytic subset of [N]. For every M ∈ [N] there exists an L ∈ [M] such that either [L] ⊂ A or else [L] ⊂ [M]\A. We start with the following consequence of Theorem III.1.1 that makes use of some ideas due to Elton [40]. T HEOREM III.1.2. Let M be a compact family of ﬁnite subsets of N. Then for every M ∈ [N] there exists an L ∈ [M] such that M[L] is a hereditary family.

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P ROOF. We assert the following: C LAIM 1. For every M ∈ [N] there exists L ∈ [M] such that the following is fulﬁlled: (∗) For every Q ∈ [L], Q = (qn )n∈N , G ∈ M such that for some n, {q2 , . . . , qn } ⊂ G, there exists G ∈ M such that q1 ∈ / G and {q2 , . . . , qn } ⊂ G . To see that Claim 1 is valid, we let A denote the collection of all L ∈ [M] satisfying property (∗). The set A is a closed subset of [M] hence Theorem III.1.1 yields that there exists L = (n )n∈N such that either [L] ⊂ A or [L] ⊂ [M]\A. We assert that only the ﬁrst alternative occurs. If not, ﬁx n ∈ N and for i = 1, . . . , n set Li,n = {i } ∪ {n+1,... }. Since Li,n ⊂ L property (∗) fails for this set. Therefore there exist Gi,n ∈ M and ki ∈ N such that {n+1 , . . . , n+ki } ⊂ Gi,n and every G ∈ M containing {n+1 , . . . , n+ki } must also contain i . Let ki0 = max{ki }ni=1 and observe that the set G = Gi0 ,n contains the set {1 , . . . , n }. Since n is arbitrary and M is a compact family of ﬁnite sets we derive a contradiction which completes the proof of Claim 1. C LAIM 2. There exists L ∈ [M], L = (n )n∈N such that the following holds: For n ∈ N, F1 ⊂ {1 , . . . , n }, F2 ⊂ {n+2,... } and G ∈ M satisfying G ∩ {1 , . . . , n } = / G . F1 and F2 ⊂ G, there exists G ∈ M with G ∩ {1 , . . . , n } = F1 , F2 ⊂ G and n+1 ∈ Let us observe that if such an L exists then repeated applications of Claim 2 yield that M[L] is indeed a hereditary family and the proof of the theorem is complete. To obtain the set L we inductively choose a decreasing sequence (Ln )n∈N of inﬁnite subsets of M such that setting n = min Ln , the sequence (n )n∈N is strictly increasing and for every F1 ⊂ {1 , . . . , n }, F2 ⊂ Ln+1 \{n+1 }, G ∈ M as in Claim 2 there exists G ∈ M with G ∩ {1 , . . . n } = F1 , F2 ⊂ G and n+1 ∈ / G . Let us see how we accomplish this by induction. Assume that L1 ⊃ · · · ⊃ Ln has been obtained. Choose F ⊂ {1 , . . . , n } and set MF = {G ∈ M: G ∩ {1 , . . . , n } = F }. Choose any QF ∈ [Ln \{n }] such that condition (∗) of Claim 1 is fulﬁlled for [QF ] and the compact family MF . It is easy to check that our desired property is satisﬁed for any Ln+1 ∈ [QF ] and F1 = F . A ﬁnite induction running over all subsets F of {1 , . . . , n } leads to a set Ln+1 that works simultaneously for all F and this completes the proof of Claim 2, and the entire proof. The proof of the above theorem describes the essential combinatorial part of the proofs of two results concerning restricted forms of unconditionality that occur in weakly null sequences. Namely J. Elton’s near unconditionality (cf. [40]) and Argyros–Mercourakis– Tsarpalias convex unconditionality (cf. [11]). Their proofs are more complicated than the present one due, mainly, to the fact that in the later case we have to deal with functions instead of sets. These results will be discussed later. Now we give the following consequence of Theorem III.1.2 due to Rosenthal (cf. [99]). C OROLLARY III.1.3. Let (fn )n∈N be a sequence of non-zero characteristic functions in C(K) with K a compact Hausdorff space. If (fn )n∈N converges pointwise to zero, then it contains an unconditional basic subsequence.

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P ROOF. Let Φ : K → {0, 1}N be deﬁned as: Φ(k) = (fn (k))n∈N . Clearly Φ is a continuous function and Φ[K] is a countable compact subset of {0, 1}N naturally identiﬁed with a compact family M of ﬁnite subsets of N. Theorem III.1.2 yields that there exists an L ∈ [N] such that M[L] is hereditary. It is easy to verify that (fn )n∈L is an unconditional basic sequence. Corollary III.1.3 indicates that it is not possible to obtain weakly null sequences with no unconditional basic sequences through constructions of (non-hereditary) compact families of ﬁnite subsets of N. The ﬁrst example of a normalized weakly null sequence with no unconditional subsequence is due to Maurey and Rosenthal [93]; the construction there necessarily uses functions rather than sets (because of III.1.3). For the existence of reﬂexive Banach spaces with no unconditional basic sequences at all, see [63]. We close this part with the next result due to Gasparis which, roughly speaking, asserts that any two hereditary families are comparable. T HEOREM III.1.4 (cf. [53]). Let M, N be two hereditary families of ﬁnite subsets of N. Then there exists an L ∈ [N] such that either M[L] ⊂ N [L] or vice versa.

III.2. Schreier families and the repeated averages hierarchy A well known property of weakly null sequences is that they admit convex blocks tending to zero in the norm topology. Schreier [122] with a classical example has shown that there exist weakly null sequences whose averages are bounded away from zero in norm. This result raises the problem of an effective description of those convex combinations which tend to zero in norm. In recent years this description has been accomplished through two hierarchies; the Schreier families hierarchy and the Repeated Averages Hierarchy that we exhibit in this paragraph.

Schreier families Schreier families (Sξ )ξ <ω1 introduced by Alspach and the ﬁrst author in [1]. Their recursive deﬁnition has been given in Deﬁnition II.4.1. The following four properties of (Sξ )ξ <ω1 are established by induction. 1. Each Sξ is a compact, hereditary and spreading family of ﬁnite subsets of N. 2. If G ∈ Sξ and max G < min{n, m} then G ∪ {n} ∈ Sξ if and only if G ∪ {m} ∈ Sξ . 3. If L ∈ [N], L = (j )j ∈N and d ∈ N are so that {1 , . . . , d } ∈ Sξ while {1 , . . . , d , d+1 } ∈ / Sξ , then {1 , . . . , d } is a maximal element of Sξ . As a consequence of this we obtain that every L ∈ [N] has a unique decomposition L = ∞ i=1 Fi such that is a collection of consecutive maximal elements of S . {Fi }∞ ξ i=1 4. For all ξ, ζ with ζ < ξ there exists n(ζ, ξ ) = n0 ∈ N such that Sζ [N\{1, . . . , n0 − 1}] ⊂ Sξ . We recall that to each M ∈ [N] and ξ < ω1 we have associated to Sξ two compact families. Namely, the families Sξ [M] and Sξ (M). Since Sξ is hereditary and spreading

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we have that Sξ (M) ⊂ Sξ [M] ⊂ Sξ and for a M = N these inclusions are strict. The next lemma, due to Androulakis and Odell, shows that, after passing to an appropriate L ∈ [M] the families Sξ (M)[L] and Sξ [L] become almost equal. L EMMA III.2.1 ([2]). For every M ∈ [N] there exists an L ∈ [M] such that for all ξ < ω1 , F ∈ Sξ [L], the set F \{min F } ∈ Sξ (M). P ROOF. Let M = (mi )i∈N . We inductively deﬁne the set L as follows: we set 1 = m1 , and if k has been deﬁned with k = mik we set k+1 = mmik . It is veriﬁed, by induction, that the set L satisﬁes the required property. The Androulakis–Odell lemma is an unexpected and very useful result. It plays a key role in the proofs of the results of the following sections as well as in the proofs of other results related to Schreier families. In order to clarify what we mean by this and for understanding better the signiﬁcance of the families Sξ (M), we point out the following: each Sξ , being countable and compact, is homeomorphic to an ordinal and it is not difﬁcult to show ξ ξ that this ordinal is ωω . One of the advantages of the representation of ωω as Sξ is that there exist, naturally deﬁned, closed subsets, namely Sξ [M] or Sξ (M), with the same complexity as Sξ . From another point of view we may consider Sξ [M], as “subspaces” of Sξ while we may consider Sξ (M) as an “isomorphic” embedding of Sξ into Sξ [M]. Under this consideration, the Androulakis–Odell lemma describes a phenomenon similar to James’ theorem (cf. [68]) about non-distortion of the spaces 1 and c0 . Before passing to the results concerning the families (Sξ )ξ <ω1 we state Ptak’s theorem [110]. T HEOREM III.2.2. Let M be a compact and hereditary family of subsets of N. Assume that there exists an ε > 0 such that for all (αn )n∈N ∈ S+1 there exists a G ∈ M such that n∈G αn > ε. Then there exists an M ∈ [N] such that [M] ⊂ M. A quick proof of this theorem goes as follows: Assume that no such M exists. Then M consists of ﬁnite sets. Therefore the basis (en )n∈N of the space (c00 (N), · M ) is weakly null. This yields that there exists a convex combination of (en )n∈N such that ∞ n=1 an en M < ε a contradiction, yielding the proof of the result. D EFINITION III.2.3. Let D be a subset of the unit sphere of 1 , ξ < ω1 and M ∈ [N]. We say that D is (ξ, M)-large if for every ζ < ξ , ε > 0, L ∈ [M] there exists x ∈ D such that x|N\L < ε and xζ < ε (xζ = sup{ n∈G |x(n)|: G ∈ Sζ }). We shall also say that the subset M of [N]<∞ δ-norms D (δ > 0) provided that supF ∈M x|F > δ, for every x ∈ D. The following localizes Ptak’s theorem and generalizes the large families lemma, Lemma III.2.9 of the next section. T HEOREM III.2.4. Let D be a (ξ, M)-large family. Assume that M is a hereditary family that δ-norms D. Then there exists an L ∈ [M] such that Sξ (L) is contained in M.

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P ROOF. First we treat the case ξ = ζ + 1. In this case we claim that for every N ∈ [M] and ζ m ∈ N there exists Q ∈ [N] such that for every L ∈ [Q], m i=1 Fi (L) ∈ M. This follows from Theorem III.1.1 and the fact that D is (ξ, M) large family, δ-normed by M. After proving this, we use induction to derive a decreasing sequence M ⊃ Q1 ⊃ · · · ⊃ Qm ⊃ · · · such that Qm satisﬁes the claim for the number m. Choose a diagonal subset Q of the sequence (Qm )m∈N and easily check that Sξ (Q) is contained in M. If ξ is a limit ordinal, we denote by (ξn )n∈N the sequence of successor ordinals used in the deﬁnition of Sξ . From the ﬁrst part of the proof we obtain a decreasing sequence (Qm )m∈N such that Qm ∈ [M] and Sξn (Qm ) ⊂ M. Again any diagonal subset Q = (qm )m∈N such that (qk )km ⊂ Qm satisﬁes Sξ (Q) ⊂ M. This completes the proof of the theorem. We close this part with a brief presentation of the families {Aξ }ξ <ω1 deﬁned also recursively as follows. (a) We set A0 = {∅}. (b) For ξ = ζ + 1, if Aζ has been deﬁned, we set Aξ = {n} ∪ F : n < F, F ∈ Aζ ∪ {∅}. (c) For a limit ordinal ξ , we ﬁx a strictly increasing sequence (ξn )n with sup ξn = ξ and we set Aξ = {F : ∃n ∈ N, n < F, F ∈ Aξn } ∪ {∅}. This deﬁnition is similar to the deﬁnition of the families {Pξ }ξ <ω1 introduced in the previous chapter after Theorem II.4.5. The families {Aξ }ξ <ω1 share similar properties as {Sξ }ξ <ω1 . Namely each Aξ is a hereditary, spreading and compact family of ﬁnite subsets on N. Moreover it is homeomorphic to the ordinal ωξ . The later means that {Aξ }ξ <ω1 is increasing slower than Schreier families {Sξ }ξ <ω1 . In particular, for each ξ < ω1 , the compact space Aωξ and Sξ are homeomorphic. The next result makes more transparent the relation between Aωξ and Sξ . P ROPOSITION III.2.5. Let {Sξ }ξ <ω1 be deﬁned. Then for each limit ordinal ξ we may ﬁx an increasing sequence (ξn )n with sup ξn = ξ , such that the family {Aξ }ξ <ω1 deﬁned with the use of these sequences (ξn )n , satisﬁes Aωξ = Sξ for all ξ < ω1 . The proof of this result follows the same lines as the proof of Proposition 5.2 in [10].

The repeated averages hierarchy (RA-hierarchy) For ξ < ω1 and M ∈ [N] we recursively deﬁne a sequence (ξnM )n∈N in the following manner:

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(a) For ξ = 0 and M ∈ [N] with M = (mn )n∈N , ξnM = emn . Here (en )n denotes the usual basis of 1 . (b) For ξ = ζ + 1 if (ζnM )n∈N has been deﬁned for all M ∈ [N] and M = (mn )n∈N is given we set: ξ1M =

ζ1M + · · · + ζmM1 m1

.

M Then, recursively, we deﬁne ξnM = ξ1Mn where M1 = M, Mn = M\ n−1 i=1 supp ξ1 . (c) For ξ a limit ordinal, let (ξn )n∈N be the increasing sequence of successor ordinals used in the deﬁnition of Sξ . For M = (mk )k∈N we set n1 = m1 and ξ1M = (ξn1 )M 1 . Mk M Then we deﬁne Mk = Mk−1 \ supp(ξnk−1 )1 , nk = min Mk and ξk = (ξnk )1 . The following properties are inductively veriﬁed. 1. Each ξnM ∈ S+1 and if ξ > 0 ξnM (m) min1 M , m ∈ N. M and supp ξ M ∈ S . 2. supp ξnM < supp ξn+1 ξ n ∞ 3. M = n=1 supp ξnM . M 4. If (nk )k∈N is a strictly increasing sequence, setting M = ∞ k=1 supp ξnk we have that

M M ξk = ξnk for all n ∈ N. N OTATION . For s = (xk )k∈N a sequence in a Banach space we set ξnM

·s =

∞

ξnM (k) · xk .

k=1

Clearly ξnM · s deﬁnes a convex combination of the sequence (xn )k∈N . The inﬁnite Ramsey theorem combined with property 4 above, yields the following: P ROPOSITION III.2.6. For ξ < ω1 , M ∈ [N] we set, IM,n = {(ξkN1 , . . . , ξkNn ): k1 < · · · < kn , N ∈ [M]}. Then for A ⊂ IM,n there exists L ∈ [M] such that either IL,n ⊂ A or IL,n ⊂ IM,n \A. D EFINITION III.2.7. A sequence s = (xk )k∈N in a Banach space is called (ξ, M)convergent if limn ξnM · s = 0. The sequence s is called ξ -convergent if for every M ∈ [N] there exists L ∈ [M] such that s is (ξ, L)-convergent. P ROPOSITION III.2.8. Let s = (xk )k∈N be a sequence in a Banach space and ξ < ω1 , M ∈ [N]. Then one of the following alternatives holds. (i) There exists ε > 0 and N ∈ [M] such that for every L ∈ [N] and n ∈ N ξnL · s > ε. (ii) There exists N ∈ [M] such that for every L ∈ [N] the sequence s is (ξ, L)convergent. We conclude this section with the large families lemma. For this we need the following.

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P ROPOSITION III.2.9. For every ζ < ξ , M ∈ [N], ε > 0 there exists N ∈ [M] (depending on ζ, ξ, M, ε) such that ξnL ζ < ε for all L ∈ [N] and n ∈ N. Hence the family {ξnL : L ∈ M} is a (ξ, M)-large family. This proposition and Theorem III.2.4 yield the next result. L EMMA III.2.10 (Large families lemma). Let M be a hereditary family of ﬁnite subsets of N. Assume that for some ξ < ω1 , M ∈ [N] and δ > 0 the family M δ-norms the family {ξnL : L ∈ [M]}. Then there exists L ∈ [M] such that Sξ (L) is contained in M. III.3. Restricted unconditionality and dichotomies for weakly null sequences Maurey–Rosenthal’s examples of normalized weakly null sequences with no unconditional basic subsequence [93] and the recent more advanced constructions due to Gowers and Maurey of reﬂexive Banach spaces with no unconditional basic sequence, indicate that full unconditionality does not occur in every weakly null sequence. Our intention is to present forms of restricted unconditionality and some consequences of them. We also present three dichotomies concerning the c0 and 1 structure of weakly null sequences. We start with the following: D EFINITION III.3.1. Let s = (xn )n∈N be a sequence in a Banach space. (i) The sequence s is said to be nearly unconditional provided for all δ > 0 there exists C(δ) > 0 such that n∈F an xn C(δ) ∞ n=1 an xn where (an )n∈N is a sequence of scalars and F ⊂ {n ∈ N: |an | δ}. (ii) The sequence s is said to be convexly unconditional if for every δ > 0 there exists C(δ) > 0 such that for every sequence (an )n∈N ∈ S1 if an xn > δ then εn an xn C(δ) for all sequences (εn )n∈N ∈ {−1, 1}N . if there exists (iii) The sequence s is said to be Sξ unconditional C > 0 such that for every (an )n∈N , F ∈ Sξ we have that n∈F an xn C ∞ n=1 an xn . In the case of ξ = 1 we shall also use the term C-Schreier unconditional. Concerning these forms of unconditionality we have T HEOREM III.3.2. Let (xn )n∈N be a seminormalized weakly null sequence in a Banach space and ε > 0. There exists a subsequence (xn )n∈M which is: (i) Nearly unconditional. (ii) Convexly unconditional. (iii) (2 + ε) Schreier unconditional. As we have already mentioned after Theorem III.1.2, near and convex unconditionality describe phenomena similar to the content of Theorem III.1.2. Our next proposition highlights this fact and constitutes the main part of the proof of those results. P ROPOSITION III.3.3. Let W be a weakly compact subset of c0 (N) and δ, ε positive reals. Then there exists M ∈ [N] such that the following are fulﬁlled.

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(1) If F ∈ [M]<∞ , φ ∈ W such that for n ∈ F , φ(n) >0 and n∈F φ(n) > δ then there exists φ ∈ W such that n∈F φ (n) > δ − ε and n∈M\F |φ (n)| < ε.

(2) If F ∈ [M]<∞ , φ ∈ W such that for n ∈ F , φ(n) > δ then there exists φ ∈ W such

that φ (n) δ − ε and n∈M\F |φ (n)| < ε. Observe that both parts (1) and (2) are indeed analogues of Theorem III.1.2. Their proofs are also obtained by arguments similar to the corresponding proof of Theorem III.1.2. Part (1) concerns near unconditionality and part (2) convex unconditionality. The proof of (2 + ε) Schreier unconditionality uses also similar ideas. Let us point out that we should not expect to obtain Sξ unconditionality for all ξ < ω1 since this would imply that (xn )n∈N must contain an unconditional subsequence. In particular, as was observed by Odell, in the Maurey–Rosenthal examples [93], there is no subsequence which is S2 unconditional. We next present some consequences. For this we need the following. D EFINITION III.3.4. Let s = (xn )n∈N be a sequence in a Banach space. (a) The sequence s is said to be series-bounded if the sequence { nk=1 xk }n∈N of its partial sums is norm bounded. (b) The sequence s is said to be semi-boundedly complete if for every sequence of scalars (an )n∈N such that the sequence (an xn )n∈N is series-bounded we have that limn an = 0. Near unconditionality yields directly the next result, due to Elton (cf. [40,99]). T HEOREM III.3.5 (1st dichotomy). Every seminormalized weakly null sequence either contains a subsequence equivalent to the usual c0 -basis or a semi-boundedly complete subsequence. The next result (cf. [99]), due also to Elton, is another consequence of near unconditionality. P ROPOSITION III.3.6. Let (en )n∈N be a seminormalized nearly unconditional and weakly null Schauder basis for the space X. Assume that no subsequence of (en )n∈N is equivalent to the c0 basis. Then the sequence of biorthogonal functionals (en∗ )n∈N is weakly null. C OROLLARY III.3.7. A Banach space either contains an isomorph of c0 , or a subspace failing the Dunford–Pettis property. Recall that a Banach space X has the Dunford–Pettis property if for every pair (xn )n∈N , of weakly null sequences in X and X∗ respectively we have that xn∗ (xn ) → 0. The following is a generalization of the near unconditionality theorem, involving the repeated averages (cf. [9]).

(xn∗ )n∈N

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T HEOREM III.3.8. Let s = (xn )n∈N be a weakly null sequence and ξ < ω1 . There exists M ∈ [N] such that for every δ ∈ (0, 1] there exists a constant C(δ) > 0 so that the following property is fulﬁlled: if L ∈ [M], (an )n∈N are scalars in [−1, 1] then ∞ L

L ai ξi · s ai ξi · s C(δ) max δ, i∈F

i=1

for all F ⊂ {n: |an | δ}. This result is used in the proof of Theorem III.3.16 stated below. The appearance of the constant δ in the right-hand side of the above inequality is necessary because we do not have a lower estimate for the quantities ξiL · s. We continue with ξ

ξ

D EFINITION III.3.9. A sequence (xn )n∈N is said to generate an p , (resp. c0 ) spreading model, with ξ a countable ordinal, if (xn ) ξ -generates a spreading model equivalent to the usual basis of the space p (resp. c0 ) as in Deﬁnition II.4.2. ξ

ξ

In the sequel we shall restrict our attention to 1 and c0 spreading models. In such a case ξ ξ a bounded sequence (xn )n∈N generates an 1 , (resp. c0 )-spreading model provided there exists a δ > 0 such that for every F ∈ Sξ and all scalar sequences (an )n an xn |an | δ· n∈F

n∈F

resp. an xn δ max |an |: n ∈ F . n∈F ξ

The stability properties of the Schreier families yield that if (xn )n∈N generates an p spreading model then so does every subsequence (xn )n∈M . Furthermore, if ζ < ξ then ξ there exists n0 = n0 (ζ, ξ ) such that (xn )n>n0 is an p spreading model. Bourgain’s ξ p -index, deﬁned through well founded trees, yields that if a sequence (xn )n∈N admits p spreading models for ξ in an unbounded subset of [0, ω1 ) then it contains a subsequence equivalent to the p basis. (See Theorem I.1.4 above and also the discussion in I.2.6, on Section 4.) In particular if (xn )n∈N is a weakly null sequence, then there exists a countable ξ ordinal ξ0 such that for every ξ > ξ0 the sequence (xn )n∈N admits no 1 spreading model. This observation will be used in the second dichotomy (Theorem III.3.11). P ROPOSITION III.3.10. Let s = (xn )n∈N be a seminormalized weakly null sequence and ξ < ω1 such that (xn )n∈N is not ξ -convergent. Then there exists a subsequence s = (xnk )k∈N of (xn )n∈N such that ξ (i) s generates an 1 spreading model.

(ii) s is Sξ -unconditional.

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P ROOF. (i) This is a consequence of the large families lemma (Lemma III.2.10). Indeed Proposition III.2.9 yields that there exists M ∈ [N] and ε > 0 such that for all L ∈ [M] and i ∈ N ξiL s˙ > ε. Set Fε/2 = F ⊂ N: ∃ x ∗ ∈ BX∗ , x ∗ (xn ) ε/2, ∀n ∈ F and observe that Fε/2 is a compact hereditary family which ε/2 norms the family {ξiL : i ∈ N, L ∈ [M]}. From Lemma III.2.10 we obtain that for some L ∈ [M] Sξ (L) ⊂ Fε/2 . Also we may assume that (xn )n∈L is convexly unconditional (Theorem III.3.2). It readily follows ξ that (xn )n∈L generates an 1 spreading model. (ii) This is a consequence of the convex unconditionality of (xn )n∈L obtained in part (i). This proposition and Lemma III.2.1 yield the following result. T HEOREM III.3.11 (2nd dichotomy [11]). Let s = (xn )n∈N be a weakly null sequence and let ξ be a countable ordinal. Then one of the following holds exclusively: (a) The sequence s is ξ -convergent. ξ (b) There exists a subsequence s of s which generates an 1 spreading model and is Sξ unconditional. Moreover there exists a unique ordinal ξ0 such that for all ξ ξ0 the sequence s is ξ -convergent while for all ξ < ξ0 it is not. P ROOF. The fact that one of the two alternatives holds is a direct consequence of Proposition III.2.8 and Proposition III.3.10. The last part of the statement follows from the next C LAIM . If s = (xn )n∈N is ξ -convergent for some ξ , then it is ζ -convergent for all ζ ξ . Indeed, if not, then for some ζ > ξ and some ε > 0 there exists M ∈ [N] such that Sζ (M) ⊂ Fε/2 . Here Fε/2 is the same set as in the proof of Proposition III.3.10. Property 4 of Schreier families yields that for some M ∈ [M] Sξ (M ) ⊂ Fε/2 . Finally Lemma III.2.10 provides L ∈ [M ] such that for every Q ∈ [L] and i ∈ N, ξiQ s˙ > ε/2 hence s is not ξ -convergent which derives a contradiction. For ξ = 1 this theorem yields the well known dichotomy that every seminormalized weakly null sequence either contains a subsequence generating an 1 spreading model or else a Cesaro summable subsequence (cf. [41,113]). For a recent approach of this result see [95]. N OTATION . For a sequence (xn )n∈N in a Banach space X and M ∈ [N] we denote by XM the closed linear span of (xn )n∈M . ξ

ξ

The duality of c0 and 1 spreading models is described in the next result.

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T HEOREM III.3.12 ([9]). Let (xn )n∈N , (xn∗ )n∈N be weakly null sequences in X and X∗ respectively. Assume that for some ε > 0 we have that infn |xn∗ (xn )| > ε. Then, for a countable ordinal ξ , the following are equivalent. ξ (1) For every M ∈ [N] there exists an L ∈ [M] such that (xn )n∈L generates a c0 spreading model. ξ (2) For every M ∈ [N] there exists an L ∈ [M] such that (xn∗ |XM )n∈L generates an 1 spreading model. The implication (1) ⇒ (2) is trivial. (2) ⇒ (1) uses the near unconditionality and the large families lemma. D EFINITION III.3.13. The Banach space X satisﬁes the ξ -Dunford–Pettis property (ξ -DP) if for every pair (xn )n∈N , (xn∗ )n∈N of weakly null sequences in X and X∗ respectively with (xn∗ )n∈N ξ -convergent, we have that limn xn∗ (xn ) = 0. The space X is said to be hereditarily ξ -DP provided every subspace V of X satisﬁes the ξ -DP. The next result is a consequence of Theorem III.3.12. C OROLLARY III.3.14. For a Banach space X and 1 ξ < ω1 the following are equivalent. (1) Every seminormalized weakly null sequence in X has a subsequence which generξ ates a c0 -spreading model. (2) X is hereditarily ξ -DP. We pass now to the last dichotomy concerning weakly null sequences. We begin with the following: D EFINITION III.3.15. A seminormalized Schauder basic sequence (xn )n∈N is said to be boundedly convexly complete (b.c.c.) provided the following property holds for all sequences of scalars (an )n∈N such that (an xn )n∈N is series-bounded: Given (Fj )j ∈N , a sequence of consecutive ﬁnite subsets of N such that supj n∈Fj |an | < ∞, we have that limj n∈Fj an xn = 0. Let us observe that every b.c.c. sequence (xn )n∈N is also semi-boundedly complete. The converse is not valid. The following extends the 1st dichotomy (Theorem III.3.5) and its proof uses many of the ingredients present in the previous part of this section. T HEOREM III.3.16 (3rd dichotomy [9]). For every seminormalized weakly null sequence s = (xn )n∈N , one of the following alternatives holds exclusively: (a) There exists a boundedly convexly complete subsequence. (b) Every subsequence admits a convex block subsequence equivalent to the unit vector basis of c0 . Moreover (b) is equivalent to: (b ) For all M ∈ [N] there exists N ∈ [M] and ξ < ω1 such that for all L ∈ [N] the convex block subsequence (ξnL · s)n∈N is equivalent to the c0 basis.

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This dichotomy is the analogue, for weakly null sequences, of Rosenthal’s c0 -theorem [118] (Theorem II.2.2 above). However its proof depends on different techniques and it is not known how to obtain one result from the other. We conclude with C OROLLARY III.3.17. Let (xn )n∈N be a seminormalized weakly null sequence with the following property: every subsequence (xn )n∈M admits a further convex block subsequence which is seminormalized and series bounded. Then every subsequence has a convex block subsequence equivalent to the c0 basis.

III.4. Asymptotic p spaces The corner stone of what follows is one of the most important discoveries of the last decades in Banach space theory. That is Tsirelson’s space, invented by Tsirelson (cf. [132]). We recall the deﬁnition of the norm of this space which in the sequel will be denoted by T . For x ∈ c00 (N) we set: xT = max x0 ,

sup nE1 <···<En

n 1 Ei xT , 2 i=1

where {Ei }ni=1 are successive subsets of N and Ei x denotes the restriction of x on the set Ei . Observe that this is an implicit deﬁnition and the existence of such a norm follows by induction. The space T is the completion of (c00 (N), · T ). This deﬁnition is due to Figiel and Jonhson (cf. [49]) and concerns the dual of Tsirelson’s original space. The fundamental property of T is that it is a reﬂexive Banach space not containing any p , 1 < p < ∞. One of our objectives in this part is to show that “behind” the above deﬁnition there exists a method creating a large variety of completely new Banach spaces. Let us start with the following: D EFINITION III.4.1. A Banach space X with a Schauder basis (ei )i∈N is called an asymptotic p space, 1 p < ∞ (resp. asymptotic c0 space), provided that there exists a K > 0 such that every normalized block basis {xk }nk=1 of (ei )i∈N such that n < supp x1 < · · · < supp xn , is K-equivalent to the usual basis of np (resp. c0n ). Clearly p spaces are asymptotic p spaces. The Tsirelson-space T is an asymptotic 1 space while T ∗ is an asymptotic c0 space. As we will see next, Tsirelson-type norms provide examples of asymptotic p spaces not containing isomorphs of p . Among the properties that one could observe is that any spreading model of an asymptotic p space, generated by a block subsequence of the basis, is isomorphic to p (resp. c0 if p = ∞). An extensive study of asymptotic 1 spaces is presented by Odell, Tomczak and Wagner in [105]. The following interesting result, due to Odell and Schlumprecht, provides a criterion for a Banach space to contain isomorphs of 1 or c0 .

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T HEOREM III.4.2 ([103]). Let X be a Banach space with a Schauder basis (en )n∈N . Assume that every spreading model of X generated by a block subsequence of the basis is isometric to 1 . Then X contains isomorphically 1 . A similar statement holds also for c0 . The proof of this theorem is obtained by transﬁnite induction. More precisely it is shown ξ ξ that for all ξ < ω1 the space X contains 1 (resp. c0 ) spreading models. As we have mentioned before, this yields that X contains 1 or c0 respectively. Let us point out that the isometric assumption concerning the spreading models is necessary, as Tsirelson space indicates. Further it is not known if a similar statement holds for 1 < p < ∞.

Tsirelson and mixed Tsirelson norms We start with the following: D EFINITION III.4.3. Let M be a compact family of ﬁnite subsets of N. (i) A sequence {Ei }ni=1 of successive subsets of N is called M-admissible if there exists G = {mi }ni=1 ∈ M such that m1 E1 < · · · < mn En . A sequence {xi }ni=1 of vectors of c00 (N) is called M-admissible provided that the sequence {supp xi }ni=1 is M-admissible. (ii) A sequence {Ei }ni=1 of ﬁnite pairwise disjoint subsets of N is called M-allowable if the family {{min Ei }}ni=1 is M-admissible. A sequence {xi }ni=1 of vectors of c00 (N) is M-allowable provided that the sequence {supp xi }ni=1 is M-allowable. Let us observe that the condition n E1 < · · · < En appearing in the deﬁnition of Tsirelson’s norm is equivalent to saying that {Ei }ni=1 is S-admissible where S denotes the Schreier family S1 . The concept of M-admissible families was introduced in two unpublished papers. First it was deﬁned for M = Sξ by the ﬁrst author and subsequently the general case by Deliyianni jointly with the ﬁrst author. We are now ready to deﬁne Tsirelson and mixed Tsirelson norms. We begin with the ﬁrst one. (a) Norms of the form Tp (M, ϑ) Let M be a compact family of ﬁnite subsets of N, 0 < ϑ < 1 and 1 p < ∞. The space Tp (M, ϑ) is the completion of c00 (N) under the following norm. For x ∈ c00 (N) we set x(p,M,ϑ) = max x0 , sup ϑ

n

1/p p Ei x(p,M,ϑ)

.

i=1

The sup is taken over all M-admissible families {Ei }ni=1 . If p = 1 we shall denote T1 (M, ϑ) by T (M, ϑ). Let us also denote by An the compact family {F ⊂ N: #F n}. Next we list some properties of these spaces.

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T.1 The space T (S, 1/2) is Tsirelson space while Tp (S, 1/2) is what is called the p-convexiﬁcation of Tsirelson space. Each one of them deﬁnes an example of an asymptotic p space not containing p . The dual of T (S, 1/2) is an asymptotic c0 space. Similar properties hold for the spaces Tp (Sξ , ϑ). Moreover every normalized ξ block sequence in Tp (Sξ , ϑ) deﬁnes an p spreading model. T.2 For all n ∈ N, 1 p < ∞, ϑ ∈ (0, 1) the space Tp (An , ϑ) is isomorphic to some q , 1 < q < ∞ or to c0 . In particular if p = 1 and ϑ 1/n then T (An , ϑ) is isomorphic to c0 while if ϑ > 1/n then T (An , ϑ) is isomorphic to p where 1/p + 1/q = 1 and 1/n1/q = ϑ. T.3 If M is a compact family with Cantor–Bendixson index i(M) > ω then for all ϑ ∈ (0, 1) the space T (M, ϑ) is a reﬂexive space not containing any p , 1 < p < ∞. The p-convexiﬁcation of Tsirelson’s space was deﬁned and studied in [49] and modiﬁed versions of it in [69]. The isomorphisms of T (An , ϑ) and p is a result of Bellenot (cf. [14]). Results concerning the relation of Cantor–Bendixson index of M and the structure of the space T (M, ϑ) are contained in [15]. (b) Mixed Tsirelson norms of the form Tp ((Mn )n , (ϑn )n ) Let (Mn )n∈N be a sequence of compact families and (ϑn )n∈N be a null sequence with 0 < ϑn < 1. For 1 p < ∞ we deﬁne Tp ((Mn )n , (ϑn )n ) to be the completion of c00 (N) under the norm · ∗ deﬁned as follows: For x ∈ c00 (N) we set,

x∗ = max x0 , sup sup ϑn n

n

1/p p Ei x∗

,

i=1

where the inside sup is taken over all Mn -admissible families {Ei }ni=1 . Such spaces are called mixed Tsirelson spaces. The modiﬁed mixed Tsirelson spaces are denoted by TpM ((Mn )n , (ϑn )n ) and their norms are deﬁned by equations similar to the above one where the inside supremum is taken over all Mn -allowable families {Ei }ni=1 . Let us point out that TpM (M, ϑ), the modiﬁed Tsirelson-type spaces, are deﬁned in a similar manner. Mixed Tsirelson norms were introduced in [5] and modiﬁed mixed Tsirelson norms in [6]. Next we present some properties of mixed Tsirelson spaces as well as some examples. The most interesting families (Mn )n∈N for such examples are the family (An )n∈N and the family (Sn )n which is the ﬁrst sequence of the Schreier families {Sξ }ξ <ω1 . (Recall that a Banach space X is called HI (Hereditarily Indecomposable) if Y + Z is non-closed for all (inﬁnite-dimensional closed linear) subspaces Y and Z with Y ∩ Z = {0}.) 1 M.T.1. The space T ((An )n , ( log (n+1) )n ) is Schlumprecht’s space (cf. [121]). This is the 2 ﬁrst example of an arbitrarily distortable Banach space and one the main ingredients used by Gowers and Maurey in the construction of the ﬁrst HI Banach space. This space is reﬂexive and it does not contain any p . M.T.2. There exist sequences (ϑn )n∈N such that the spaces Tp ((Sn )n , (ϑn )n ) share properties similar to those of Schlumprecht’s space. Namely they are reﬂexive, not

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containing any p , arbitrarily distortable and additionally they are asymptotic p spaces. Also using these spaces we obtain HI asymptotic p spaces (cf. [5]). M.T.3. The spaces T (Sn , ϑ), T M (Sn , ϑ) are isomorphic (cf. [30]). The picture for mixed Tsirelson spaces is rather different. 1 )n ) contains 1 (i) The modiﬁed Schlumprecht space T M ((An )n , ( log (n+1) 2 isomorphically. Hence it is neither reﬂexive nor arbitrarily distortable. This result is due to Th. Schlumprecht (unpublished). For a proof see [89]. (ii) If a member Mn0 of the family (Mn )n∈N contains a Schreier family then the space T M ((Mn )n , (ϑn )n ) is a reﬂexive space not containing any p (cf. [6]). (iii) There are sequences (ϑn )n∈N such that T M ((Sn )n , (ϑn )n ) is arbitrarily distortable and totally incomparable to T ((Sn )n , (ϑn )n ) (cf. [7]). We continue this chapter with some results concerning distortion and HI (Hereditarily Indecomposable) Banach spaces. We ﬁrst recall the following concept. D EFINITION III.4.4. A Banach space X is called arbitrarily distortable if for every λ > 1, there exists an equivalent norm | · | on X such that for all inﬁnite-dimensional subspaces Y of X, |x| : x, y ∈ SY λ. sup |y| For a comprehensive discussion of distortion issues, see Chapter 31 in this Handbook by E. Odell and T. Schlumprecht, where their fundamental discovery is reviewed: p is arbitrarily distortable for all 1 < p < ∞ [104]. The reader will also ﬁnd discussed there the following remarkable result due to Milman and Tomczak-Jaegermann [97]. T HEOREM III.4.5. Let X be a Banach space containing no arbitrarily distortable subspace. Then X contains an asymptotic p space for some 1 p < ∞ or an asymptotic c0 space. It remains an open question as to whether Tsirelson’s space itself, or its dual, could satisfy the hypotheses of III.4.5. On the other hand, it is also open as to whether a space with these properties must be c0 or 1 saturated, implying by a celebrated result of James that it has no distortable subspaces. We continue with the following result due to Tomczak-Jaegermann [131]. T HEOREM III.4.6. Every HI Banach space is arbitrarily distortable. The basic argument in the proof uses transﬁnite induction and shows that every boundedly distortable Banach space contains Sξ unconditional sequences for all ξ < ω1 . This yields that the space must contain an unconditional basic sequence. Note that this theorem yields that the asymptotic p -space occurring in the conclusion of Theorem III.4.5 may be chosen to have an unconditional basis. We conclude this section with the next result which demonstrates the size and the variety of the class of HI spaces.

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T HEOREM III.4.7 ([4]). Let X be a separable Banach space containing all reﬂexive HI Banach spaces. Then X is universal, i.e., it contains an isomorphic copy of any separable Banach space. This result extends a previous theorem due to Bourgain [23] (see also [116] and Section I.5 above). Its proof is also related to transﬁnite constructions. For a Banach space Y with a basis (en )n∈N Bourgain has obtained a long sequence (Rξ (Y ))ξ <ω1 of reﬂexive separable spaces with the property: whenever a separable Banach space X contains isomorphic copies of all (Rξ (Y ))ξ <ω1 , then it also contains an isomorphic copy of Y itself. The proof of the above theorem uses interpolation methods from [8] to obtain a corresponding sequence (Hξ (Y ))ξ <ω1 such that each Hξ (Y ) is a reﬂexive HI space and this family satisﬁes the same property of Bourgain’s class (Rξ (Y ))ξ <ω1 . The entire proof is obtained by considering Y = C[0, 1].

III.5. Notes and remarks 1. There is strong interaction between Ramsey and Banach space theories. Chapter 24 in this Handbook by W.T. Gowers (cf. [62]) is an excellent presentation of some deep consequences of Ramsey theory in Banach spaces. His recent discovery of the Banach space Ramsey theorem and the famous Gower’s dichotomy are also included in the same article. Some of the results contained in the present chapter whose proofs use the inﬁnite Ramsey theorem have alternative proofs relying on transﬁnite induction. Among them is Corollary III.1.3, originally proved by transﬁnite induction (unpublished). 2. The Schreier families {Sξ }ξ <ω1 in Deﬁnition II.4.1 were introduced by the authors of [1] in 1986 when they were visitors at The University of Texas at Austin. They are so named because S1 is given by Schreier in his classical construction [122]. The initial sequence {Sn }n∈N is included implicitly in the inductive deﬁnition of Tsirelson space and appeared for ﬁrst time in an example constructed by Odell [2]. Lemma III.2.1 has extensive applications in recent results of Banach space theory. For example, it has a key role in the proof of the dichotomy for quotients of HI spaces proved by Felouzis and the ﬁrst author [8]. The family {Aξ }ξ <ω1 is a very useful tool for proofs involving transﬁnite induction and compact families with increasing complexity. The reason is that passing from Aξ to Aξ +1 , the families increase slower than the corresponding Sξ families. The repeated averages hierarchy was introduced by Mercourakis, Tsarpalias and the ﬁrst author [11]. The initial motivation was to provide a hierarchy of summability methods yielding a complete classiﬁcation of the summability of weakly null sequences as described in Theorem III.3.11. Moreover, the repeated averages have been very useful in the study of mixed Tsirelson and HI spaces providing “special convex combinations” which form a key ingredient in the study of these spaces. The large families lemma (Lemma III.2.10) is proved in [11] by transﬁnite induction and in the present form is contained in [9]. 3. Near unconditionality was invented by Elton in his Ph.D. thesis [40] which also contains the proof of his remarkable theorem (Theorem III.3.2(i)). Convex unconditionality and

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Theorem III.3.2(ii) are contained in [11]. Schreier unconditionality is stronger than a well known property of spreading models. Namely a spreading model generated by a seminormalized weakly null sequence is unconditional. Theorem III.3.2(iii) was stated in [93] and also proved in [100] (see also [9]). Theorem III.3.16 may be considered as a continuation of Theorem III.3.5 in the direction of a deeper search in the linear span of a seminormalized weakly null sequence for the existence of a block subsequence equivalent to the unit vector basis of c0 . 4. As we have mentioned, Tsirelson’s original construction concerns the dual of the space deﬁned here as Tsirelson space. The norm in T ∗ is not described by any formula. A comprehensive exposition of the results related to Tsirelson space and its variations is presented by Casazza and Shura in [31]. It is worth noticing that the isomorphism between T (An , 1/n1/q ) and p (1/p + 1/q = 1) yields a uniﬁed description of the norms of the classical lp spaces and Tsirelson space. Mixed Tsirelson spaces were deﬁned by Deliyianni and the ﬁrst author [5] to provide examples of arbitrarily distortable and HI asymptotic 1 spaces. Recent results [12] show that the quotients of asymptotic 1 spaces not containing 1 are all separable Banach spaces not containing 1 . In particular the following is proved: if a separable Banach space X does not contain 1 then there exists an asymptotic 1 HI Banach space Y which has as a quotient the space X. The same paper also contains the construction of a non-separable HI Banach space which actually is the dual of an asymptotic 1 HI space. Chapter 29 in this Handbook by B. Maurey [92] is devoted to a beautiful exposition of the construction of HI spaces, which is not an easy task, and to the study of Banach spaces with small spaces of operators. It is an easy observation that every normalized block sequence in an asymptotic 1 space generates an n1 spreading model. Certain mixed Tsirelson spaces of the form T ((Sn )n , (ϑn )n ) have normalized block sequences generating ω1 spreading models. (The corresponding result for Schlumprecht’s space S is that it admits an 1 spreading model [80].) This is a consequence of the ﬁnite representability of c0 in T ((Sn ), (ϑn )) (cf. [6,7].) A remarkable result due to Gasparis [54] asserts that for a certain sequence (ϑn ) the space T ∗ ((Sn )n , (ϑn )n ) admits a c0ω spreading model. (It is not known whether the dual of Schlumprecht’s space admits a c0 spreading model.) Manoussakis has recently provided a criterion for the existence of an arbitrarily distortable subspace of an asymptotic 1 space ξ determined by the structure of 1 spreading models that the space admits [90]. The proof of this result uses the fundamental work by Odell, Tomczak and Wagner [105] on the structure of asymptotic 1 spaces. We recommend to the interested reader the article by Odell [101] for a comprehensive and beautiful presentation of the theory of asymptotic p spaces. References [1] D. Alspach and S.A. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992), 1–44. [2] D. Alspach and E. Odell, Subspaces of Lp , 1 p < ∞, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 123–160. [3] G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125–149.

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[99] E. Odell, Applications of Ramsey theorems to Banach space theory, Notes in Banach Space, H.E. Lacey, ed., University of Texas Press (1980), 379–404. [100] E. Odell, On Schreier unconditional sequences, Contemp. Math. 144 (1993), 197–201. [101] E. Odell, On Subspaces, Asymptotic Structure and Distortion of Banach spaces, Connections with Logic, Analysis and Logic, Part Three, LMS Cambridge University Press. [102] E. Odell and H.P. Rosenthal, A double-dual characterization of separable Banach spaces not containing 1 , Israel J. Math. 20 (1975), 375–384. [103] E. Odell and Th. Schlumprecht, A problem on spreading models, J. Funct. Anal. 153 (1998), 249–261. [104] E. Odell and Th. Schlumprecht, Distortion and asymptotic structure, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003) (this Handbook). [105] E. Odell, N. Tomczak-Jaegermann and R. Wagner, Proximity to 1 and distortion in asymptotic 1 spaces, J. Funct. Anal. 150 (1) (1997), 101–145. [106] A. Pełczy´nski, A connection between weakly unconditional convergence and weak completeness of Banach spaces, Bull. Acad. Polon. Sci. 6 (1958), 251–253. [107] A. Pełczy´nski, Universal bases, Studia Math. 32 (1969), 247–268. [108] R. Phelps, Lectures on Choquet’s Theorem, D. van Nostrand Co., Princeton, NJ (1966). [109] R. Pol, Note on pointwise convergence of sequences of analytic sets, Mathematika 36 (1989), 290–300. [110] V. Ptak, A combinatorial lemma on the existence of convex means and its application to weak compactness, Proc. Symp. Pure Math. VII, Convexity, V. Klee, ed., Amer. Math. Soc., Providence (1963). [111] M. Rogalski, Opérateurs de Lion, projecteurs boréliens et simplexes analytiques, J. Funct. Anal. 2 (1968), 458–488 (French). [112] H.P. Rosenthal, A characterization of Banach spaces containing 1 , Proc. Nat. Acad. Sci. USA 71 (1974), 2411–2413. [113] H.P. Rosenthal, Weakly independent sequences and the Banach–Saks property, Bull. London Math. Soc. 8 (1976), 22–24. [114] H.P. Rosenthal, Point-wise compact subsets of the ﬁrst Baire class, Amer. J. Math. 99 (1977), 362–378. [115] H.P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (5) (1978), 803–831. [116] H.P. Rosenthal, On applications of the boundedness principle to Banach space theory, according to J. Bourgain, Seminaire d’ Initiation à l’Analyse, 18e , Paris VI, 1978–79 (1979), 5-01–5-14. [117] H.P. Rosenthal, Weak*-Polish Banach spaces, J. Funct. Anal. 76 (1988), 267–316. [118] H.P. Rosenthal, A characterization of Banach spaces containing c0 , J. Amer. Math. Soc. 7 (1994), 707– 748. [119] H.P. Rosenthal, Differences of bounded semi-continuous functions I, to appear. [120] H.P. Rosenthal, The Banach spaces C(K), Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003) (this Handbook). [121] Th. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), 81–95. [122] J. Schreier, Ein gegenbeispiel zur theorie der schwachen konvergence, Studia Math. 2 (1930), 58–62. [123] A. Sersouri, A note on the Lavrientiev index for quasi-reﬂexive Banach spaces, Contemp. Math. 85 (1989), 497–508. [124] J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970), 60–64. [125] I. Singer, On biorthogonal systems and total sequences of functionals, II, Math. Ann. 201 (1973), 1–8. [126] M.M. Souslin, Sur une déﬁnition des ensembles mesurables B sans nombres transﬁnis, C.R. Acad. Sci. Paris 141 (1917), 88–91. [127] C. Stegall, The Radon–Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213–223. [128] W. Szlenk, The non existence of a separable reﬂexive Banach space universal for all separable Banach spaces, Studia Math. 30 (1968), 53–61. [129] M. Talagrand, A new type of afﬁne Borel function, Math. Scand. 54 (1984), 183–188. [130] S. Todorˇcevi´c, Compact subsets of the ﬁrst Baire class, J. Amer. Math. Soc. 12 (4) (1999), 1179–1212. [131] N. Tomczak-Jaegermann, Banach spaces of type p have arbitrarily distortable subspaces, GAFA 6 (1996), 1074–1082. [132] B.S. Tsirelson, Not every Banach space contains p or c0 , Functional Anal. Appl. 8 (1974), 138–141. [133] M. Zinnmeister, Les dérivations analytiques, Publ. Math. Univ. Paris-Sud 86T10 (1986).

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CHAPTER 24

Ramsey Methods in Banach Spaces W.T. Gowers Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 OWB, UK E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . 2. Applications of Ramsey’s theorem 3. Inﬁnite Ramsey theory . . . . . . . 4. Structural Ramsey results . . . . . 5. Banach-space dichotomies . . . . . References . . . . . . . . . . . . . . .

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1. Introduction To mathematicians unfamiliar with Banach space theory, the existence of a chapter on Ramsey theory in this Handbook might well seem somewhat paradoxical. For surely Banach space theory is a branch of analysis, while Ramsey theory belongs to combinatorics, which is almost the antithesis of analysis. Nonetheless, as many other chapters also demonstrate, tools from discrete mathematics are very useful to Banach space theorists and this introduction will attempt to dispel the air of paradox in the case of Ramsey theory. We shall begin with a statement and quick proof of Ramsey’s original theorem. Given a set X, let X(k) stand for the set of all subsets of X of size k. An r-colouring of a set A means a function from A to the set {1, 2, . . . , r}. If a set A has an r-colouring and B ⊂ A, then B is said to be monochromatic if the image of B is constant. The theorem of Ramsey is the following. T HEOREM 1.1. Let k and r be positive integers. Then for every r-colouring of the set N(k) there exists an inﬁnite subset X of N such that X(k) is monochromatic. P ROOF. When k = 1 the result is obvious, so let us suppose that k > 1 and apply induction (noting that in the statement of the theorem we could replace N by any countably inﬁnite set). Let f be an r-colouring of N(k) , and construct a sequence of integers 1 = x1 < x2 < · · · and nested sets N = X0 ⊃ X1 ⊃ X2 ⊃ · · · of N as follows. When x1 , . . . , xm−1 and X0 , X1 , . . . , Xm−1 have been constructed, let xm be the least element of Xm−1 and let (k−1) Ym−1 = Xm−1 \ {xm }. Deﬁne an r-colouring gm of Ym−1 by gm (A) = f ({xm } ∪ A). The (k−1) inductive hypothesis implies that Ym−1 contains an inﬁnite subset Xm such that Xm is monochromatic under the colouring gm . Now let Y be the set {x1, x2 , x3 , . . .}. Because our construction guarantees that xr ∈ Xs whenever r > s, we ﬁnd that the value (or colour) f (A) of any set A ⊂ Y of size k depends only on the least element of A. Therefore, by the pigeonhole principle we may pass to an inﬁnite subset X of Y such that X(k) is monochromatic.

The above technique of constructing simultaneously a sequence of elements and a nested sequence of subsets, guaranteeing a good property at each stage, appears over and over again in Ramsey theory. It is quite extraordinary how much can be done with it. To see how the above result can in principle be used in continuous problems, we now prove a simple corollary. C OROLLARY 1.2. Let K be a compact metric space. For any ε > 0, any positive integer k and any function f : N(k) → K there is an inﬁnite subset X ⊂ N such that d(f (A), f (B)) < ε for every pair of sets A, B ∈ X(k) . P ROOF. Since K is totally bounded, we may cover it with disjoint sets B1 , . . . , Br , each of diameter less than ε. Given A ∈ N(k) , let us deﬁne g(A) to be the unique j such that f (A) ∈ Bj . Applying Theorem 1.1 we obtain an inﬁnite subset X such that X(k) is monochromatic for the colouring g. This set clearly satisﬁes the conclusion of the corollary.

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We can think of the function f above as being a sort of continuous colouring. It is then too much to ask for X(k) to be monochromatic (in the sense of the restriction of f being constant) but the corollary shows that we can ﬁnd X such that X(k) is approximately monochromatic, which is usually enough in applications. This indicates that the distinction between discrete and continuous problems is not an interesting one, at least in this context. While most mathematicians have heard of Ramsey’s theorem, it is much less well known that it has given rise to a large and thriving branch of mathematics. A theorem is usually said to belong to Ramsey theory if it is of the following type. Given any ﬁnite colouring of some mathematical object, there is a subobject of a certain type which is monochromatic. Here, for example, is a famous theorem of Graham and Rothschild [13]. T HEOREM 1.3. For every pair of positive integers r, d and every ﬁnite ﬁeld F , there is a positive integer N such that, for any r-colouring of the vector space V = F N , there is a d-dimensional afﬁne subspace W ⊂ V which is monochromatic. Notice that we could not have taken a vector subspace above, because we could have coloured elements of V according to whether or not they are zero. Even if we remove zero, we could colour according to the value of the ﬁrst non-zero coordinate. Although the Graham–Rothschild theorem itself has not been applied to Banach spaces, it concerns vector spaces and subspaces. This is very suggestive, and leads one to wonder whether there might be Ramsey theoretic results about Banach spaces and their subspaces. This turns out to be a fascinating question, which will be discussed later in this article. There are in fact three (not completely distinct) ways in which Ramsey theory is useful in the study of Banach spaces. The ﬁrst, and most obvious, is direct application of existing results from Ramsey theory. This will be illustrated in Sections 2 and 3. The second is the discovery of new results in Ramsey theory, obtained with the speciﬁc aim of applying them to problems in Banach space theory. An example of this will be given in Section 4. The third way is the proof of pure Banach-space results that are Ramsey theoretic in character and inspired by the methods of Ramsey theory. This is illustrated in Section 5.

2. Applications of Ramsey’s theorem Let us begin with a theorem of Brunel and Sucheston [6], which looks simple now but which was unexpected at the time and has had many applications. T HEOREM 2.1. Let x1 , x2 , . . . be a normalized basic sequence in a Banach space X, let ε > 0 and let k be a positive integer. Then there is an inﬁnite subsequence y1 , y2 , . . . of x1 , x2 , . . . such that, given any sequence (a1 , . . . , ak ) of scalars, and any pair of sequences m1 < · · · < mk and n1 < · · · < nk of integers, we have k k ai ymi (1 + ε) ai yni . i=1

i=1

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P ROOF. Let the basis constant of x1 , x2 , . . . be C. If n1 < n2 < · · · < nk , then we certainly know that C

−1

k max |ai | ai xni k max |ai |. i=1

Let F0 be the set of all norms · on k∞ such C −1 x x∞ kx for every x ∈ k∞ , let K be the unit sphere of k∞ , and let F be the set of restrictions of norms in F0 to K. Clearly F is a subset of C(K). It is easy to see also that F is compact (in the uniform metric on C(K)) since K is totally bounded, every norm · in F is determined to within ε by its values on an εk −1 -net of K, and these values lie in the closed interval [C −1 , k]. Now, given a sequence of integers n1 < · · · < nk , let A be the set {n1 , . . . , nk } and let f (A) be the norm · A in F deﬁned by k (a1 , . . . , ak ) = a x i n i . A i=1

By Corollary 1.2 we can ﬁnd an inﬁnite set Z = {n1 , n2 , . . .} of positive integers such that, for any pair of sets A, B ∈ Z (k) , the distance d( · A , · B ) (in C(K)) is at most ε/C. Since there is no loss of generality in assuming that the sequence of scalars (a1 , . . . , ak ) in the statement of the theorem belongs to the unit sphere of k∞ , and since for any such sequence and any norm · in F we have (a1 , . . . , ak ) C −1 , the subsequence xn1 , xn2 , . . . satisﬁes the conclusion of the theorem. Using a further diagonalization, Brunel and Sucheston obtained a stronger seeming result. T HEOREM 2.2. Let x1 , x2 , . . . be a normalized basic sequence. Then there is a subsequence y1 , y2 , . . . of x1 , x2 , . . . with the following property. Given any positive integer (k) k, any sequence (a1 , . . . , ak ) of scalars and any sequence k A1 , A2 , . . . of sets in N such that min Ai tends to inﬁnity, the sequence ci = j =1 aj ynij converges, where Ai = {ni1 , . . . , nik } with ni1 < · · · < nik . P ROOF. By Theorem 2.1 we can choose subsequences Si = (xi1 , xi2 , . . .) of the sequence S0 = (x1 , x2 , . . .) with the following properties. First, Si is a subsequence of Si−1 for every i 1 (that is, the subsequences are nested). Secondly, for each i 1, Si satisﬁes the conclusion of Theorem 2.1 with k = i and ε = i −1 . Now let S be the diagonal subsequence (x11, x22 , . . .). Given a sequence of sets A1 , A2 , . . . as in the statement of this theorem, our construction guarantees that the sequence ci is Cauchy, and hence converges. A few remarks about the above theorem are in order. To begin with, notice that the rate of convergence in the above theorem depends only on k and min Ai . Moreover, by passing to a further subsequence, we can speed up the rate of convergence as much as we like.

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However, there will always be a function ω : N → N tending to inﬁnity such that we can conclude nothing about the sum kj =1 aj ynj if n1 < ω(k). We can deﬁne a norm on c00 (see [15] for this deﬁnition) by letting (a1 , . . . , ak ) be the limit of the ci as ni1 tends to inﬁnity. This is clearly well-deﬁned. Let us deﬁne a basis x1 , x2 , . . . to be C-spreading if k k ai xmi C ai xni i=1

i=1

whenever k is a positive integer and n1 < · · · < nk and m1 < · · · < mk . The unit vector basis of c00 is clearly 1-spreading in the limiting norm we have just constructed. Let Z be the completion of c00 in this norm. It is easy to see that any space which is (block) ﬁnitely represented in Z (with respect to the obvious basis) is (block) ﬁnitely represented in the space generated by x1 , x2 , . . . . This means that when considering questions about (block) ﬁnite representability, we can often conﬁne ourselves to spaces with a 1-spreading basis. This observation motivates the following deﬁnition of Brunel and Sucheston. D EFINITION 2.3. Let (x1 , x2 , . . .) be a normalized basic sequence in a Banach space X. A spreading model of (x1 , x2 , . . .) is a normalized basic sequence (y1 , y2 , . . .) (not necessarily in X) for which there exists a sequence (δn ) tending to zero such that k k ai xmi − ai yi < δn i=1

i=1

whenever the scalars ai have modulus at most 1 and n m1 < · · · < mn . Note that a spreading model need not exist, is unique if it does, and automatically has a 1-spreading basis. We have shown that every normalized basic sequence has a subsequence with a spreading model. It follows that every Banach space contains a sequence with a spreading model. This is an important step in the proof of Krivine’s theorem, which we discuss brieﬂy in Section 4, and which is covered more fully in [16].

3. Inﬁnite Ramsey theory In this section we shall state and prove a Ramsey theorem of Nash-Williams [17] and use it to deduce one of the most famous results in Banach space theory, Rosenthal’s 1 -theorem [20]. Given a countably inﬁnite set X, let us write X(ω) for the set of all inﬁnite subsets of X. Nash-Williams’s theorem concerns colourings of N(ω) . One might initially hope that, given any r-colouring of N(ω) , there would be an inﬁnite subset X ⊂ N such that X(ω) was monochromatic. However, this is not the case, at least if the axiom of choice is assumed. For we may deﬁne inﬁnite sets A, B ⊂ N to be equivalent if their symmetric difference is ﬁnite, and from each equivalence class we can pick

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a representative. Then we may colour A red or blue according to whether the symmetric difference of A and the representative of its equivalence class is of odd or even size. Since adding an element to a set always changes its colour, there is no hope of ﬁnding a monochromatic X(ω) . (I cannot resist giving an alternative argument, which assumes the existence of a nonprincipal ultraﬁlter, which is known to be a strictly weaker assumption than the axiom of choice. Let α be such an ultraﬁlter. Given an inﬁnite set A, let f (n) = (−1)|A∩{1,2,...,n}| and let the colour of A be the limit along α of f . If the element N is added to A, then all values of f (n) change when n N , so the colour changes and once again there is no monochromatic X(ω) .) Such counterexamples are initially discouraging, but they are certainly not the end of the story. Instead, one considers colourings where the colours are in some sense “simple”, and obtains positive results for these. It is known, for example, that if the colour classes are all Borel in the product topology, then one can ﬁnd a monochromatic X(ω) . In fact, the situation is as good as one could possibly hope for, in that any colouring which can be described in a reasonable way gives a positive answer, and this applies to any set that comes up in applications. Nash-Williams’s theorem is the simplest result along these lines, a positive result for open sets. We shall state and prove an equivalent result, and then deduce Nash-Williams’s theorem from it. For a more general theorem of Ellentuck [8], which essentially characterizes those colourings for which a positive result holds, see Chapter 20 of [5]. Let A be a ﬁnite set and let Y be an inﬁnite set. We shall say that A is an initial segment of Y if Y ∩ {1, 2, . . ., max A} = A. (We shall also say that A is an initial segment of Y if A is empty. This is an important technicality.) Notice that this is much stronger than saying that A is a subset of Y . Now let A be any collection of ﬁnite subsets of N. If X ∈ N(ω) , we shall say that A is large for X if every Y ∈ X(ω) has a ﬁnite subset A ∈ A. We shall say that A is very large for X if every Y ∈ X(ω) has an initial segment in A. For a ﬁnal deﬁnition, given a collection A of ﬁnite sets, and a ﬁnite set F (not necessarily in A), let us write A(F ) for the set of all ﬁnite sets B such that max F < min B and F ∪ B ∈ A. T HEOREM 3.1. Let A be a collection of ﬁnite sets and suppose that A is large for N. Then there is an inﬁnite subset X of N such that A is very large for X. P ROOF. Let us suppose that no such set X exists. Then for every X ∈ N(ω) there is some set Y ∈ X(ω) with no initial segment in A. We shall now construct inductively sequences x1 < x2 < · · · of positive integers and N = X0 ⊃ X1 ⊃ X2 ⊃ · · · of inﬁnite subsets of N such that, for every n, two properties hold. First, xn ∈ Xn−1 and second, for no set (ω) F ⊂ {x1 , . . . , xn } and for no set Y ∈ Xn is A(F ) very large for Y . The induction starts with X0 = N and the assumption that A is not very large for any Y ∈ N(ω) . Suppose now that we have found x1 , . . . , xn and X1 ⊃ · · · ⊃ Xn , and suppose that we are unable to ﬁnd suitable choices for xn+1 and Xn+1 . This means that, for any y ∈ Xn such that xn < y and for any Y ∈ Xn(ω) , there is some subset F of {x1 , . . . , xn , y} such that A(F ) is very large for some Z ∈ Y (ω) . Using this fact, we can construct inductively sequences xn < y1 < y2 < · · · and Xn = Y0 ⊃ Y1 ⊃ Y2 ⊃ · · · such that, for every k, yk ∈ Yk−1 and there is a subset F ⊂ {x1 , . . . , xn , yk } such that A(F ) is very large for Yk . Notice that the

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construction of x1 , . . . , xn and Xn guarantees that such an F must contain yk , so let us write it as E ∪ {yk }, with E ⊂ {x1, . . . , xn }. By the pigeonhole principle, we can ﬁnd an inﬁnite subset W ⊂ {y1 , y2 , . . .} such that the same set E is used for every yi ∈ W . But then A(E) is very large for W , because, given any V ∈ W (ω) , let yi = min V . We know that A(E ∪ {yi }) is very large for Yi , which contains V \ {yi } as a subset, so V \ {yi } has an initial segment in A(E ∪ {yi }), which implies that V has an initial segment in A(E). We have thus contradicted our main (outer) inductive hypothesis. It follows that the sequences x1 < x2 < x3 < · · · and X1 ⊃ X2 ⊃ · · · can indeed be constructed. Now let X = {x1 , x2 , . . .}. We claim that no ﬁnite subset of X belongs to A. Indeed, if A ∈ A is a ﬁnite subset of X, then let xn > max A and let Y = {xn , xn+1 , xn+2 , . . .}. Obviously A(A) is very large for Y , because the empty set is an initial segment of any inﬁnite subset of Y . This contradicts our initial assumption that A is not very large for any X ∈ N(ω) . We conclude that A is not large for N, and the theorem is proved. Notice that Theorem 3.1 implies Ramsey’s theorem. Indeed, let N(k) be coloured with two colours, red and blue. If there is no subset X such that X(k) contains only red sets, then the set of all blue sets of size k is large for N. Therefore, there is an inﬁnite set Y such that the set of blue sets is very large for Y . But this means that the ﬁrst k elements of any inﬁnite subset of Y form a blue set, and therefore that Y (k) consists only of blue sets. The statement for r colours can be deduced easily. C OROLLARY 3.2. Let U ⊂ N(ω) be an open set. Then there is an inﬁnite subset X ⊂ N such that either X(ω) ⊂ U or X(ω) ∩ U = ∅. P ROOF. A basic open neighbourhood in the product topology is a set of the form {Y ∈ N(ω) : Y ∩ {1, 2, . . . , n} = A}. Since every such Y has a least element greater than n, it is easy to see that such a neighbourhood is a union of neighbourhoods of the form N(A) = Y ∈ N(ω) : A is an initial segment of Y . Given our open set U , let A be the collection of all sets A such that N(A) is a subset of U . That is, A ∈ A if and only if every inﬁnite set with A as an initial segment belongs to U . Since U is open, it is the union of all N(A) such that A ∈ A. Suppose that we cannot ﬁnd any X ∈ N(ω) such that X(ω) ∩ U = ∅. Then for every X ∈ N(ω) we can ﬁnd Y ∈ X(ω) ∩ U . By our above remarks, we can then ﬁnd A ∈ A such that Y ∈ N(A). Since A is an initial segment of Y and Y ⊂ X, we ﬁnd that A ⊂ X. Since X was arbitrary, we have shown that A is large. Applying Theorem 3.1, we obtain a set X ∈ N(ω) such that every Y ∈ X(ω) has an initial segment in A. But this means that every Y ∈ X(ω) belongs to N(A) for some A ∈ A, and therefore to U . In other words, X ∈ U . The above proof differs from that of Nash-Williams, who used ordinals. It is more usual to deduce 3.2 from a lemma of Galvin and Prikry, which is very similar to 3.2 but a bit more general (see [5] for this approach). We have given a direct argument in order to emphasize the similarity with 5.5 below.

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Let us now turn to Rosenthal’s theorem, which is the following beautiful characterization of Banach spaces that contain 1 . Though Ramsey-theoretical ideas were implicit in Rosenthal’s argument, the connection was ﬁrst brought out explicitly by Farahat [9]. The proof we give here is similar in spirit to that of [9], but in its details is closer to an approach of Behrends [3]. T HEOREM 3.3. Every bounded sequence x1 , x2 , . . . of vectors in a Banach space has a subsequence which is either weakly Cauchy or equivalent to the unit vector basis of 1 . P ROOF. Let X be the Banach space and let B be the unit ball of the dual space X∗ . We can regard each xn as a continuous function from B to C, and the norm on X as the supremum norm on C(B). To say that the sequence (xn ) is weakly Cauchy is to say that the sequence (xn (s)) is Cauchy for every s ∈ B, which is the same as saying that the functions xn converge pointwise. Let us now suppose that this does not happen for any subsequence of (xn ), and try to ﬁnd instead a subsequence equivalent to the unit vector basis of 1 . Let D be the set of all pairs of open discs (D1 , D2 ) in the complex plane with (complex) rational centres (z1 , z2 ) and equal rational radii r, with the additional assumption that 4r < |z1 − z2 |. We are supposing that, for every inﬁnite set M ⊂ N, the sequence (xn )n∈M fails to converge pointwise. This gives us some s ∈ B such that the sequence (xn (s))n∈M has at least two accumulation points, and hence a pair (D1 , D2 ) ∈ D such that (xn (s)) lies inﬁnitely often in D1 and inﬁnitely often in D2 as n ranges over the set M. An easy diagonal argument shows that, after passing to an appropriate subsequence of (xn ), we can choose the same pair (D1 , D2 ) for every set M. Indeed, suppose this is not possible. Then for every k, every inﬁnite set M and every (D1 , D2 ) ∈ D we can ﬁnd an inﬁnite subset L ⊂ M such that the sequence (xn (s))n∈L does not visit both D1 and D2 (k) (k) inﬁnitely often for any s ∈ S. If we enumerate the countable set D as ((D1 , D2 ))k∈N , we can therefore construct a nested sequence of sets M1 ⊃ M2 ⊃ · · · such that, for every k, (k) (k) the sequence (xn (s))n∈Mk does not visit both D1 and D2 inﬁnitely often for any s ∈ B. Let M = {m1 < m2 < · · ·} be a diagonal sequence, that is, one with mk ∈ Mk for every k. By construction, there is no (D1 , D2 ) ∈ D such that (xn (s))n∈M visits both D1 and D2 inﬁnitely often for some s ∈ B, which is a contradiction. So we may now ﬁnd a ﬁxed (D1 , D2 ) ∈ D and assume (after passing to a subsequence and relabelling it as the whole sequence) that for every M ∈ N(ω) there is some s ∈ B such that the sequence (xn (s))n∈M visits both D1 and D2 inﬁnitely often. We are now ready to apply Nash-Williams’s theorem theorem, which we shall do in the form of Theorem 3.1. Let us deﬁne a ﬁnite set A = {n1 < n2 < · · · < nk } to be possible if we can ﬁnd s ∈ B such that xni (s) is in D1 when i is odd and D2 when i is even. Otherwise, say that A is impossible. Let A be the collection of all impossible sets. We know from the previous paragraph that, for every M ∈ N(ω) , there is some s ∈ B such that the sequence (xn (s))n∈M visits D1 and D2 inﬁnitely often, which means that there is some set L ∈ M (ω) , L = l1 < l2 < · · ·, such that xli (s) is in D1 when i is odd and D2 when i is even. In other words, every M ∈ N(ω) has an inﬁnite subset L with no impossible initial segments. This tells us that the set A is not very large for any inﬁnite subset of N.

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By Nash-Williams’s theorem, we can therefore ﬁnd an inﬁnite set M ∈ N(ω) such that no ﬁnite subset of M belongs to A, which means that every ﬁnite subset of M is possible. Let M = {m1 < m2 < m3 < · · ·} and let L = {m2 , m4 , m6 , . . .}. We shall now show that (xn )n∈L is equivalent to the unit vector basis of 1 . We chose alternate elements of M for the following reason. Let n ∈ N and let f {m2 , m4 , . . . , m2n } → {1, 2} be an arbitrary function. Deﬁne a set A ⊂ {m1 , . . . , m2n } by saying that m2j ∈ A for every j n, m1 ∈ A if and only if f (m2 ) = 2 and m2j −1 ∈ A if and only if f (m2j ) = f (m2j −2 ). Since A is a possible set, we can ﬁnd s ∈ B such that xm2j (s) belongs to D1 when f (m2j ) = 1 and to D2 when f (m2j ) = 2. Now consider the sum nj=1 cj xm2j , with cj = aj + ibj . Let z1 be the centre of D1 , z2 the centre of D2 and r the common radius of the two discs. Without loss of generality (after multiplying by appropriate scalars) we may assume that z2 − z1 is real and positive, and that nj=1 |aj | nj=1 |bj |. Let f1 (m2j ) equal 2 if aj 0 and 1 otherwise, and choose s1 as in the paragraph above. Let f2 (m2j ) equal 1 if aj 0 and 2 otherwise, and choose s2 as in the paragraph above. Let A = {aj : aj 0} and B = {aj : aj < 0}. Then !

n

aj x2j (s1 ) ! A(z2 − r) + B(z1 + r)

j =1

and !

n

aj x2j (s2 ) ! A(z1 − r) + B(z2 + r) .

j =1

It follows that n n

! aj x2j (s1 − s2 )/2 (A − B)(z1 − z2 )/2 2r |aj |. j =1

j =1

It is not hard to see that n n

! bj x2j (s1 − s2 )/2 r |bj |. j =1

j =1

It follows that n n n cj xm2j r |aj | (r/2) |cj |. j =1

j =1

j =1

−1 This shows that the sequence (xm2j )∞ j =1 is 2r C- equivalent to the unit vector basis of 1 , where C is the upper bound on the norms of the vectors xn .

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We end this section by remarking that Rosenthal has a companion theorem characterizing spaces that contain c0 [21]. This theorem is even more closely connected with descriptive set theory, and is discussed in [1].

4. Structural Ramsey results So far, we have applied Ramsey theorems to sequences in Banach spaces, using them to ﬁnd subsequences with good properties. However, there are many questions in Banach space theory concerning the existence of subspaces with good properties, and although the subsequence results are useful, one begins to yearn for something more. Is there, for example, a result resembling Theorem 1.3 but concerning Banach spaces and closed linear subspaces? Before pursuing this question, one must make certain obvious restrictions. First of all, it is important to eliminate simple colourings according to the norm of a vector (such as red for vectors of norm at most one and blue for the rest). This is easily done by restricting attention to the unit sphere. Second, one must not ask for the result to be too exact. Consider, for example, the colouring of the unit sphere of 2 where a vector x is red if its ﬁrst non-zero coordinate is positive and blue otherwise. Then x never has the same colour as −x. However, every vector with ﬁrst coordinate zero is arbitrarily close to a vector of the opposite colour, which shows that the unit sphere of the subspace {x: x1 = 0} is at least approximately monochromatic. The correct formulation of the question turns out to be as follows. P ROBLEM 4.1. Let X be a Banach space, let f : S(X) → {1, 2, . . . , r} be an r-colouring of the unit sphere of X and let ε > 0. Must there exist an inﬁnite-dimensional subspace Y and a colour j such that for every vector y in the unit sphere of Y there is a vector z in the unit sphere of Y such that y − z < ε and f (z) = j ? In other words, can Y be found such that every point in the sphere of Y is close to a point of some given colour? The Banach space X in the above problem may either be a general one or something speciﬁc such as 2 . In both cases the answer is far from obvious. An equivalent and often more useful formulation is the following. P ROBLEM 4.2. Let X be a Banach space, let f : S(X) → R be a uniformly continuous function and let ε > 0. Must there exist an inﬁnite-dimensional subspace Y such that, given any two vectors x, y ∈ S(Y ), |f (x) − f (y)| < ε? These problems are discussed fully in Chapter 31 on distortion [19], so we shall conﬁne ourselves to a few remarks. First, it is an easy consequence of Milman’s proof of Dvoretzky’s theorem that the problems have positive answers if we ask merely that Y should have arbitrarily large ﬁnite dimension. Moreover, using the proof of Krivine’s theorem instead of that of Dvoretzky’s (for all these results see [16]), one can ask for Y to be generated by a block basis of any given basic sequence in X. In particular, if X is c0 or p , Y can be chosen isometric to n∞ or np .

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Second, there are results in Ramsey theory which are encouraging. One of them is a theorem of Hindman [14], which can be formulated as follows. T HEOREM 4.3. Let the ﬁnite subsets of N be coloured with ﬁnitely many colours. Then there exist ﬁnite subsets A1 , A2 , A3 , . . . of N, with max Ai < min Ai+1 for every i, such that all non-empty unions of ﬁnitely many of the Ai have the same colour. If one identiﬁes ﬁnite subsets of N with 01-sequences in the natural way, then the above theorem says that for any ﬁnite colouring of a sort of discrete sequence space there is a monochromatic “inﬁnite-dimensional block subspace”. This seems to be evidence in favour of a positive answer to Problems 4.1 and 4.2. The theorems of Dvoretzky and Krivine show that any counterexample to the problems has to be very deﬁnitely inﬁnite-dimensional. Hindman’s theorem suggests that it must also be non-combinatorial in an essential way if it is to defeat known Ramsey-theoretic arguments. Despite these two restrictions, counterexamples are known to exist, but because of the restrictions, they are very mysterious. The ﬁrst known explicit counterexample was, in the second formulation, a simple renorming, due to Odell, of a small variant of Tsirelson’s space (see [15]). The famous distortion problem concerns the case X = p and was solved in the negative by Odell and Schlumprecht [18]. The ideas they introduced have now been used to show that for almost no space X is there a positive answer: this is in striking contrast with the ﬁnite-dimensional situation. This article is concentrating on positive results, so we shall now consider the one case where there is a Ramsey result for Banach spaces and their subspaces, which is when the space is c0 . This space is sufﬁciently like a space of zeros and ones, and addition of disjointly supported vectors is sufﬁciently like the union of disjoint sets, that ideas from the proof of Hindman’s theorem can be used successfully. The ﬁrst step of the argument is to deﬁne a sort of combinatorial approximation of c0 . To begin with, we do this just for the positive part of the unit sphere (i.e., the set of vectors with non-negative coordinates). Given ε > 0 and k = k(ε) sufﬁciently large, let A (for alphabet) be the set {0} ∪ {(1 − ε)r : 0 r k}. Then any vector y in the positive part of the sphere of c0 can be approximated to within ε by a ﬁnitely supported vector x all of whose coordinates belong to A. In addition, we can insist that at least one coordinate of x is 1. Let X be the set of all ﬁnitely supported vectors with coordinates in A, at least one of which is 1. The advantage of the alphabet A over a more obvious one like {0, k −1 , 2k −1 , . . . , 1} is that it is easy to deal with scalar multiplication. Indeed, if we deﬁne a binary operation ∗ on A by letting a ∗ b be ab if ab ∈ A and 0 otherwise, then |ab − a ∗ b| < (1 − ε)k , which we have assumed to be less than ε. Given a ∈ A and x = (x1 , x2 , . . .) ∈ X, let us deﬁne a ∗ x to be the vector (a ∗ x1 , a ∗ x2 , . . .). Although a ∗ x does not belong to X (unless a = 1), we still have the property that if x and y are in X and have disjoint supports, then x + a ∗ y and a ∗ x + y belong to X. Because X is closed under the addition of disjointly supported vectors, it has the structure of a partial semigroup. This allows us to use ultraﬁlter techniques, which also yield a very neat proof of Hindman’s theorem (due to Glazer). Let us begin with a lemma which is fundamental in this kind of argument.

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L EMMA 4.4. Let (S, +) be a compact Hausdorff semigroup such that addition is rightcontinuous, meaning that for every x ∈ S the map y → y + x is continuous. Then S contains an idempotent; that is, an element x such that x + x = x. P ROOF. By an easy Zorn’s lemma argument there is a minimal non-empty compact subsemigroup T of S. Let x ∈ T be arbitrary. Since T is compact and addition is rightcontinuous, the set T + x is compact. Moreover, (T + x) + (T + x) = (T + x + T ) + x ⊂ T + x, so T + x is a subsemigroup. But T + x ⊂ T , so by the minimality of T we must have T + x = T . It follows that there exists z ∈ T such that z + x = x. Now the set of all such z is compact, as it is the inverse image of {x} under the continuous map z → z + x. Moreover, it is closed under addition. Therefore, by minimality again, it is the whole of T . It follows that x + x = x. We shall apply Lemma 4.4 to a certain semigroup of ultraﬁlters. Let us say that an ultraﬁlter α on the set X deﬁned above is coﬁnite if every set of the form Xn = {x ∈ X: x1 = · · · = xn−1 = 0} belongs to α. (This is the appropriate non-triviality condition, and is stronger than saying merely that α is non-principal.) The following notation is incredibly useful, not just to save writing but also as an aid to thought. If P (x) is a statement about vectors x ∈ X, we shall write (αx) P (x) for the statement {x ∈ X: P (x)} ∈ α. One can think of α as a (ﬁnitely additive) measure on X and read (αx) as “for α-almost every x”. Thus, α becomes a quantiﬁer. The ultraﬁlter properties of α convert into the rules (αx) P (x) and (αx) Q(x)

if and only if (αx) P (x) and Q(x)

and not (αx) P (x)

if and only if

(αx) not P (x),

which we shall use repeatedly. If α and β are two coﬁnite ultraﬁlters on X, we deﬁne their sum to be α + β = A ⊂ X: (αx) (βy) x + y ∈ A . We interpret the statement x + y ∈ A to be false if x and y are not disjointly supported. Notice, however, that because β is coﬁnite and x is ﬁnitely supported we have that (αx) (βy) x and y are disjointly supported so in the ultraﬁlter world, questions of support are more or less irrelevant. It may make it easier to understand ultraﬁlter addition if we state a third quantiﬁer rule, which is equivalent to the deﬁnition (as may be seen by replacing “P (x)” by “x ∈ A”):

(α + β)x P (x)

if and only if

(αx) (βy) P (x + y).

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It is easy to check (and this is one of the advantages of the quantiﬁer notation) that ultraﬁlter addition is associative. We shall also deﬁne “linear combinations” of coﬁnite ultraﬁlters on X in a similar way. Let α and β be two such ultraﬁlters and let a ∈ A. Then α + a ∗ β = A ⊂ X: (αx) (βy) x + a ∗ y ∈ A and a ∗ α + β = A ⊂ X: (αx) (βy) a ∗ x + y ∈ A . Clearly this deﬁnition can be extended to general combinations a1 ∗ α1 + · · · + an ∗ αn , provided that at least one ai equals 1. Once again, there are corresponding quantiﬁer rules. For example,

(α + a ∗ β)x P (x)

if and only if

(αx) (βy) P (x + a ∗ y).

It is possible and useful to interpret a ∗ α as an ultraﬁlter itself, not on X buton a ∗ X, which is the combinatorial version of the set of points in c0 of norm a. Let Y = a∈A a ∗ X (the combinatorial unit ball rather than unit sphere). Then Y is still a partial semigroup, and if we deﬁne a ∗ α to be {A ⊂ a ∗ X: (αx) a ∗ x ∈ A}, then a ∗ α + β is simply the sum of two ultraﬁlters deﬁned on Y . We state the next lemma without proof, since it is basically an exercise to check everything. L EMMA 4.5. Let S be the set of all coﬁnite ultraﬁlters on X. Then S is a compact Hausdorff semigroup, under ultraﬁlter addition. Moreover, for every a ∈ A and every β ∈ S, the maps α → a ∗ α + β and α → α + a ∗ β are continuous. Notice that Lemmas 4.4 and 4.5 imply the existence of an idempotent ultraﬁlter in S. The next result is a strengthening of this fact, and is the key to the whole argument. L EMMA 4.6. There exists an ultraﬁlter α ∈ S such that α + a ∗ α = a ∗ α + α = α for every a ∈ A. P ROOF. Set a = (1 + ε)−1 and for j = 0, 1, 2, . . ., k let us write Xj for a k−j ∗ X and Sj for the set of all coﬁnite ultraﬁlters on Xj . Lemma 4.5 tells us that Sj is a compact Hausdorff semigroup and that linear combinations are right continuous. (This is not quite directly true, but follows from the fact that in Lemma 4.5 there is an implicit dependence on k, and there is an isomorphism between Xj and the case k = j of X.) We shall prove by induction that for every j k there is a coﬁnite ultraﬁlter α on Xj such that a i ∗ α + α = α + a i ∗ α = α for i = 0, 1, . . . , j . The result is true when j = 0, because X0 consists of ﬁnitely supported non-zero vectors all of whose coordinates are either 0 or a k . Therefore, all we need to ﬁnd is an idempotent ultraﬁlter. The existence of this follows from Lemmas 4.4 and 4.5. So let us now suppose

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that j > 0, and that we have a coﬁnite ultraﬁlter β on Xj −1 such that β + a i ∗ β = a i ∗ β + β = β for i = 0, 1, . . . , j − 1. Let T = {α ∈ Sj : a ∗ α = β}. It is easily checked that the map α → a ∗ α is a continuous map from Sj to Sj −1 . Therefore T is compact, as is the set T + β. This second set is also closed under addition, since if a ∗ γ = a ∗ δ = β, then a ∗ (γ + β + δ) = a ∗ γ + a ∗ β + a ∗ δ = β + a ∗ β + β = β + β = β. (The equality a ∗ β + β = β follows even when j = 1 because then β ∈ S0 and a ∗ β = 0.) By Lemma 4.4 we can ﬁnd an idempotent in T + β. That is, there is an ultraﬁlter γ ∈ Sj such that a ∗ γ = β and γ + β + γ + β = γ + β. Now set α = β + γ + β. If 1 i j , then α + a i ∗ α = β + γ + β + a i ∗ β + a i−1 ∗ β + a i ∗ β = β + γ + β = α (when i = j we are using the fact that a j β = 0) while α + α = β + γ + β + β + γ + β = β + (γ + β) + (γ + β) = β + (γ + β) = α. Therefore, the ultraﬁlter α gives us the inductive step.

By a block basis of X, we shall mean a sequence in X that forms a normalized block basis of c0 . If x1 , x2 , . . . is a block basis of X, then by the subspace generated by x1 , x2 , . . . we shall mean the set of all points in X of the form ni=1 ai ∗ xi . (Notice that ai must equal 1 for at least one i.) A subspace generated by a block basis is a block subspace. The ultraﬁlter we have just constructed implies the following combinatorial Ramsey result. T HEOREM 4.7. Let X be ﬁnitely coloured. Then X contains a monochromatic inﬁnitedimensional block subspace. We do not give the proof here, because it is formally almost identical to the proof of Theorem 4.13 which is more central to our concerns and will be given later. Notice, however, that if k = 1, then Theorem 4.7 is equivalent to Hindman’s theorem. One can use Theorem 4.7 to prove a positive answer to 4.2 in the case of functions f that depend only on the (pointwise) modulus of a vector. (One could think of these as “unconditional” functions.) We now turn to the more general situation. Let Z be the set of all ﬁnitely supported vectors in the unit sphere of c0 with coordinates in A ∪ −A, and note that Z is an ε-net of the entire unit sphere of c0 . The deﬁnitions of multiplication in A ∪ −A, scalar multiplication in Z, block subspaces and so on are all obvious. The next lemma will allow us to use the method of Lemma 4.6 to ﬁnd an ultraﬁlter appropriate for results about Z. One cannot hope for exact results about Z because of the colouring according to the sign of the ﬁrst non-zero coordinate. Accordingly, even though Z is discrete, we must consider approximations. The alphabet A ∪ −A is a totally ordered set. Let us say that two elements a, b ∈ A are neighbours if they are next to each other in this ordering. Given

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vectors x, y ∈ Z, let us say that they are neighbours if xi is a neighbour of yi for every i. Let the neighbourhood N(x) of x be the set of all neighbours of x. Given a subset E ⊂ Z, let us deﬁne the expansion E of E to be the set of all neighbours of elements of E, or equivalently the union of all neighbourhoods of elements of E. L EMMA 4.8. There exists a coﬁnite ultraﬁlter α on Z such that, for any set E ∈ α, the set −E also belongs to α. P ROOF. Let β be any coﬁnite ultraﬁlter on Z, let a = (1 − ε) and let α = a k ∗ β − a k ∗ β + a k−1 ∗ β − a k−1 ∗ β + · · · + a ∗ β − a ∗ β + β − β + a ∗ β − a ∗ β + a 2 ∗ β − a 2 ∗ β + · · · + a k ∗ β − a k ∗ β. We shall see that α has the required property. For notational convenience, let a1 , a2 , . . . , am be the sequence a k , −a k , . . . , a, −a, 1, −1, . . ., a k , −a k of coefﬁcients used above. Then, using our quantiﬁer rules for linear combinations, the statement that E belongs to α is equivalent to the statement (βx1 ) . . . (βxm )

m

ai ∗ xi ∈ E.

i=1

By relabelling some of the dummy variables, we can say also that

(βx0 ) . . . (βxm−1 )

m−1

ai+1 ∗ xi ∈ E.

i=0

Now the sequence a1 , . . . , am has been chosen with the property that, for every i < m, ai is a neighbour of −ai+1 , and also so that a1 and am are neighbours of 0. Therefore, from the second statement, we can deduce that (βx1 ) . . . (βxm )

m

ai ∗ xi ∈ −E

i=1

which shows that −E ∈ α, as required.

The above trick is very similar to an argument used by Brunel and Sucheston to ﬁnd an unconditional spreading model in any Banach space, using invariance under spreads. Let us deﬁne an approximate ultraﬁlter on Z to be a ﬁlter γ such that, whenever Z = C1 ∪ · · · ∪ Cr , there exists i such that the expansion C i belongs to γ . We shall say that

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a ﬁlter γ is sign-invariant if E ∈ γ implies that −E ∈ γ . Notice that sign-invariance is equivalent to the quantiﬁer rule (γ x) P (x)

if and only if (γ x)P (−x).

C OROLLARY 4.9. There exists a coﬁnite sign-invariant approximate ultraﬁlter γ on Z. P ROOF. Let α be the ultraﬁlter constructed in Lemma 4.8. Deﬁne γ to be the set of all E such that E ∈ α and −E ∈ α. Then the sign-invariance of γ is immediate, and all we need to do is show that γ is an approximate ultraﬁlter. This is also easy. If Z = C1 ∪ · · · ∪ Cr , then Ci belongs to α for some i, and by Lemma 4.8 so does −C i . Obviously C i belongs to α as well, so it belongs to γ . The next lemma is another exercise in checking uninteresting facts, so again we state it without proof. L EMMA 4.10. The set S of all coﬁnite sign-invariant approximate ultraﬁlters on Z is a compact Hausdorff semigroup under ﬁlter addition. Moreover, the maps α → α + a ∗ β and α → a ∗ α + β are continuous for every a ∈ A ∪ −A and every β ∈ S. The next lemma is deduced from Lemma 4.10 in a similar way to how Lemma 4.6 is deduced from Lemma 4.5. However, the proof is not quite identical: instead of deﬁning Zj to be a k−j ∗ Z and letting Sj be the set of coﬁnite sign-invariant approximate ultraﬁlters on Zj , one should instead deﬁne Sj to be a k−j ∗ S. (If one does not do this, then it is not true that a ∗ Sj = Sj −1 .) In fact, the proof given in [10] is for this reason not quite correct. A correct proof, with full details, can be found in [4] on pages 318–319. L EMMA 4.11. There exists a coﬁnite sign-invariant approximate ultraﬁlter α on Z such that α + a ∗ α = a ∗ α + α = α for every a ∈ A. Given a block basis x1 , . . . , xn in Z we shall write x1 , . . . , xn for the (combinatorial) subspace generated by x1 , . . . , xn . C OROLLARY 4.12. There exists an approximate ultraﬁlter α on Z such that E ∈ α implies that (αx) (αy) x, y ⊂ E. P ROOF. Let α be the ﬁlter from Lemma 4.11 and let E ∈ α. For every j ∈ {0, 1, . . . , k} we know that (αx) (αy) x + a j ∗ y ∈ E and (αx) (αy) a j ∗ x + y ∈ E.

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By the quantiﬁer rule for sign-invariance, we can deduce that (αx) (αy) ± x ± a j ∗ y ∈ E and (αx) (αy) ± a j ∗ x ± y ∈ E and the required property of α follows from the quantiﬁer rule for conjuctions of statements. Notice that the property of α in Corollary 4.12 translates into the quantiﬁer rule (αx) P (x)

if and only if

(αx) (αy) ∀z ∈ x, y P (z).

We can now deduce a Ramsey theorem for colourings of Z. As commented earlier, almost exactly the same argument proves Theorem 4.7. T HEOREM 4.13. Let Z be ﬁnitely coloured with colours C1 , . . . , Cr . Then there exists i such that C i contains an inﬁnite-dimensional block subspace of Z. P ROOF. Let α be the ﬁlter from Corollary 4.12, let i be such that C i ∈ α and let C = C i . We shall apply our quantiﬁer rules over and over again. To begin with we know that (αx) x ∈ C. By the rule stated just before this theorem we can deduce that (αx) (αy) x, y ⊂ C. It follows (after renaming x and y) that there exists x1 such that (αx) x1 , x ⊂ C. Suppose inductively that we can ﬁnd x1 , . . . , xn such that (αx) x1 , . . . , xn , x ⊂ C. Then by the quantiﬁer rule we ﬁnd that (αx) (αy) ∀z ∈ x, y x1 , . . . , xn , z ⊂ C which is easily seen to be equivalent to the statement (αx) (αy) x1 , . . . , xn , x, y ⊂ C. Hence, we can ﬁnd xn+1 to continue the induction. We therefore construct a sequence x1 , x2 , . . . with the property that x1 , . . . , xn ⊂ C for every n, which means that the block subspace generated by x1 , x2 , . . . is a subset of C. It is easy to deduce from this a result about uniformly continuous functions on the sphere of c0 .

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T HEOREM 4.14. Let f be a uniformly continuous function from the unit sphere of c0 to R and let δ > 0. Then there is an inﬁnite-dimensional subspace W of c0 such that |f (v) − f (w)| < δ for every pair of vectors v, w in the unit sphere of W . P ROOF. Without loss of generality, f takes values between 0 and 1. Choose ε and k such that the set Z is an η-net of the unit sphere of c0 , where η is small enough that |f (x) − f (y)| < δ/5 whenever x − y 2η, and partition [0, 1] into sets I1 , . . . , Ir of diameter at most δ/5. For 1 j r let Cj be the set of x ∈ Z such that f (x) ∈ Ij . By Theorem 4.13 there is an inﬁnite-dimensional block subspace (in the combinatorial sense) V of Z contained in C j for some j . It is easy to check that the distance between two neighbours in Z is at most η, so for every v ∈ V there exists x ∈ Z such that v − x 2η and f (x) ∈ Ij . It follows that f (v) lies strictly within δ/5 of a point in Ij . Now let W be the block subspace in c0 generated by V . It is easy to check that V is an η-net of the unit sphere of W . Therefore, for every w in the sphere of W , f (w) lies strictly within 2δ/5 of a point in Ij . It follows easily that W has the required property.

5. Banach-space dichotomies While it is now known that a general Banach space does not need to contain a subspace with nice symmetry properties (see [15]), there is still scope for dichotomy theorems, that is, results that assert the existence of a subspace with one of two extreme properties. Rosenthal’s 1 -theorem is an example of such a theorem. In this section, we shall consider some others. Typically, they involve saying that if a Banach space fails to contain a subspace with some good symmetry property, then it must have a subspace which lacks symmetry in a very extreme way. It is natural to try to use Ramsey theory to obtain such results, colouring appropriate objects according to whether they are good (helping to produce symmetry) or bad (showing no symmetry at all). However, it is not quite so clear what the objects should be, or how to formulate and prove an appropriate result, especially given the counterexamples mentioned at the beginning of the previous section. In order to help us arrive at a good formulation, it will be useful to state a theorem from [11], which was the initial motivation for a more abstract Banach-space Ramsey result. For the deﬁnitions in the statement of the theorem, see [15]. All spaces and subspaces in this section will be inﬁnite-dimensional, unless we explicitly say otherwise. T HEOREM 5.1. Every Banach space X has a subspace Y which either has an unconditional basis or is hereditarily indecomposable. Now by standard arguments, we may as well let X be a space of the form (c00 , · ), where c00 is the space of all ﬁnitely supported sequences and · is a norm on c00 making the natural basis into a normalized monotone basis (at least after the space is completed, but this technicality is unimportant here). The following lemma, an easy exercise, is stated without proof.

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L EMMA 5.2. A space X = (c00 , · ) contains no unconditional basic sequence if and only if, for every C 1, every block subspace of X contains a sequence y1 < · · · < yn (for some n) of vectors of norm at most 1 such that n n n (−1) yi > C yi . i=1

i=1

Let us call such a sequence y1 < · · · < yn C-conditional. If X contains no unconditional basic sequence, then every block subspace of X contains a ﬁnite C-conditional sequence for every C. Encouraged by the proof of Theorem 3.1, let us deﬁne a set of ﬁnite block sequences in c0 to be large for a block subspace Y of c00 if every block subspace of Y contains one of them. In this terminology, for every C 1 the set of C-conditional sequences in X is large for c00 = X. Can we now give a Banach-spaces version of Theorem 3.1? The answer is no, until we can give a suitable deﬁnition of “very large”. The most obvious idea is to say that a collection σ of ﬁnite block sequences is very large for X if every block basis of X has an initial segment in σ . However, the distortability results mentioned earlier (and proved in [19]) show that this is too strong a deﬁnition, even for sequences of length one. Indeed, let X be a space such that the unit sphere contains asymptotic sets A, B with x − y α > 0 for every x ∈ A and y ∈ B. Let σ be the set of sequences (x) such that x ∈ A. Then σ is large, but every block subspace of X has a point not in A (or even close to A) which can obviously be extended to an inﬁnite block basis. Let us return to our motivating result, Theorem 5.1. We are hoping to use a Ramsey result for ﬁnite sequences of blocks, applied to conditional sequences, to obtain a hereditarily indecomposable subspace. The next lemma, also stated without proof as it is an easy exercise and is essentially covered in [15], gives us a clue about what to do. L EMMA 5.3. Let X = (c00 , ·) and let Y be a block subspace of X. Then Y is hereditarily indecomposable if and only if, for every C 1 and any pair Z, W of block subspaces of Y , there is a sequence z1 < w1 < z2 < w2 < · · · < zn < wn (for some n) such that n n (zi + wi ) > C (zi − wi ). i=1

i=1

Notice that the property we require of Y certainly implies that the set of C-conditional sequences is large for every C. (Just let Z = W .) It is not hard to show that it is strictly stronger. Let Σ = Σ(X) be the set of all ﬁnite block sequences in X consisting of vectors of norm at most 1. It would now be natural to deﬁne a set σ ⊂ Σ to be very large for Y if, for any pair of block subspaces Z, W of Y there was a sequence (z1 , w1 , z2 , w2 , . . . , zn , wn ) in σ such that zi ∈ Z and wi ∈ W for every i. Instead, we shall give what appears to be the strongest deﬁnition for which a positive result can be obtained. Given a set σ ⊂ Σ, one can deﬁne a two-player game as follows. One player, S (for subspace), chooses a block subspace X1 of X. The other player, V (for vector), chooses a vector x1 ∈ X1 of norm at most 1. Then S chooses a subspace X2 and V chooses x2 ∈ X2

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of norm at most 1. Play continues like this. V wins the game if, at some point, the sequence (x1 , . . . , xn ) resulting from the play belongs to σ . It is an easy but instructive exercise to give an example of a set σ of sequences such that V wins the game, but such that for any n, S can force the game to last for at least n moves. D EFINITION 5.4. A set σ ⊂ Σ of ﬁnite block sequences is strategically large, or s-large for Y if V wins the above game when S is restricted to subspaces of Y . As in Section 4 any Ramsey result must be an approximate one, so we now introduce appropriate notation for the approximations. If Δ = (δ1 , δ2 , . . .) is a sequence of non-negative real numbers and σ ⊂ Σ, then σΔ will stand for the set of all sequences (x1 , . . . , xn ) ∈ Σ such that there exists a sequence (y1 , . . . , yn ) ∈ σ with xi − yi δi for every i n. We shall also write σ−Δ for the complement of (σ c )Δ . An equivalent way of deﬁning σ−Δ is as the set of all sequences (x1 , . . . , xn ) ∈ Σ such that (y1 , . . . , yn ) ∈ σ whenever xi − yi δi for every i n. It follows, of course, that (σ−Δ )Δ ⊂ σ . It is also easy to see that (σΔ )−Δ ⊃ σ . We are now ready for our Nash-Williams-type theorem. In order to deal with the approximations, it is necessary for the statement to be a little untidy. However, as we shall show in Corollary 5.6, a tidier version follows easily. T HEOREM 5.5. Let X = (c00 , · ) be a sequence space with basis constant C and let ρ ⊂ Σ(X) be a set of ﬁnite block sequences in the unit ball of X. Let Θ = (θ1 , θ2 , . . .) with every θ i > 0 and let Δ = (δ1 , δ2 , . . .) be another sequence of positive real numbers such that C ∞ i=N δi θN for every N . If σ−Θ is large for X, then X has a block subspace Y such that σ2Δ is strategically large for Y . P ROOF. We shall need an important trick (or at least it seems to be important) which was not necessary for the proof of Theorem 3.1. This is to assume that σ is minimal in the following sense. If (x1 , . . . , xn ) is in σ and (y1 , . . . , ym ) is a sequence in Σ such that yi ∈ x1 , . . . , xn for every i, then (y1 , . . . , ym ) is not in σ unless n = m, which implies that yi is a multiple of xi for every i. It is easy to see that if σ is large for X, then σ contains a subset which is minimal in this sense and still large for X, so we lose no generality by making this assumption. As far as possible, the rest of the proof will be organized along similar lines to the proof of Theorem 3.1. Accordingly, given a sequence (x1 , . . . , xn ) ∈ Σ, let us deﬁne σ (x1 , . . . , xn ) to be the set of all (possibly null) sequences (y1 , . . . , ym ) such that (x1 , . . . , xn , y1 , . . . , ym ) belongs to σ . Given n ∈ N, let Δn be the sequence (δ1 , . . . , δn , 0, 0, . . .) and let Γn be the sequence (δ1 , . . . , δn , 2δn+1 , 2δn+2 , . . .). We now attempt to construct sequences of vectors x1 < x2 < · · · and block subspaces X = X0 ⊃ X1 ⊃ X2 ⊃ · · · with the following properties, for every n. (i) xn ∈ Xn−1 . (ii) σΔn (x1 , . . . , xn ) is large for Xn . (iii) σΓn (x1 , . . . , xn ) is not strategically large for any subspace of Xn . If the theorem is false, then the induction starts with the null sequence and X0 = X, since σ is large for X and not strategically large for any subspace of X. (We omit the

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word “block” from now on.) As in the proof of Theorem 3.1, let us now suppose that we have constructed x1 , . . . , xn and X1 ⊃ X2 ⊃ · · · but are unable to ﬁnd xn+1 and Xn+1 to continue the induction. This tells us that, for every y ∈ Xn such that y 1 and xn < y, and for any subspace Y ⊂ Xn , there is a subspace Z of Y such that the following statement holds: either no sequence in σΔn+1 (x1 , . . . , xn , y) is contained in Z, or σΓn+1 (x1 , . . . , xn , y) is strategically large for Z. Let us write P (y, Z) for this statement. Notice that P (y, Z) implies P (y, W ) for every subspace W of Z. Using this fact, we shall construct a sequence z1 < z2 < · · · of unit vectors in Xn and subspaces Xn = Z0 ⊃ Z1 ⊃ Z2 ⊃ · · · with the following properties, for every k. (a) zk ∈ Zk−1 . (b) For every y ∈ z1 , . . . , zk , either no sequence in σΔn (x1 , . . . , xn , y) is contained in Zk or σΓn (x1 , . . . , xn , y) is strategically large for Zk . Note that (b) is not quite the statement ∀y P (y, Zk ) because the approximations are different. The induction starts with the null sequence and Z0 . To see that it can always be continued, let us suppose that we have constructed z1 , . . . , zk and Z1 ⊃ · · · ⊃ Zk . Let zk+1 be any unit vector in Zk such that zk < zk+1 . We need yet another induction to deﬁne Zk+1 , but fortunately it is not too complicated. Let w1 , . . . , wN be a δn+1 -net of the unit ball of the subspace z1 , . . . , zk . Now choose subspaces Zk ⊃ W1 ⊃ W2 ⊃ · · · ⊃ WN such that, for each i, the statement P (wi , Wi ) holds. This we can do because for every y ∈ Xn such that y 1 and xn < y and for every subspace Y ⊂ Xn there is a subspace Z ⊂ Y such that P (y, Z) is true, as has already been mentioned. Let Zk+1 = Wn . By the hereditary property of P (y, Z), we ﬁnd that P (wi , Zk+1 ) for every i n. If we now let y ∈ z1 , . . . , zk be any vector of norm at most one, then statement (b) (with k replaced by k + 1) follows easily on approximating y by some wi with y − wi δn+1 . We are now ready to diagonalize. Let Z be the subspace generated by the block basis z1 , z2 , . . . and let Y be any subspace of Z. Because σΔn (x1 , . . . , xn ) is large but not strategically large for Y (by (i) and (ii)), we know that there must be some y ∈ Y such that Y and hence Z contains a sequence belonging to σΔn (x1 , . . . , xn , y). Let k be minimal such that y ∈ z1 , . . . , zk . It follows that Zk contains a sequence in σΔn (x1 , . . . , xn , y). By (b), we deduce that σΓn (x1 , . . . , xn , y) is strategically large for Zk , and hence for Z. To summarize, we have shown that, for every subspace Y ⊂ Z, there exists y ∈ Y such that σΓn (x1 , . . . , xn , y) is strategically large for Z. But this implies that σΓn (x1 , . . . , xn ) is strategically large for Z, which contradicts (iii). This shows that our original (outer) induction continues for ever. So now let us diagonalize again, letting V be the subspace generated by x1 , x2 , . . . . We claim that no sequence in V belongs to σ−Θ . To see this, let (v1 , . . . , vr ) ∈ Σ consist of vectors in V and choose n such that every vi belongs to the subspace x1 , . . . , xn and at least one xi is not a multiple of a vi . Because σΔn (x1 , . . . , xn ) is large for Xn , we can ﬁnd a sequence (y1 , . . . , yN ) ∈ σ such that yi − xi δi whenever i n. Let φ be the linear map from x1 , . . . , xn to y1 , . . . , yn that takes xi to yi for every i. Since each vi has norm at most 1, no coefﬁcient can exceed C, so by the triangle inequality and our choice of Δ, we ﬁnd that φ(vi ) − vi θ i for every i. But because (y1 , . . . , yN ) ∈ σ and σ satisﬁes the minimality

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condition stated at the beginning of the proof, we know that (φ(v1 ), . . . , φ(vk )) ∈ / σ , which / σ−Θ . implies that (v1 , . . . , vk ) ∈ We have therefore contradicted the assumption that σ−Θ was large, which proves the theorem. C OROLLARY 5.6. Let X = (c00 , · ) be a sequence space, let σ ⊂ Σ(X) be a collection of ﬁnite block sequences, let Δ be a sequence of positive real numbers and suppose that σ is large for X. Then there exists a subspace Y of X such that σΔ is strategically large for Y . P ROOF. Let Θ = Δ/2 and let ρ = σΘ . Then ρ−Θ is large for X. Theorem 5.5 implies that, for a sufﬁciently small sequence Γ , there is a subspace Y such that ρΓ is strategically large for Y . Since this remains true when Γ increases, we can of course take Γ to be Θ. However, ρΘ = σΔ . A further reformulation is possible in terms of colourings: if Σ(X) is coloured with two colours, σ and τ , then there is a subspace Y of X such that either every ﬁnite block sequence in the unit ball of Y belongs to τ , or σΔ is strategically large for Y . Put this way, the Ramsey-theoretic nature of the result is emphasized, but also the difference between this result and conventional results in Ramsey theory. The two most important differences are that it is weaker (in that being strategically large is weaker than being very large) and that there is an asymmetry between σ and τ . About the second point we shall have more to say later. Let us now return to our motivating theorem, and see how it follows easily from Corollary 5.6. P ROOF OF T HEOREM 5.1. Let X = (c00 , · ) be a sequence space containing no unconditional basic sequence, and for N ∈ N let σN be the set of all block sequences (x1 , . . . , xn ) of vectors in X of norm at most one such that n n n (−1) xi > N xi i=1

i=1

and such that at least one xi has norm 1. By Lemma 5.2, we know that, for every N , σN is large for X. It is an easy exercise to ﬁnd, for every N , a positive sequence ΔN such that n n (−1)n yi > (N/2) yi i=1

i=1

for every (y1 , . . . , yn ) ∈ (σN )ΔN . Let us now apply Corollary 5.6 repeatedly, to obtain a nested sequence X1 ⊃ X2 ⊃ · · · of block subspaces of X such that, for every N , (σN )ΔN is strategically large for XN . Let Y be a diagonal subspace, that is, a subspace generated by a block basis y1 , y2 , . . . with yn ∈ Xn for every n. It is easy to see that (σN )ΔN is strategically large for Y , whatever the value of N .

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Now let Z and W be arbitrary block subspaces of Y . If S plays the strategy Z, W, Z, W, . . . , then V can win, whatever the value of N . This shows that the condition of Lemma 5.3 is satisﬁed, and hence that Y is hereditarily indecomposable. Theorem 5.5 has an extension from sets of ﬁnite sequences to analytic sets of inﬁnite sequences. To be precise, let Σ(X) now stand for the set of all inﬁnite block sequences of X, with the blocks of norm at most 1, and give Σ(X) the (metrizable) topology of pointwise convergence. (This is a slight oversimpliﬁcation, as we need Σ(X) to be complete. One can deal with this technicality by considering sequences (xn , λn )∞ n=1 where the xn form a normalized block basis and the λn belong to the interval [0, 1].) It is obvious how to extend to sets of inﬁnite sequences of the deﬁnitions of large, strategically large and so on. T HEOREM 5.7. Let X = (c00 , · ) be a sequence space with basis constant C and let σ be an analytic subset of Σ(X). Let Δ = (δ1 , δ2 , . . .) be a sequence of positive reals. If σ is large for X, then X has a block subspace Y such that σΔ is strategically large for Y . A proof of this theorem can be found in [11]. The slightly weaker statement where Σ(X) consists of normalized block sequences is also proved in [2]. Theorem 5.7 ought to be a very useful tool for proving further Banach-space dichotomies. However, the results so far have been a little disappointing. In particular, no natural theorem is known that uses anything like the full strength of the theorem. However, there is one consequence (which can be deduced from the much simpler special case where σ is assumed to be closed) of some interest, as it provides a dichotomy for Banach spaces with unconditional bases, and thus converts Theorem 5.1 from a dichotomy to a trichotomy. To appreciate the result it is necessary to know of the existence of a space constructed in [12]. This space has an unconditional basis, but is otherwise as free of structure as possible. What does this mean? Well, it is an easy exercise to show that if X is a Banach space ∞ with an unconditional basis (xn )n=1 , then every diagonal map, that is, linear map of the form Λ : an xn → λn an xn , is continuous, provided that it satisﬁes the trivial necessary condition that the λn are bounded. (Such maps are also called multipliers.) The space XU in [12] has the property that the diagonal maps are essentially the only continuous maps deﬁned on it. To be precise, every map in L(XU ) is the sum of a diagonal map and a strictly singular map. (It would be interesting to know what can be said about maps from subspaces of XU to XU . See Problem 5.13 below for a concrete question.) In particular, the space XU is not isomorphic to any proper subspace of itself. On the other hand, most spaces with unconditional bases have far more structure than this, which suggests that some sort of dichotomy might be true. Let us deﬁne a Banach space X to be quasi-minimal if any two subspaces Y , Z of X have further subspaces U , V that are isomorphic. This is true in particular if X is minimal, which means that every subspace Y of X has a further subspace isomorphic to X. However, there are non-minimal examples of quasi-minimal spaces, of which Tsirelson’s space is a famous one. (See [7] for a proof of this.) Thus, in a certain sense, quasi-minimal spaces are rich in isomorphisms between subspaces. The following result concerning quasi-minimal spaces is a consequence of Theorem 5.7 for closed sets σ .

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T HEOREM 5.8. Every Banach space X with an unconditional basis has a subspace Y such that either Y is quasi-minimal or no subspace of Y is isomorphic to any proper subspace of itself. As it stands, the above theorem is not a proper dichotomy, since it is likely that there are quasi-minimal Banach spaces Y such that no subspaces are isomorphic to further proper subspaces. (There is even a suggestion for how to construct such a space, but nobody has checked that it really works.) However, it is a consequence of the following stronger result which is a genuine dichotomy. T HEOREM 5.9. Every Banach space X with an unconditional basis has a block subspace Y such that either Y is quasi-minimal or no two subspaces of Y generated by disjointly supported block bases are isomorphic to each other. The deduction of Theorem 5.8 from Theorem 5.9 uses standard facts about block bases and a criterion of Casazza. Details can be found in [11]. These results suggest further lines of investigation, but it seems that dichotomies as tidy as those of Theorem 5.1 and Theorem 5.9 are hard to come by. For example, if it is true that a quasi-minimal space X may have the property that no subspace of X is isomorphic to further proper subspace, it is tempting to think of such a space X as ‘bad’ and to try to prove a dichotomy along the following lines: every quasi-minimal space has a subspace which is either bad or, in some appropriate sense to be determined, very good. But what should a good space be like? If the previous results are anything to go by, one might expect such a space to be very rich in isomorphisms from subspaces to proper subspaces. Let P be the set of Banach spaces that are isomorphic to some proper subspace. If X has no bad subspace, then it follows immediately that every subspace Y of X has a subspace in P . That is, X is P -saturated. What stronger property could we ask for from a very good subspace? One possibility is that every subspace should belong to P . However, it seems quite likely that this deﬁnition does not lead to a dichotomy theorem. Indeed, here is a conjecture that would, if true, place considerable restrictions on any positive statements one might hope to prove. C ONJECTURE 5.10. There exists a quasi-minimal Banach space X with an unconditional basis such that every block subspace Y has a subspace Z generated by a block basis z1 , z2 , . . . such that the shift operator with respect to this basis is an isomorphism, and also has a subspace W which is isomorphic to no proper subspace of itself. Even a weaker statement, where it is assumed only that Z is isomorphic to some proper subspace, would contradict many natural conjectures about P -saturated spaces. Proving the above conjecture would require a breakthrough in the technology for producing counterexamples. The methods so far have tended to be all-or-nothing, in the sense that if one uses them to deﬁne a space with certain bad qualities, then those qualities will saturate the space, or at least the part of the space where the methods are used. This is a somewhat vague statement, but the following unsolved problem (which I heard from Haskell Rosenthal) may clarify it.

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P ROBLEM 5.11. Let X be a Banach space such that every subspace of X has a further subspace with an unconditional basis. (Such a space is called unconditionally saturated.) Does it follow that X is decomposable? Equivalently (given Theorem 5.1), does there exist an indecomposable Banach space with no hereditarily indecomposable subspace? To construct such a space, one would have to remove a great deal of structure from the space while somehow leaving subspaces relatively untouched. A positive solution to the following problem would improve the analogy between Theorem 5.1 and Theorem 5.9. P ROBLEM 5.12. Let X be a Banach space with an unconditional basis such that no two subspaces Y and Z generated by disjointly supported block bases are isomorphic. Does it follow that every continuous linear map from a subspace Y of X into X is a strictly singular perturbation of the restriction of a diagonal map on X? Even the following special case of Problem 5.12 is open. P ROBLEM 5.13. Let XU be the Banach space constructed in [12] with an unconditional basis. Is every continuous linear map from a subspace Y of XU into XU a strictly singular perturbation of the restriction of a diagonal map on XU ? A FINAL REMARK . Very recently Problem 5.11 was solved by Argyros and Manoussakis. As one might expect, the answer is no. It will be very interesting to see whether their construction leads to a new generation of Banach spaces with notable “non-hereditary” properties.

References [1] S.A. Argyros, G. Godefroy and H.P. Rosenthal, Descriptive set theory and Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1007–1069 (this Handbook). [2] J. Bagaria and J. Lopez Abad, Weakly Ramsey sets in Banach spaces, Adv. Math. 160 (2001), 133–174. [3] E. Behrends, New proofs of Rosenthal’s 1 -theorem and the Josefson–Nissenzweig theorem, Bull. Polish Acad. Sci. Math. 43 (1995), 283–295. [4] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Amer. Math. Soc. Colloq. Publ. 48, Amer. Math. Soc., Providence, RI (2000). [5] B. Bollobás, Combinatorics. Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability, Cambridge Univ. Press, Cambridge (1986). [6] A. Brunel and L. Sucheston, On B-convex Banach spaces, Math. Systems Theory 7 (1974), 294–299. [7] P. Casazza and T.J. Shura, Tsirelson’s Space, Lecture Notes in Math. 1363, Springer, New York (1989). [8] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. [9] J. Farahat, Espaces de Banach contenant 1 , d’apreès H.P. Rosenthal, Espaces Lp , Applications Radoniﬁantes et Géométrie des Espaces de Banach, Exp. No. 26, Centre de Math., École Polytech., Paris (1974). [10] W.T. Gowers, Lipschitz functions on classical spaces, European J. Combin. 13 (1992), 141–151. [11] W.T. Gowers, An inﬁnite Ramsey theorem and some Banach-space dichotomies, Ann. of Math, to appear.

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[12] W.T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997), 543– 568. [13] R.L. Graham, K. Leeb and B.L. Rothschild, Ramsey’s theorem for a class of categories, Adv. in Math. 8 (1972), 417–433. [14] N. Hindman, Finite sums from sequences within cells of a partition of N, J. Combin. Theory Ser. A 17 (1974), 1–11. [15] B. Maurey, Banach spaces with few operators, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1247–1297 (this Handbook). [16] B. Maurey, Type, cotype and K-convexity, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1299–1332 (this Handbook). [17] C.St.J.A. Nash-Williams, On well quasi-ordering transﬁnite sequences, Proc. Cambridge Philos. Soc. 61 (1965), 33–39. [18] E. Odell and Th. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281. [19] E. Odell and Th. Schlumprecht, Distortion and asymptotic structure, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1333–1360 (this Handbook). [20] H.P. Rosenthal, A characterization of Banach spaces containing 1 , Proc. Nat. Acad. Sci. 71 (1974), 241– 243. [21] H.P. Rosenthal, A characterization of Banach spaces containing c0 , J. Amer. Math. Soc. 7 (1994), 707–748.

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CHAPTER 25

Quasi-Banach Spaces Nigel Kalton∗ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 3. Linear subspaces and basic sequences . . . . . . . . . . . 4. The three-space problem and minimal extensions . . . . . 5. The Krein–Milman theorem . . . . . . . . . . . . . . . . 6. Operators and the structure of Lp -spaces when 0 < p < 1 7. Lattices and natural spaces . . . . . . . . . . . . . . . . . 8. Analytic functions and applications . . . . . . . . . . . . 9. Tensor products and algebras . . . . . . . . . . . . . . . . 10. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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∗ Supported in part by NSF DMS-9870027.

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1. Introduction The theory of the geometry of Banach spaces has evolved very rapidly over the past ﬁfty years. By contrast the study of quasi-Banach spaces has lagged far behind, even though the ﬁrst research papers in the subject appeared in the early 1940’s ([18,6]). There are very sound reasons to want to develop understanding of these spaces, but the absence of one of the fundamental tools of functional analysis, the Hahn–Banach theorem, has proved a very signiﬁcant stumbling block. However, there has been some progress in the non-convex theory and arguably it has contributed to our appreciation of Banach space theory. A systematic study of quasi-Banach spaces only really started in the late 1950’s and early 1960’s with the work of Klee, Peck, Rolewicz, Waelbroeck and Zelazko. The efforts of these researchers tended to go in rather separate directions. The subject was given great impetus by the paper of Duren, Romberg and Shields in 1969 which demonstrated both the possibilities for using quasi-Banach spaces in classical function theory and also highlighted some key problems related to the Hahn–Banach theorem. This opened up many new directions of research. The 1970’s and 1980’s saw a signiﬁcant increase in activity with a number of authors contributing to the development of a coherent theory. An important breakthrough was the work of Roberts in 1976 [73] and [75] who showed that the Krein–Milman Theorem fails in general quasi-Banach spaces by developing powerful new techniques. Quasi-Banach spaces (Hp -spaces when p < 1) were also used signiﬁcantly in Alexandrov’s solution of the inner function problem in 1982 [4]. During this period three books on the subject appeared by Turpin [86], Rolewicz [77] (actually an expanded version of a book ﬁrst published in 1972) and the author, Peck and Roberts [56]. In the 1990’s it seems to the author that while more and more analysts ﬁnd that quasi-Banach spaces have uses in their research, paradoxically the interest in developing a general theory has subsided somewhat. In this short article we will only give a glimpse of the theory, and we have tried to make the subject accessible for an audience which is primarily interested in and familiar with Banach space theory. There is no attempt to be encyclopaedic. Thus we will look carefully at problems related to the existence of closed subspaces which are very much in the spirit of the recent work of Gowers and Maurey [29] in Banach space theory. We will also consider the problem of characterizing the complemented subspaces of Lp (0, 1) when 0 < p < 1. In the last few sections we consider how the theory of lattices, analytic functions and tensor products alters in the non-convex setting. Although this article is devoted to quasi-Banach spaces, much of the early theory was developed in the context of more general topological vector spaces or sometimes F -spaces (complete metric linear spaces). In some cases (such as the study of the Krein–Milman theorem for compact convex sets, see Section 5) restricting to quasi-Banach spaces loses nothing in terms of generality, and in most cases there is relatively little loss.

2. Preliminaries In this section we will review a few elementary concepts and deﬁnitions. Further details can be found in one of the books [78,56].

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Let us recall ﬁrst that a quasi-norm · on vector space X over the ﬁeld K = R or C is a map X → [0, ∞) with the properties: • x = 0 if and only if x = 0. • αx = |α|x if α ∈ K, x ∈ X. • There is a constant C 1 so that if x1 , x2 ∈ X we have x1 + x2 C(x1 + x2 ). The constant C is often referred to as the modulus of concavity of the quasi-norm. A very basic and important result is the Aoki–Rolewicz theorem ([6,77]). This result can be interpreted as saying that if 0 < p 1 is given by C = 21/p−1 then there is a constant B so that for any x1 , . . . , xn ∈ X we have n 1/p n p xk B xk . k=1

(2.1)

k=1

It is then possible to replace · by an equivalent p-subadditive quasi-norm ||| · ||| so that

1/p |||x1 + x2 ||| |||x1 |||p + |||x2 |||p . X is said to p-normable if (2.1) holds. We will say that X is p-normed if the quasi-norm on X is p-subadditive. In general it is convenient to assume unless otherwise mention that a quasi-Banach space is p-normed for some p > 0. The quasi-norm · induces a metric topology on X: in fact a metric can be deﬁned by d(x, y) = |||x − y|||p , when the quasi-norm is p-subadditive. X is called a quasi-Banach space if X is complete for this metric. Note that if we assume X is p-normed for some p > 0 then the quasi-norm is a continuous function for the metric topology. It is important to emphasize that the standard basic results of Banach space theory such as the Uniform Boundedness Principle, Open Mapping Theorem and Closed Graph Theorem which depend only on completeness apply to this type of space; however applications of convexity such as the Hahn–Banach theorem are not applicable. If X and Y are quasi-Banach spaces then L(X, Y ) denotes the space of bounded linear operators T : X → Y under the quasi-norm T = sup{T x: x 1}. A special and important case is the dual space X∗ = L(X, K) which is always a Banach space. The best known examples of quasi-Banach spaces are the spaces p and Lp (0, 1), when 0 < p < 1. These spaces are p-normable. It is readily seen that ∗p = ∞ but Lp (0, 1)∗ = {0}. Notice that p has a separating dual while Lp has a trivial dual. This latter result is due to Day [18] in what is arguably the ﬁrst paper on quasi-Banach spaces. Another important example is the Hardy space Hp , i.e., the closed subspace of dθ Lp (T, 2π ) spanned by the functions {einθ : n 0}. Although Lp has trivial dual, Hp has a separating dual ([22]). If X has a separating dual then we can deﬁne an associated norm on X by the formula xc = sup x ∗ (x): x ∗ 1 . It can be easily shown that · c is the largest norm on X dominated by the original quasinorm. The completion of X with this norm Xc is called the Banach envelope of X. In a natural sense Xc and X have the same dual space.

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Many of the standard notions in Banach space theory can be carried through to quasiBanach spaces. For example, the notions of (Rademacher) type and cotype (see [32]) can be deﬁned in exactly the same way. T HEOREM 2.1. Let X be a quasi-Banach space of type p where 0 1 then X is normable (i.e., a Banach space). We remark that there are non-locally convex spaces of type one which are not Banach spaces [40]. The Krivine–Maurey–Pisier theorems on ﬁnite representability of np ’s have analogues in this setting (we refer to [36,7] and [9]). In particular Dvoretzky’s theorem always has an appropriate generalization (see [36] and [19]). Let us also mention an important substitute for convexity in complex quasi-Banach spaces. We will say that a quasi-norm (which we assume r-subadditive for some r < 1) is plurisubharmonic if for any x, y ∈ X then 1 x 2π

2π

x + eiθ y dθ.

0

For a discussion of this condition see [17]. It is important to note that the spaces Lp and their subspaces have plurisubharmonic norms. We will discuss this condition further in Section 8.

3. Linear subspaces and basic sequences In this section we will discuss some very fundamental structure problems for quasi-Banach spaces concerning linear subspaces of quasi-Banach spaces. Many of the results and problems in this section are interesting in the category of F -spaces but we will restrict ourselves to quasi-Banach spaces for clarity. A fundamental and still unresolved problem is the following: P ROBLEM 3.1 (The atomic space problem). Does every quasi-Banach space have a proper closed inﬁnite-dimensional subspace? A quasi-Banach space X is called atomic if it has no proper closed inﬁnite-dimensional subspaces. Very little is known about this problem. For a recent contribution in the context of F -spaces see [70]. Although Problem 3.1 remains elusive much progress has been made in understanding the structure of subspaces of quasi-Banach spaces. Before reviewing this progress we discuss the historical context for some of these ideas. Since the failure of the Hahn–Banach theorem is a characteristic of non-locally convex spaces, it is natural that much of the early research in the area was devoted to trying to understand this phenomenon. A Banach space has a very rich dual space and this also

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means that it has a very rich class of closed subspaces (each non-trivial continuous linear functional gives rise to a closed subspace of codimension one). Therefore an associated problem for quasi-Banach spaces is to ﬁnd (inﬁnite-dimensional) proper closed subspaces. It is clear from Day’s result that it is possible to ﬁnd a closed subspace E of Lp when 0 < p < 1 and a continuous linear functional e∗ ∈ E ∗ which cannot be extended; indeed E can be taken to be one-dimensional. This construction will work in any space X which fails to have a separating dual. In the space p for 0 < p < 1 more work is required but Peck [68] gave a similar example of the failure of the Hahn–Banach theorem. Later, Duren, Romberg and Shields [23] found a representation of the dual of Hp for 0 < p < 1 and used it to show that the Hahn–Banach theorem also fails in these spaces. Their work led them to formulate a conjecture. They deﬁned a quasi-Banach space X (or more generally an F-space) to have the Hahn–Banach Extension property (HBEP) if whenever e∗ is a continuous linear functional on a closed subspace E of X then e∗ has an extension x ∗ ∈ X∗ . They also deﬁned the notion of a proper closed weakly dense (PCWD) subspace as a proper closed subspace E so that the quotient X/E has trivial dual. They then asked whether a quasiBanach space with (HBEP) is necessarily locally convex and whether a any non-locally convex quasi-Banach space has a PCWD-subspace. It is easy to see that if X has HBEP then it must have a separating dual and every quotient must have HBEP; hence if X has a PCWD-subspace it cannot have HBEP. These two questions had a considerable impact on the theory because they focused attention on the problem of subspaces. In effect HBEP is equivalent to the statement that the weak and norm topologies have the same closed subspaces. After important contributions in [79] and [71] the ﬁrst of these problems was resolved in [33]: T HEOREM 3.2. A quasi-Banach space X has HBEP if and only if X is locally convex (i.e., a Banach space). The method of proof of Theorem 3.2 relies on the construction of basic sequences. Of course, there is no guarantee that quasi-Banach spaces will contain basic sequences (unlike Banach spaces). In fact an atomic space (if it exists) would be an immediate counterexample; but we will later show how to construct a quasi-Banach space without a basic sequence. However it is natural to start by imitating as far as the possible the classical Bessaga–Pełczy´nski basic sequence selection techniques. It soon becomes clear that the role of the weak (or weak∗ ) topology can be replaced by any weaker Hausdorff vector topology τ on X so that X has an equivalent τ -lower-semi-continuous quasi-norm. We will call such a topology polar. P ROPOSITION 3.3 (Basic sequence selection criterion). Let X be a quasi-Banach space and suppose (xn ) is a sequence so that lim xn = 0 for some polar vector topology τ but inf xn > 0. Then (xn ) has a subsequence which is basic. We recall that a sequence (xn ) in a quasi-Banach space X is called a Markushevich basis if [xn ] = X and there is a bi-orthogonal sequence (xn∗ ) so that (xn∗ ) separate the points of X.

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We will say that (xn ) is a Markushevich basic sequence if it is a Markushevich basis for its closed linear span. An immediate corollary of this proposition is: P ROPOSITION 3.4 (Markushevich basic sequence selection criterion). Let X be a quasiBanach space and suppose (xn ) is a sequence so that lim xn = 0 for some weaker Hausdorff vector topology τ but inf xn > 0. Then (xn ) has a subsequence (yn ) which is a Markushevich basic sequence and whose bi-orthogonal sequence (yn∗ ) in [yn ]∗ satisﬁes sup yn∗ < ∞. An alternative approach to this result was given by Drewnowski [21]. We are now a position to indicate a proof of Theorem 3.2: P ROOF OF T HEOREM 3.2. Since X has (HBEP) it is clear that X∗ separates points and therefore the Banach envelope norm , c induces a weaker Hausdorff vector topology on X. We argue that it cannot be a strictly weaker topology than the quasi-norm topology. Indeed, if it is strictly weaker, then using Proposition 3.4 one can ﬁnd a sequence (xn ) such that xn c < 4−n but xn = 1 for all n and (xn ) is a Markushevich basis for its closed linear span E with bi-orthogonal functionals (xn∗ ) satisfying sup xn∗ < ∞. Then we can ∞ −n ∗ ∗ ∗ deﬁne e ∈ E by e = n=1 2 xn∗ . Suppose e∗ can be extended to a bounded linear functional f ∗ ∈ X∗ . Then f ∗ (4n xn ) = 2n but 4n xn c 1 for all n. This contradiction shows that X coincides with its Banach envelope. Once this is established it is not too difﬁcult to prove a companion result for PCWD subspaces [37]: T HEOREM 3.5. Let X be a quasi-Banach space with a separating dual. If X has no PCWD subspace then X is locally convex. Notice however that the hypothesis of a separating dual is required here. We will see later that this hypothesis is necessary: there exist non-locally convex quasi-Banach spaces which do not have any quotient with trivial dual. Let us now return to the discussion of basic sequences. Theorems 3.3 and 3.4 yield some characterizations of spaces with basic sequences: T HEOREM 3.6. Let X be a separable inﬁnite-dimensional quasi-Banach space. Then the following conditions on X are equivalent: (i) X contains a basic sequence. (ii) Thereis descending sequence (Ln ) of inﬁnite-dimensional closed subspaces of X with ∞ n=1 Ln = {0}. (iii) There is a family L of inﬁnite-dimensional closed subspaces such that {L: L ∈ F } is inﬁnite-dimensional for any ﬁnite subset F of L but {L: L ∈ L} = {0}. (iv) There is a strictly weaker Hausdorff vector topology on X. These implications are relatively easy. The equivalence of (ii) and (iii) simply follows from the Lindelof property for separable metric spaces. That (i) implies (ii) is trivial. For

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(ii) implies (iv) simply consider the vector topology on X induced by the semi-quasinorms x → d(x, Ln ) for n = 1, 2, . . . . Thus the only implication with any difﬁculty here is that (iv) implies (i). Let τ be a Hausdorff vector topology on X, which is strictly weaker than the original quasi-norm topology qn. Let τ ∗ be a maximal Hausdorff vector topology on X strictly weaker than qn (such a topology must exist). Let τ ∗∗ be the quasi-norm topology on X deﬁned by taking the τ -closure of the original unit ball as a new unit ball. Then the maximality of τ means that either τ ∗∗ = τ ∗ or τ ∗∗ = qn. But the former case means that the identity i : (X, τ ∗ ) → (X, qn) is almost continuous and a form of the Closed Graph Theorem comes into play: one deduces that τ ∗ = qn a contradiction. It follows that τ ∗∗ = qn and so τ ∗ is a polar topology. One can use the Lindelof property to construct a weaker metrizable Hausdorff vector topology ρ which is still polar. Then an application of Theorem 3.3 completes the proof. The last condition leads to the deﬁnition of a minimal space as any quasi-Banach space which does not have any weaker Hausdorff vector topology. A separable quasi-Banach space is minimal if and only if it contains no basic sequence (separability is redundant here, but that requires a little more explanation). Obviously an atomic space must be minimal but we shall see that the converse is false. Let us now illustrate the problem by considering an arbitrary separable Banach space X. Let L be a maximal collection of inﬁnite-dimensional closed subspaces of X with the property that any ﬁnite intersection is inﬁnite-dimensional. Let E = {L: L ∈ L}. There are three possibilities: • E = {0}. Then by Theorem 3.6 X is non-minimal. • E is inﬁnite-dimensional. Then E is atomic. • dim E < ∞. In this case X/E contains a basic sequence, but X could still be minimal. The third possibility suggests a way of constructing a minimal space with no atomic subspace. It is even possible to hope for an example where dim E = 1 and X/E is a Banach space. Obviously one needs that X is not a Banach space: this brings into focus a distinct problem which also received a considerable amount of attention in the 1970’s: the three space problem for Banach spaces, which is discussed in the next section. It will turn out that there is a counterexample of this nature and it is closely related to the recent work of Gowers and Maurey [29,28]. T HEOREM 3.7 ([52]). There is a quasi-Banach space which does not contain a basic sequence. We will postpone discussion of this theorem to Section 4. In view of Theorem 3.7 it is possible to ask whether such examples can be created in classical spaces such as Lp when p < 1. In fact there are two positive results which show that every subspace of Lp has a basic sequence. The ﬁrst result is due to Bastero [8] who show that the theory of Krivine–Maurey stability can be extended to quasi-Banach spaces. This shows that: T HEOREM 3.8. If X is closed subspace of Lp when 0 < p < 1 then X contains a subspace isomorphic to r for some p r 2. The second result of Tam [84] gives a general and important criterion for the existence of basic sequences. Notice that this result also includes the case of subspaces of Lp .

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T HEOREM 3.9. Let X be a complex quasi-Banach space with a plurisubharmonic quasinorm. Then X contains a basic sequence. To conclude this section, we note that in [30] an example is created of a sequence (fn ) contained in Lp when 0 0 and every subsequence (fn )n∈M is fundamental in Lp .

4. The three-space problem and minimal extensions We now turn our attention to a central problems of the area in the 1970’s, the three-space problem for local convexity which asked if there is a non-locally convex quasi-Banach space X with a closed subspace E such that both E and X/E are locally convex. This problem belongs to a family of three-space problems for which we refer to [12]. T HEOREM 4.1. There is a non-locally convex quasi-Banach space X with a subspace E of dimension one so that X/E is isomorphic to 1 . Theorem 4.1 is due independently (and essentially simultaneously) to the author, Ribe and Roberts [38,72] and [75]. However the examples created in each case were very different. Suppose X is a quasi-Banach space. We will say that Y is a minimal extension of X if there is a subspace E of Y with dim E = 1 and Y/E ≈ X. We will say that Y is the trivial extension (or that Y splits) if L is complemented, i.e., Y ≈ L ⊕ X in the natural way. We say ([55]) that X is a K-space if every minimal extension of X is trivial. We now describe a general construction of a minimal extension (ﬁrst used in [38] and [72]). Let us suppose X is a quasi-Banach space (over the ﬁeld K). Let X0 be any ﬁxed dense linear subspace of X (of course X0 = X is a possible choice). We say that a map F : X0 → K is quasilinear if: (1) F (αx) = αF (x) for x ∈ X0 and α ∈ K. (2) There is a constant K so that

F (x1 + x2 ) − F (x1 ) − F (x2 ) K x1 + x2 ,

x1 , x2 ∈ X.

We then deﬁne a quasi-norm on K ⊕ X0 by (α, x) = α − F (x) + x, F

x ∈ X0 , α ∈ K.

The completion of K ⊕ X0 for this quasi-norm is a minimal extension of X which we denote K ⊕F X. Conversely every minimal extension of X is isomorphic (as an extension) to K ⊕F X for a suitable quasilinear map F (see [37]). It is clear that if F and G are any two quasilinear maps on X0 then F and G deﬁne equivalent quasi-norms on K ⊕ X0 if and only if there is a constant C so that |F (x) − G(x)| Cx for every x ∈ X0 . In this case

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we will say that F and G are equivalent. The minimal extension K ⊕F X splits if and only if there is a linear map G : X0 → K equivalent to F , i.e., G satisﬁes estimate of the form F (x) − G(x) Cx,

x ∈ X0 .

(4.2)

In this way K-spaces are characterized in terms of an approximation property. We refer to [11] for related results. We also note the connection with the concept of Hyers–Ulam functional stability (see [31]); the question is essentially is whether a functional which satisﬁes a perturbation of the functional equation for linear maps is itself a perturbation of a linear map. Let us note that if X is a Banach space then (4.2) is equivalent to an estimate of the form n n

F (xi ) C xi , x1 , . . . , xn ∈ X0 . (4.3) F (x1 + · · · + xn ) − i=1

i=1

P ROOF OF T HEOREM 4.1. To prove Theorem 4.1 we follow the construction of Ribe [72] of a space now known as the Ribe space. According to the preceding discussion, it is enough to exhibit a function F : c00 → K deﬁned on the dense subspace c00 of all ﬁnitely supported sequences in 1 , which is quasilinear and fails to satisfy (4.3). Ribe’s example is the functional ∞ ∞ ∞ F (x) = xk log |xk | − xk log xk k=1

k=1

k=1

(where 0 log 0 := 0). A slight modiﬁcation yielding an equivalent quasi-norm is the functional Λ(x) =

∞ k=1

xk log

|xk | . x

To see that (4.3) does not hold it is enough to compute F (e1 + · · · + en ) − −n log n.

(4.4) n

k=1 F (ek ) =

The Ribe space immediately produces the necessary example for Theorem 4.1. Also as observed by Roberts [75] it gives an example of a non-locally convex space with no quotient with trivial dual; this gives a counterexample to complement Theorem 3.5. At this point let us mention an important open problem: P ROBLEM 4.2. Classify those Banach spaces which are K-spaces (i.e., so that every minimal extension is trivial). Is it is true that a Banach space X is a K-space if and only if X∗ has non-trivial cotype? There is some body of evidence to support the conjecture in Problem 4.2. The known results are:

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T HEOREM 4.3 ([38]). Suppose X is a Banach space with non-trivial type. Then every minimal extension of X is trivial. T HEOREM 4.4 ([58]). Suppose X is a Banach space which is the quotient of an L∞ -space. Then every minimal extension of X is trivial. It is perhaps worth noting that the latter theorem can be restated in terms of a stability theorem for set functions. T HEOREM 4.5 ([58]). There is a universal constant so that whenever A is an algebra of subsets of some set Ω and F : A → R is a set function satisfying: F (A ∪ B) − F (A) − F (B) 1 if A ∩ B = ∅ then there is an additive set function μ with F (A) − μ(A) K for every A ∈ A. There are some non-locally convex K-spaces: T HEOREM 4.6 ([38]). If 0 < p < 1 then every minimal extension of p or Lp splits. Let us return to the case of 1 . As we have seen it is possible to characterize minimal extensions of 1 via quasilinear maps on c00 . It turns out that it is possible up to equivalence to characterize quasilinear maps in a very convenient form. To understand this let us ﬁrst + of note that it is only necessary to specify a quasilinear map F on the positive cone c00 c00 since any map obeying the conditions for quasilinearity on the positive cone can be extended by the formula

F (x) = F x + − F (x − ), where x + = max(x, 0) and x − = max(−x, 0). This extension is then unique up to equivalence. Let X be a Banach sequence space, i.e., a space of sequences equipped with a norm · X such that • The basis vectors en ∈ X. • If ξ ∈ X and |ηk | |ξk | for every k then η ∈ X and ηX ξ X . • For every n ∈ N the linear functional η → ηn is continuous. • If ξ is a sequence such that n ∈ N Sn ξ = (ξ1 , . . . , ξn , 0, . . .) ∈ X and sup Sn ξ X < ∞ then ξ ∈ X and ξ X = supn∈N Sn ξ X .

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The last condition here is usually called the Fatou property. We can now deﬁne an associ+ ated quasilinear map on c00 by the formula ΦX (x) = sup

∞

ξ X 1 k=1

xk log |ξk |.

(4.5)

This functional was introduced under the name indicator function of X in [51] and later under the name entropy function of X in [66] where it plays an important role in the solution of the distortion problem. The fact that it is quasilinear is ﬁrst observed in [51]. By way of illustration consider the case X = 1 , when as easy calculation gives that ΦX = Λ where Λ is deﬁned by Eq. (4.4). Note that Φp = p1 Λ and Φ∞ = 0. The entropy functions yield an important source of minimal extensions of 1 . They do not completely classify minimal extensions because each is convex on the positive cone. A complete classiﬁcation is however obtained in [51]: T HEOREM 4.7. Let F : c00 → K be a quasilinear map (for the 1 -norm). Then there exists a positive α and a Banach sequence space X so that F is equivalent to α(ΦX − ΦX∗ ) where X∗ is the Köthe-dual of X. P ROOF OF T HEOREM 3.7. We return to Theorem 3.7. As suggested in the discussion it is reasonable to hope for an example of a minimal extension of 1 with no basic sequence. If Y is this minimal extension and L is the kernel of the quotient map onto 1 this requires that every inﬁnite-dimensional closed subspace of Y contains L. Clearly this is very much related to the construction of Gowers and Maurey [29] of a Banach space where any two inﬁnite-dimensional subspaces almost intersect. Now if we write Y in the form K ⊕F 1 where F is deﬁned on c00 , then we can translate our requirement to a condition on F . This is that F cannot be equivalent to a linear functional on any inﬁnite-dimensional subspace of c00 . This in turn is equivalent to the requirement that for every inﬁnite-dimensional subspace E of c00 we have: sup F (x): x ∈ E, x 1 = ∞. In this language, this is a type of distortion problem similar to distortion problem for Banach spaces solved in [66]. Clearly Theorem 4.7 suggests we should try use a functional of the type F = ΦX where X is a suitably exotic Banach sequence space. The correct choice is a space used by Gowers [28] which is a modiﬁcation of the original Gowers–Maurey construction in [29]. The proof that such an example works requires technical calculations similar in spirit to work in [29]; we refer to [52]. One ﬁnal remark on this example is in order: one can ﬁnd a dense subspace which has HBEP. Thus in Theorem 3.2 it is necessary to assume that X is complete; it should be noted that in [71] a number of results are proved for metrizable topological vector spaces with HBEP, without the assumption of completeness.

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5. The Krein–Milman theorem A classic problem in non-locally convex spaces asks whether every compact convex subset of a quasi-Banach space has an extreme point. This can be traced back as least as far as [61], and probably much further. Although, at ﬁrst sight, this problem is unrelated to the three-space problem of the preceding section, in retrospect their negative solutions require very much the same constructions. T HEOREM 5.1. There is a compact convex subset of Lp when 0 < p < 1 which has no extreme points. Theorem 5.1 is due to Roberts [73] and [74]. Roberts’s original construction is contained in [74] and this only gives a compact convex subset of some quasi-Banach space without extreme points, while [73] contains a much simpliﬁed approach and the theorem as stated. Since [73] is not readily available, a good reference for this argument is [56]. Roberts then used the key ideas in his proof of Theorem 5.1 to prove Theorem 4.1. In this article we will take the opposite direction, which in hindsight seems the right way to look at things. We now turn to the proof of Theorem 5.1. The approach used by Roberts in both [73] and [75] is through the notion of a needlepoint. If X is a quasi-Banach space we say that x ∈ X is a needlepoint if given ε > 0 there exists a ﬁnite set F so that x ∈ co F , v < ε if v ∈ F and for any y ∈ co F there exists 0 α 1 with y − αx < ε. X is called a needlepoint space if every point of X is a needlepoint. The main ingredients of the proof are: P ROPOSITION 5.2. If X is a needlepoint space then X contains a compact convex set K with no extreme points. P ROPOSITION 5.3. The space Lp for 0 < p < 1 is a needlepoint space. In [75] it is simply shown that there is a needlepoint space: the fact that Lp has this property is proved in [73]. For the proof of Proposition 5.2 we refer to [56]. The construction is in fact is a quite logical inductive argument using at each stage that every point of the space is a needlepoint. We will describe an approach to Proposition 5.3 which uses the notion of minimal extensions. In fact, it is very easy to see that if Y is a non-trivial minimal extension of a Banach space X so that X ≈ Y/L where dim L = 1 then every e ∈ L is a needlepoint. Thus we have an easy direct construction of a non-zero needlepoint by using the Ribe space. Of course, the Ribe space is not a needlepoint space; in fact any needlepoint space must have trivial dual. However we next note: T HEOREM 5.4 ([42]). The Ribe space is isomorphic to a subspace of Lp when 0 < p < 1.

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It is in fact instructive to see that the Ribe space is a rather natural subspace of Lp . Let (ξn ) be a sequence of independent random variables each with the Cauchy distribution, i.e., 1 λ(ξn ∈ B) = π

B

dx 1 + x2

so that

eisξn (t ) dt = e−|s| .

Then consider the space E generated by the constant function 1 and the sequence {|ξn |}∞ n=1 . It may be shown that if α0 , α1 , . . . ∈ R is a ﬁnitely supported sequence then ∞ ∞

∞ αk |ξk | ∼ α0 − Λ (αk )k=1 + |αk |, α0 + k=1

p

k=1

where Λ is deﬁned in (4.4). For details we refer to [42]. This implies that E is isomorphic to the Ribe space and the constant function 1 is a needlepoint of Lp . To complete the argument we need only note that Lp is a transitive space, i.e., given any f, g ∈ Lp with f = 0 there is a bounded linear operator T : Lp → Lp with Tf = g. As the image of a needlepoint is necessarily a needlepoint this means that Lp is a needlepoint space. This then completes the proof of Proposition 5.3 and hence of Theorem 5.1. There has been some investigation of geometric properties of a compact convex set K which guarantee the existence of extreme points. One way to formulate this idea is to consider a Banach space X and a compact linear operator T : X → Y where Y is a quasiBanach space. Let K = T (BX ). Then K is a symmetric compact convex set in Y . We can then consider geometric conditions on X so that K is afﬁnely homeomorphic to a compact convex subset of a locally convex space; this is equivalent to the existence of a separating family of continuous afﬁne functions on K. Two results of this nature are known: T HEOREM 5.5. Suppose X is a Banach space and T : X → Y is bounded linear operator. Then the collection of continuous afﬁne functions on K = T (BX ) separates the points of K if either of the following conditions hold: (1) ([26]) X contains no subspace isomorphic to 1 . (2) ([58]) T is compact and X is an L∞ -space. Finally an example is created in [50] of a compact convex set which is not afﬁnely homeomorphic to a subset of L0 [0, 1]. It should be pointed out that Proposition 3.2 of [50] has an error: the set K is not convex (see [5]); however the original set K0 or its symmetrization can be used in its place.

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6. Operators and the structure of Lp -spaces when 0 < p < 1 In this section we will treat some results on operators and their representations and discuss the isomorphic structure of the spaces Lp [0, 1]. If X is a quasi-Banach space with trivial dual then the algebra L(X) may, in fact, be rather small, as one does not have the rich class of ﬁnite-rank operators. Let us say that a space X is rigid if L(X) = CI . In [57] the following result is proved (see also [90] for a quasi-normed but incomplete example of a rigid space): T HEOREM 6.1. If 0 < p < 1 then Lp has a subspace X so that every quotient of X is rigid. In a recent preprint, Roberts shows there are many rigid spaces by showing: T HEOREM 6.2 ([76]). Every separable p-normable quasi-Banach space with trivial dual is the quotient of a separable rigid p-normable quasi-Banach space. One of the classical results on non-locally convex spaces is a theorem of Williamson [92] which says essentially that the theorem of the Fredholm alternative remains valid. If X has trivial dual so that X∗ reduces to {0} then this implies that any compact operator K : X → X has only zero in its spectrum. Later Pallaschke [67] observed that if X is also transitive then K must itself be the zero operator, since if K = 0 one can ﬁnd an endomorphism T : X → X so that T K has one in its spectrum. In particular this result applies to Lp when p < 1. This result was generalized by Turpin [86] to a wider class of spaces and in [56] Pallaschke’s result is extended to strictly singular operators. These results suggested a question (due to Pełczy´nski): is it possible to ﬁnd a compact endomorphism of a space with trivial dual? In effect, this is equivalent to a more general question: does there exist a quasi-Banach space X with a trivial dual and a non-zero compact operator K : X → Y where Y is any quasi-Banach space? If such an example can be constructed we can suppose K(X) dense in Y so that Y ∗ = {0} and consider the map (x, y) → (0, Kx) on X ⊕ Y . Let us say then that a space X admits compact operators if there is a non-zero compact operator T : X → Y where Y is some quasi-Banach space. Pełczy´nski’s question was resolved in [59]: T HEOREM 6.3. There is a quasi-Banach space with trivial dual which admits compact operators. The proof used some classical function theory and the earlier results of Duren, Romberg and Shields [23]. We would like however to indicate a slightly different proof based on [60] and the results of Section 3. The key observation is that if X is a quasi-Banach space whose unit ball BX is compact for some (Hausdorff) vector topology τ , then one can mimic the proof of the Banach–Dieudonné theorem to show that the topology τˆ which is deﬁned to be the ﬁnest topology on X agreeing with τ on bounded sets is a vector topology. Let us take X = 2 (np ), where 0 < p < 1. Then X has certain features of a reﬂexive Banach space; in particular, its unit ball is weakly compact. If we use the weak topology w for τ then τˆ

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is the “bounded weak” topology bw. We show next that every closed subspace E of X is bw-closed. In fact if not we can ﬁnd a sequence xn ∈ BX converging in the bw-topology to a point x ∈ / E. Using Theorem 3.3 or even a simple gliding hump argument we can suppose (xn − x) is a basic sequence equivalent to a block basis of the original basis. But then passing to a further subsequence we can suppose it is equivalent to the canonical basis of 2 . But there is a linear functional ϕ on its closed linear span with ϕ(xn −x) = 1 for all n, which produces a contradiction. Now an application of Theorem 3.5 gives a subspace E so that X/E has trivial dual; in this space the bw-topology factors to a quotient topology for which the unit ball is compact. We remark that this topology by its construction is locally p-convex and so easily provides examples of compact operators into p-normable quasi-Banach spaces. For full details see pp. 140–146 of [56]. Let us note that Sisson [81] showed that there is an example of a rigid space admitting compact operators; this can now also be deduced from Theorem 6.2. The examples indicated above leave open the question of whether speciﬁc spaces admit compact operators. Of course the obvious example is the space Lp when 0 < p < 1. This space does not admit compact operators; in fact more is true: T HEOREM 6.4 ([35]). Suppose 0 < p < 1. Let T : Lp → Y be a non-zero operator into some quasi-Banach space Y (or even a topological vector space). Then there is an inﬁnitedimensional subspace H of Lp which is isomorphic to 2 and so that T |H is an isomorphism. This result also holds in a wide class of non-locally convex Orlicz spaces. Let us sketch a very simple proof that there cannot be a compact operator K : Lp → Y where we assume Y has an r-subadditive quasi-norm. Let f ∈ Lp and suppose rn are the Rademacher functions. If K(f rn ) has any cluster point it must be zero since for any subsequence 1 n f (rk1 + · · · + rkn ) converges to zero. Hence limn→∞ K(f rn ) = 0. Let An = {rn = 1} and Bn = {rn = −1}. Then Kf = 2K(f χAn ) − K(f rn ) = K(f rn ) − 2K(f χBn ). Thus

1/r Kf 21−1/r lim inf Kf χAn r + Kf χBn r n→∞

1/p 21−1/p lim inf Kf χAn p + Kf χBn p n→∞

2

1−1/p

Kf p .

This yields K 21−1/p K, i.e., K = 0. It is very possible that Theorem 6.4 is not the best result here. Under special hypotheses one can do much better: T HEOREM 6.5. Suppose 0 < r < p < 1. Then:

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(1) ([39]) Let T : Lp → Lp be a non-zero operator. Then there is a subspace E of Lp so that E ≈ Lp and T |E is an isomorphism. (2) ([41]) Let T : Lp → Lr be a non-zero operator. Then if p < q 2 there is a subspace E of Lp with E ≈ Lq and T |E is an isomorphism. These results follow from representation theorems which we discuss shortly. Let us remark at this point that part (2) extends to operators taking values in natural which we will discuss in Section 7. P ROBLEM 6.6. Suppose 0 < p < 1 and let T : Lp → Y be any non-zero operator. Suppose p < q 2. Does it follow that there is a subspace E ≈ Lq (or even q ) so that T |E is an isomorphism? We will now turn to some basic questions on the structure of the spaces Lp [0, 1] when 0 < p < 1. Some of these questions can be regarded as analogues of similar questions for the Banach spaces Lp when p 1. However, the theory has a different ﬂavor. A simple example is the fact that the quotient of Lp by a one-dimensional subspace is never isomorphic to Lp when p < 1 [55]. In fact this is an immediate consequence of the observation in Theorem 4.6 that every minimal extension of Lp splits. A key result is the concrete representation of operators on Lp [0, 1] given in [39]. This result which was inspired by an earlier similar result for the case p = 0 due to Kwapie´n [62] has a somewhat similar form also in the case p = 1 [39]. Results of similar type have been studied for operators T : Lp → L0 in [41] and also for operators on general rearrangementinvariant spaces [44] and [47]. T HEOREM 6.7. Suppose T : Lp [0, 1] → Lp [0, 1] is a bounded operator where 0 < p < 1. Then there is a sequence of Borel functions an : [0, 1] → K and Borel maps σn : [0, 1] → [0, 1] so that: (1) |an (s)| |an+1 (s)| for n = 1, 2, . . . and s ∈ [0, 1]. (2) σn (s) = σm (s) when m = n and s ∈ [0, 1]. ∞ p (3) n=1 |an (s)| < ∞ for almost every s ∈ [0, 1]. ∞ an (s)p ds T p λ(B) for every Borel set B ⊂ [0, 1]. (4) −1 n=1 σn B

(5) If f ∈ Lp then Tf (s) =

∞

an (s)f (σn s)

s-a.e.

n=1

Conversely if (an ) and (σn ) are given satisfying (1)–(4) then (5) deﬁnes a bounded operator on Lp . This theorem rather easily yields Theorem 6.5(1). It also gives a simple proof that Lp is a primary space, i.e., if Lp = X ⊕ Y then either X or Y must be isomorphic to Lp . (In fact essentially the same proof can be given in the case p = 1; see [39] and [24].)

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However an intriguing question, closely related to a similar question in the case p = 1, is: P ROBLEM 6.8. Is Lp [0, 1] a prime space when 0 < p < 1, i.e., is any inﬁnite-dimensional complemented subspace isomorphic to Lp ? It is well-known that the spaces p are prime when 0 < p < 1. This is due to Stiles [82], by a proof similar to that of Pełczy´nski for the case p 1. In [39] it is shown that the space L0 [0, 1] is prime by using Kwapie´n’s representation of operators for this case. The case 0 < p < 1 is however more difﬁcult and remains open. The natural way to attack this problem is to take an arbitrary projection P on Lp and use Theorem 6.7 to represent it. It turns out that one must show that there is a Borel set B of positive measure and n ∈ N so that |an | > 0 on B and σn is one-to-one on B. In the case p = 0 this ﬁnal step can be completed rather easily but in the case 0 < p < 1 it is not quite so clear. In [42] a detailed study of this and related problems was undertaken and a curious but unsatisfactory result was obtained: T HEOREM 6.9. Suppose 0 < p < 1. Then Lp has at most two non-trivial complemented subspaces up to isomorphism. If the second mysterious complemented subspace were to exist it would have some remarkable properties. For example, there would be an averaging projection of the space of vector-valued functions Lp (Z) = Lp ([0, 1]; Z) onto its subspace of constant functions. P ROBLEM 6.10. Suppose 0 < p < 1 and X is any quasi-Banach space. Is it possible that there is an averaging projection on Lp (X)? P ROBLEM 6.11. Suppose 1 p < ∞ and suppose X is a quasi-Banach space so that there is an averaging projection on Lp (X). Is X locally convex? We remark that in [39] it is shown that there is no averaging projection on Lp (Lp ). Some partial results on Problem 6.11 are given in [45].

7. Lattices and natural spaces We next present some basic facts in the theory of quasi-Banach lattices. It turns out that some interesting complications arise in the theory, associated with problems of convexity. Let us note ﬁrst that the discussion of pp. 40–41 of [64] of homogeneous functions applies verbatim to quasi-Banach lattices. In this way we can deﬁne the notions of p-convexity and q-concavity as in [64] or [32] for quasi-Banach lattices. It is clear, for example, that the fundamental examples Lp when 0 0. These issues are discussed in [46] and [16].

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The issues involved relate to the Maharam submeasure problem, for which we refer to the article [25]. Suppose Ω is a set and Σ is a σ -algebra of subsets of Ω. A set function φ : Σ → [0, ∞) is called a submeasure if we have • φ(∅) = 0, • φ(A) A ⊂ B, ∞ φ(B) if ∞ • φ( n=1 An ) ∞ n=1 φ(An ) for every (An )n=1 ⊂ Σ. φ is called a continuous or Maharam submeasure if An ↓ ∅ implies φ(An ) ↓ 0. The unsolved Maharam submeasure problem asks whether every Maharam submeasure is equivalent to a measure, in the sense that if φ is a Maharam submeasure, then there is a measure μ so that μ(A) = 0 if and only if φ(A) = 0. See [58] for a partial result. A submeasure φ is called pathological if whenever μ is a measure with μ φ then μ = 0. This is equivalent to the fact that given any ε > 0 there exist B1 , . . . , Bn ∈ Σ so that 1 χBk (1 − ε)χΩ n

(7.6)

max φ(Bk ) ε.

(7.7)

n

k=1

and

1kn

Pathological submeasures have been constructed by several authors, but the simplest example seems to be that due to Talagrand [83]. The Maharam problem quoted above is equivalent to asking whether a pathological submeasure can be continuous. Let us suppose then that (Ω, Σ, φ) is a submeasure space. Suppose 0 < r < ∞. Let X be the space of all Σ-measurable functions f so that:

∞

f X =

φ |f | > t r dt

1/p < ∞.

0

It is not hard to show that X is a quasi-Banach lattice. However if φ is pathological then X cannot be p-convex for any p > 0; this follows routinely from (7.6) and (7.7). It is clear that if X has a p-subadditive quasi-norm then we have (p, 1)-convexity, i.e., n n 1/p p |xk | xk , k=1

x1 , . . . , xn ∈ X.

k=1

It is shown in [16] that we then have P ROPOSITION 7.1. Suppose 0 < p < 1 and that X is a p-normable quasi-Banach space (with a p-subadditive quasi-norm). Then if 0 < q < r 1 with 1/q − 1/r = 1/p − 1 we

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have: n 1/r n 1/q r q |xk | xk , k=1

x1 , . . . , xn ∈ X.

k=1

In [46] the notion of lattice-convexity or L-convexity is introduced. A quasi-Banach lattice X lattice-convexity orL-convex if there exists C > 0 so that if u, x1 , . . . , xn ∈ X with maxk |xk | |u| but n1 nk=1 |xk | |u| then u C maxk xk . T HEOREM 7.2 ([46]). Let X be a p-normable quasi-Banach lattice. The following conditions are equivalent: (1) X is lattice-convex. (2) X is r-convex for some r > 0. (3) X is r-convex for every 0 < r < p. There are special situations which guarantees lattice-convexity: T HEOREM 7.3 ( [46,54]). Let X be a quasi-Banach lattice with non-trivial cotype. Then X is lattice-convex. We remark that in [46] this theorem is deduced from a result on the Maharam submeasure problem proved in [58]. In [54] a simpler direct proof is given. T HEOREM 7.4 ([46]). Let X be a quasi-Banach lattice which is isomorphic to a subspace of a lattice-convex quasi-Banach lattice. Then X is lattice-convex. Based on this theorem, we introduce the class of natural spaces. A quasi-Banach space X is called natural if it is linearly isomorphic to a subspace of a lattice-convex quasiBanach lattice. In practice this implies that X is isomorphic to an ∞ -product of spaces of the type Lp (μ) where 0 < p < ∞ is ﬁxed. It may be shown that a p-normable space is natural if and only if it is ﬁnitely representable in the space weak Lp or Lp,∞ . The motivation for this deﬁnition is that most spaces that arise in analysis are natural. Natural spaces are relatively good spaces to work in. We mention, for example, that natural spaces must always contain a basic sequence. In fact it is not difﬁcult to see that if X is separable and natural then there is a one-one operator T : X → Lp (0, 1). If T is not an isomorphism then X is not minimal and we can apply Theorem 3.6 while if T is an isomorphism then Theorem 3.8 applies. Of course this means that the example in Theorem 3.7 is not natural. There are some simpler examples of non-natural spaces. For example, the spaces Lp (T)/Hp for 0 < p < 1 (essentially proved in [41]) and the Schatten ideals Cp for 0 < p < 1 fail to be natural [49]. The importance of lattice-convexity is that this assumption allows us to use many of the powerful techniques available in the study of Banach lattices. Let us mention the example

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of square-function arguments. It is well-known that in a Banach lattice with non-trivial cotype one has an estimate of the form (see [32] and [64]): n 1/2 n εk xk ≈ |xk |2 E . k=1

k=1

It follows from Theorem 7.3 above and similar arguments that this estimate works in any quasi-Banach lattice with cotype. We also mention the Krivine generalization of Grothendieck’s theorem (see [64], p. 93), which remains valid (with a different constant) even for operators between lattice-convex quasi-Banach lattices. One of the most interesting lines of application of these ideas is in the study of uniqueness of unconditional bases in certain natural quasi-Banach spaces (see [88]). By combining well-established techniques from Banach space theory with the additional information that if X has an unconditional basis (un ) then it simultaneously an unconditional basis in the Banach envelope, it is possible to prove some quite powerful uniqueness results for a wide class of spaces with unconditional bases. This line of research was initiated in [34] where it is shown that the spaces p and many Orlicz sequence spaces for 0 < p < 1 have unique unconditional basis. A general uniqueness criterion was developed in [53] which was improved in a recent paper of [93]: T HEOREM 7.5. Let X be a natural quasi-Banach space with a normalized unconditional basis (un ). Suppose that: (1) X is isomorphic to X ⊕ X. ∞ q (2) There exists q < 1 so that if ∞ n=1 an un converges then n=1 |an | < ∞. Then for any normalized unconditional basis (vn ) of X there is a permutation π of N so that (un ) and (vπ(n) ) are equivalent. A weaker predecessor of this result was used in [53] to show that Hp (Tm ) and Hp (T n ) are isomorphic for 0 < p < 1 if and only if m = n. For further results on uniqueness see [63,1] and [2].

8. Analytic functions and applications Let X be a complex quasi-Banach spaces. Let Ω be an open subset of the complex plane C. Then a function F : Ω → X is called analytic if for every z0 ∈ Ω there exists δ > 0 and xn ∈ X for n 0 so that if |z − z0 | < δ then F (z) =

∞

xn z n .

n=0

This deﬁnition of an X-valued analytic function was ﬁrst employed by Turpin [86]. It is rather easy to see that other possible deﬁnitions based on complex differentiability do not work satisfactorily in quasi-Banach spaces. For example, if D is the open unit

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disk with standard area measure then the map F : C → Lp (D) where p < 1 deﬁned by F (z)(w) = (z − w)−1 is actually complex differentiable but does not have a local power series expansion, and indeed is not inﬁnitely differentiable. The basic theory of analytic functions was developed by Turpin [86], who noticed that if K is a compact subset of Ω then the convex hull of F (K) remains bounded in X. This enables one to show that there is a factorization of f through a Banach space. More precisely, if Ω0 is an open relatively compact subset of Ω then there is a Banach space Y , a one-one bounded injection j : Y → X and an analytic function G : Ω0 → Y so that F = j ◦ G. Since the theory of Banach space-valued analytic functions is very well understood one can use this device to prove many of the basic desired properties of analytic functions. For example, if f is analytic on a disk {z: |z − z0 | < r} then f has a (necessarily unique) power-series expansion valid throughout the disk. However, the theory of analytic functions is by no means as clean as for Banach spaces. The ﬁrst obstacle is the Maximum Modulus Principle. A simple example due to Alexandrov [3] shows what can happen. Consider the function F : D → Lp (T) deﬁned by F (z)(eiθ ) = e−iθ (1 − e−iθ z)−1 . This map is plainly analytic and extends continuously toa function on the closed unit disk D. Its power series expansion is given −i(n+1)θ zn . Consider the subspace H (T) and let Q : L → L /H be by F (z) = ∞ p p p p n=0 e the quotient map. Then Q ◦ F is analytic into Lp /Hp and Q(F (0)) = 1. However on the boundary if |z| = 1 then QF (z) = 0, i.e., F (z) ∈ Hp . To see this rewrite F (z) as −¯z(1 − z¯ eiθ )−1 . Of course if X has an equivalent plurisubharmonic quasi-norm then for any analytic function F : Ω → X the map z → F (z) is subharmonic and so we have a Maximum Modulus Principle. In fact this property essentially characterizes spaces for which a form of Maximum Modulus Principle holds. Let us say that X is A-convex if there is a constant C so that for every X-valued polynomial F (z) = nk=0 xk zk we have F (0) C max F (z). |z|=1

Of course this will imply that the same conclusion holds for any continuous function F on the closed unit disk D which is analytic in the interior, so that the space Lp /Hp is an example of a non-A-convex space. It is shown in [49] that X is A-convex if and only if X has an equivalent plurisubharmonic quasi-norm. Since the spaces Lp and p are A-convex it follows trivially that natural spaces are A-convex. Of course by Theorem 3.9 A-convex spaces always contain basic sequences and so this yields another proof that natural spaces also must contain basic sequences. However, it should be noted that the Schatten classes Cp for 0 < p < 1 are A-convex but fail to be natural (see [49]). The treatment of analytic functions valued in a non-A-convex space requires different techniques, but it turns out that the theory is still quite rich. The key ingredient is an atomic decomposition theorem due to Coifman and Rochberg [15]. In this paper, the authors proved some very general atomic decompositions for certain Bergman spaces. As a by-product they extended the results of [23] and [80] to calculate the p-envelope of Hr when 0 < r < 1 (this is the p-normed analogue of the Banach envelope). Let us denote this

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space Br,p . It turns out that Br,p consists of the space of all analytic functions on D such that 1/p

f (w)p 1 − |w|2 p/r−2 dA(w) < ∞, f r,p = D

where dA is standard area measure dx dy. The key to their proof of this is the following atomic decomposition: T HEOREM 8.1. There is a constant r) so that if ψ ∈ Br,p then there exist zk ∈ D C = pC(p, 1/p Cψ and αk ∈ C for k ∈ N so that ( ∞ r,p and k=1 |αk | ) ψ(w) =

∞

ν+1−σ αk 1 − |zk |2 (1 − wzk )−(ν+2) ,

k=1

where σ = 1/r − 1 and ν = [σ ]. Let us illustrate how this can be used to establish some basic estimates (cf. [48]). Suppose X is a p-normable space and that F : D → X is a polynomial. Suppose F (z) = x0 + x1 z + · · · + xn zn . Fix 0 < r < p and deﬁne a linear operator T : Br,p → X by T (f ) =

n

xk

k=0

(ν + 1)! f (k) (0). (ν + k + 1)!

Then

T (1 − wzk )−ν+2 = F (zk ). It follows by applying Theorem 8.1 rather crudely we can get an estimate that T C max F (z). z∈D

However

k!(ν + 1)! xk T wk = (ν + k + 1)! so that xk C

ν +k+1 T wk Br,p . k

If we choose r so that σ ∈ N so that σ = ν then this gives a Cauchy-type estimate which is valid for any function F analytic in the open unit disk, (k) F (0) Ck!k 1/p−1 maxF (z). (8.8) z∈D

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Once one has these Cauchy estimates certain other basic principles of complex analysis follow. For example, it is clear that Liouville’s theorem holds (i.e., a bounded analytic function is constant). This was ﬁrst noted in [48] although an earlier weaker version for functions analytic on the Riemann sphere was observed by Turpin [86]. Let us also note there is an annular Maximum Modulus Principle: T HEOREM 8.2 ([49]). For any 0 < r < 1 and any 0 < p 1 there is a constant C = C(r, p) so that if X is a p-normed quasi-Banach space and F : D → X is an analytic function then F (0) C sup F (z). r|z|<1

P ROOF OF T HEOREM 8.2. We will indicate a proof quite different from that of [49]. Suppose the theorem is false for some 0 < r < 1 and 0 < p < 1. Then we may ﬁnd a sequence of analytic functions Fk : D → Xk where each Xk is p-normed and such that sup Fk (z) = Fk (wk ) = 1

|z|<1

for some wk with |wk | < r but sup Fk (z) < 2−k . r|z|<1

Deﬁne

wk − z Gk (z) = F . 1 − w¯ k z A routine power-series calculation shows these functions are analytic on D and now we have: sup Gk (z) = Gk (0) = 1

|z|<1

and Gk (z) < 2−k ,

if |z| >

2r . 1 + r2

Now deﬁne G : D → ∞ (Xk ) by G(z) = (G(zk ))∞ k=1 . The fact that G is also analytic follows from the Cauchy estimates 8.8 which imply that the obvious power series expansion converges to G. Now let c0 (Xk ) be the subspace of the ∞ -product of sequences (xk )∞ k+1 with lim xk = 0 and let Q be the quotient map onto ∞ (Xk )/c0 (Xk ). Then Q ◦ G is also analytic on D and vanishes on the annulus 2r(1 + r 2 )−1 < |z| < 1. However the standard fact that the zeros of an analytic function are isolated unless the function vanishes

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1123

identically remains valid in a quasi-Banach space; this is an easy consequence of the factorization principle of Turpin used above. Hence Q ◦ G vanishes identically on D which gives a contradiction. In fact much more detailed information about analytic functions is obtained in [48] and [49]. It is shown that there is an intimate relationship with the theory of integration in quasi-normed spaces which was developed by Turpin and Waelbroeck (see, e.g., [86] or [89]). We will not go into these topics here, but we would like to isolate one speciﬁc result on the degree of failure of the Maximum Modulus Principle. Let us recall the example of Alexandrov in the space Lp /Hp namely G(z) = Q ◦ F (z) where Q : Lp (T) → Lp /Hp is the quotient map and F (z)(eiθ ) = e−iθ (1 − ze−iθ )−1 . As remarked above this function G extends continuously to the closed disk and vanishes identically on the boundary. One can ask for the rate of decay of G near the boundary. It can shown that

G(z) C 1 − |z| 1/p−1 ,

z ∈ D,

where C is some constant.

The point of interest here is G is the “worst” possible such function. More precisely (see [48]): T HEOREM 8.3. Suppose 0 < p < 1 and that X is a p-normable quasi-Banach space. Suppose F : D → X is an analytic function. Then: (1) If lim|z|→1 (1 − |z|)1−1/p F (z) = 0 then F vanishes identically. (2) If F does not vanish identically and F (z) C(1 − |z|)1/p−1 for some constant C and all z ∈ D then there is a non-zero linear operator T : Lp /Hp → X.

9. Tensor products and algebras In this section we apply some of the ideas in Sections 7 and 8. Let us ﬁrst note that if X, Y, Z are Banach spaces then the collection of all continuous bilinear maps B : X × Y → Z is quite rich, because of the presence of continuous linear functionals. Clearly some restrictions must be imposed when we pass to quasi-normed spaces. In fact there are no non-zero linear operators from Lp to a space which is q-normable if q > p so to construct a non-zero bilinear map B : Lp × Lp → X requires that X must already be p-normable at best. In this case there is an easy example namely B : Lp (0, 1) × Lp (0, 1) → Lp (0, 1)2 given by B(f, g) = f ⊗ g where f ⊗ g(x, y) = f (x)g(y). Let us make the problem more precise. Suppose X is p-normed and Y is q-normed where 0 < p, q 1. For 0 < r < 1 and u = nk=1 xk ⊗ yk ∈ X ⊗ Y deﬁne the r-tensor (semi-)quasi-norm by n ur = sup B(xk , yk ) , k=1

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where B runs over all bilinear maps B : X × Y → Z where Z is r-normed and B 1. From the above remarks, it is clear that · r may reduce to 0 for certain choices of X, Y and r, e.g., if X = Y = Lp and p < r < 1. The question is to determine the optimal condition on r which ensures · r is a tensor quasi-norm, i.e., ur = 0 implies u = 0 and x ⊗ y = xy. This problem was ﬁrst considered by Turpin [87] who proved the following theorem: T HEOREM 9.1. Suppose 0 < p 1 and 0 < q 1. Then · r is a tensor quasi-norm on X ⊗ Y provided 1/r 1/p + 1/q − 1. In [48] this result was shown to be best possible: T HEOREM 9.2. Suppose 0 < p 1 and 0 < q 1. Suppose 0 < r < 1 and that there is a non-zero bilinear map B : Lp /Hp × Lq /Hq → Z where Z is r-normable. Then 1/r 1/p + 1/q − 1. P ROOF OF T HEOREM 9.1. Let us indicate the proof following [16]; we consider the case of real spaces, but the argument extends to complex spaces. We need only consider the case 1/r = 1/p + 1/q − 1. Suppose X is p-normed. Let X# denote the algebraic dual of X denote the lattice of all (i.e., all linear functionals with no continuity restrictions). Let X real functions ϕ on X# so that ϕX = inf

n

1/p xk

p

n # # # # x (xk ) ∀x ∈ X < ∞. : ϕ x

k=1

k=1

is a p-normed quasi-Banach lattice (strictly speaking one must ﬁrst One then checks that X quotient by the ideal of functions ϕ with ϕX = 0). Furthermore we can embed X iso via the map x → xˆ where x(x metrically into X ˆ # ) = x # x. We make a similar construction of functions on Y . of a q-normed quasi-Banach lattice Y Next we create an r-normed quasi-Banach lattice Z of functions on X# × Y # by deﬁning hZ = inf

n

1/r xk yk r

k=1

r

n # # # # # # x (x)y (y) ∀x , y . : h x , y k=1

There is an obvious bilinear map B : X × Y → Z given by B(x, y) = xˆ ⊗ yˆ where xˆ ⊗ y(x ˆ # , y # ) = x(x ˆ # )y(y ˆ # ). Clearly B = 1. The main point to establish is that B(x, y) = xy. This will show that x ⊗ yr = xy and this can be used to show that · r is a quasi-norm on X ⊗ Y (we will omit this part of the argument). Suppose then that n xˆk ⊗ yˆk . xˆ ⊗ yˆ k=1

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we have This implies that if y # ∈ Y # then in X n # # y (y)xˆ y (yk )xˆ k k=1

so that # y (y)x

n # y (yk )p xk p

1/p

k=1

which translates as a statement in Y # , i.e., xyˆ

n

1/p xk |yk | p

p

.

k=1

Now Y is q-normed and so we can use Proposition 7.1 to deduce that xy

n

1/r xk r yk r

k=1

which proves that B(x, y) = xy.

P ROOF OF T HEOREM 9.2. We recall the construction of a function Gp : D → Lp /Hp which is analytic and satisﬁes Gp (z) C(1 − |z|)1/p−1. In fact it is easy to show that the function G = Gp constructed in Section 8 also has the property that the closed linear span of its range is the entire space Lp /Hp . Assume B : Lp /Hp × Lq /Hq → Z is a bounded bilinear form when Z is s-normed for s > 1/p + 1/q − 1. Then if |ζ | = 1 then analytic function H (z) = B(Gp (z), Gq (ζ z)) on D satisﬁes lim|z|→1 H (z)(1 − |z|)1−1/s = 0 and so by Theorem 8.3 vanishes identically. Hence B(Gp (w), Gq (z)) = 0 if |w| = |z| and w, z ∈ D. If |w| < 1 then z → B(Gp (w), Gq (z)) in analytic in D and vanishes on the circle |z| = |w| so again vanishes identically. Combining we obtain that B vanishes identically. Now let us turn our attention to the problem of (complex) quasi-Banach algebras. The fact that much of the basic theory of Banach algebras can be carried over to quasi-Banach algebras was discovered in the 1960’s by Zelazko [94] and [95]. A key fact here (see [56], p. 124) is that if A is a commutative quasi-Banach algebra then the spectral radius 1/n ρ(x) = lim x n n→∞

(9.9)

is actually a seminorm. From this it is relatively easy to build the standard theory of the spectrum and in particular to justify the name spectral radius. In particular note that in any commutative quasi-Banach algebra A with identity we must have that A∗ is non-trivial (and indeed there are continuous multiplicative linear functionals).

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Let us now turn to non-commutative algebras. We always assume that A has an identity. It is clear, in view of the preceding discussion of tensor products, that since multiplication deﬁnes a bilinear map A × A → A there must be some restrictions on A. For example, we cannot have A isomorphic to Lp /Hp when p < 1. We can explain some of the restrictions by noticing that even in the non-commutative case the spectral radius is a “nice” function (although obviously not in general a seminorm). T HEOREM 9.3 ([49]). Let A be a quasi-Banach algebra with identity. Then the spectral radius x → ρ(x) is a plurisubharmonic function. P ROOF. This is quite easy. Suppose F : C → X is a polynomial. We use the Annular Maximum Modulus Principle of Theorem 8.2. If r < 1 there is a constant C = C(r, A) so that F (0)n C sup F (z)n r|z|1

and so F (0)n 1/n C 1/n sup F (z)n 1/n . r|z|1

From this it follows that

ρ F (0) max F (z). r|z|1

Now by standard techniques using outer functions this implies the stronger inequality

log ρ F (0)

2π 0

dθ . log F eiθ 2π

Applying this to F (z)n and taking limits yields that log ρ is plurisubharmonic and hence also ρ is plurisubharmonic. The existence of a non-trivial plurisubharmonic function on A is a signiﬁcant restriction (cf. [49]). Once one has this information one can go further in extending the theory of Banach algebras to this more general setting. For example, Dilworth and Ransford showed that the set of invertible elements is pseudo-convex and extended the Johnson theorem on the uniqueness of the complete norm topology to quasi-Banach algebras [20]. We conclude by raising a question which seems rather interesting, and relates to the preceding discussion: P ROBLEM 9.4. Let A be any quasi-Banach algebra with identity. Can A have trivial dual? Of course we may assume that A = L(X) for some quasi-Banach space X. In fact in all known examples L(X) has non-trivial dual (even if X = Lp /Hp when p < 1). One reason

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we would like to understand this question is that if Problem 9.4 has a negative solution then Problem 6.10 must also have a negative solution and this would imply that Lp (0, 1) is a prime space. However our evidence for a negative solution is rather slim at present.

10. Final remarks To conclude let us mention a few topics that perhaps fall into domain of this chapter but we have been forced to omit for lack of space. Local theory. One area of increased interest recently has been the local theory of quasiBanach spaces. It is rather surprising that many of the major achievements of the local theory of Banach spaces can be extended in a reasonable form to quasi-Banach spaces (so that convexity is not really relevant!). See, for example, [27,10] and [65]. Topological classiﬁcation. The problem of topological classiﬁcation for separable quasiBanach spaces is open. Cauty [13] has shown that there are examples of separable F -spaces which are not homeomorphic to a Hilbert space. It is apparently unknown if this can be done for a quasi-Banach space. Very recently Cauty [14] has shown that, surprisingly, the Schauder Fixed Point theorem holds for any compact convex set in an F -space; this settles a problem going back over seventy years. The uniform classiﬁcation of quasi-Banach spaces is largely unexplored, but see [91] for a recent result.

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CHAPTER 26

Interpolation of Banach Spaces Nigel Kalton∗ and Stephen Montgomery-Smith† Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail: [email protected]; [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . 2. The basics . . . . . . . . . . . . . . . . . . . 3. The K-functional and the (θ, p)-methods . . 4. The complex method . . . . . . . . . . . . . 5. Properties preserved by interpolation . . . . 6. Calderón couples . . . . . . . . . . . . . . . 7. Interpolation spaces for (Lp , Lq ) . . . . . . 8. Extensions . . . . . . . . . . . . . . . . . . . 9. Self-extensions of Hilbert spaces . . . . . . . 10. Analytic families of Banach spaces . . . . . 11. Entropy functions and extensions . . . . . . 12. Commutator estimates and their applications Acknowledgement . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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1. Introduction There are several excellent books now available treating the general theory of interpolation from various points of view (see, for example, Bergh and Löfström [8], Bennett and Sharpley [5], Krein, Petunin and Semenov [69] and Brudnyi and Kruglyak [12]). The aim of this chapter is to consider the interaction between interpolation theory and the geometry of Banach spaces, and so we will not treat many topics that can be found elsewhere. Historically, interpolation theory as an abstract concept was developed by Lions, Peetre and Calderón in the 1960s. The ﬁrst real application in Banach space theory seems to be the celebrated Davis–Figiel–Johnson–Pełczy´nski factorization theorem [35] from 1974, although at the time the language of interpolation was not used; we can now see in retrospect that this result belongs to interpolation theory. This result will be discussed below (Theorem 3.4). In the 1980’s Pisier played a pioneering role in bringing interpolation techniques into the mainstream of Banach space theory. Interpolation played a role (at least implicitly) in the development of Pisier’s work on the Grothendieck program [93] and in the local theory of Banach spaces [94]. More recently Pisier, Kislyakov and Xu have studied interpolation of Hardy spaces and non-commutative analogues: we refer to [65,67,68,97,95] and [96]. Some of these ideas are covered in [66]. See also [58]. In this article we will treat some rather different topics. We will concentrate on the real (θ, p)-method and the complex method. We ﬁrst introduce these and discuss the Davis– Figiel–Johnson–Pełczy´nski factorization theorem. This leads us naturally to consider the general problem of interpolation of Banach space properties and properties of operators by these methods. In this area Theorem 5.2 is very useful; it gives a general construction to give a counterexample to many possible conjectures. We also draw attention to the Cwikel problem: is compactness of an operator preserved by complex interpolation? This problem, we believe is quite challenging for Banach space theorists. Next we discuss Calderón couples. The characterization of pairs of r.i. space which form Calderón couples curiously involves conditions (shift properties) which have natural meaning in the context of Banach space theory. We then devote much of the remainder of the article to developing differential methods in interpolation theory. This theory was initiated by Rochberg and Weiss in 1983 [101] and is very relevant to the construction of twisted sums or extensions of Banach spaces (see also [60]). We develop this theory specializing to the case of interpolation of Banach sequence spaces and relate it to the theory of entropy functions [42,55] and [87]. We discuss applications in harmonic analysis and in operator theory. 2. The basics In this section, we will introduce the basic ideas of interpolation. First we deﬁne the notion of a Banach couple. Let W be a Hausdorff topological vector space (the ambient space). Suppose X0 and X1 are two Banach spaces which are continuously embedded into W . We refer to X = (X0 , X1 ) as a Banach couple. We can then deﬁne the sum and intersection spaces. The sum space Σ(X) = X0 + X1 is equipped with the norm: xΣX = inf x0 X0 + x1 X1 : x = x0 + x1 , x0 ∈ X0 , x1 ∈ X1 .

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The intersection space Δ(X) = X0 ∩ X1 is a Banach space under the norm: xΔ(X) = max xX0 , xX1 . Once we have deﬁned these spaces, note that it is always possible to replace the ambient space W by the sum space Σ(X). If X and Y are two Banach couples, then a linear operator T : X → Y (we write T ∈ L(X, Y )) is a bounded linear map T : Σ(X) → Σ(Y ) such that T (X0 ) ⊂ Y0 and T (X1 ) ⊂ Y1 . From our assumptions, it then follows that T is bounded from Xj to Yj for j = 1, 2. We deﬁne T X→Y = max T X0 →Y0 , T X1 →Y1 . If X = Y we simply write T X . For any Banach couple X, an intermediate space Z is a Banach space such that Δ(X) ⊂ Z ⊂ Σ(X). Z is then called an interpolation space if for every T ∈ L(X) we have T (Z) ⊂ Z. We then have from the Closed Graph Theorem that T Z CT X for some C. If C = 1 we say that Z is an exact interpolation space. Notice that Δ(X) and Σ(X) immediately give two exact interpolation spaces. A motivating example for this set-up, and indeed the original Banach couple, is obtained by taking an arbitrary σ -ﬁnite measure space (Ω, μ), and letting X0 = L∞ (μ), and X1 = L1 (μ). The ambient space can be either the space L0 of all measurable functions (with the topology of convergence in measure on subsets of ﬁnite measure) or L∞ + L1 . It is sometimes convenient to impose mild extra conditions on a Banach couple. The simplest such requirement is that the intersection space Δ(X) is dense in both X0 and X1 in their respective topologies. This allows to form a dual Banach couple. If we take Δ(X)∗ as an ambient space then X0∗ , X1∗ can be considered as continuously embedded ∗ into Δ(X)∗ . Thus (X)∗ = (X0∗ , X1∗ ) is also a Banach couple with Σ(X ) = Δ(X)∗ and ∗ Δ(X ) = (ΣX)∗ in a natural way. Note that it does not then necessarily follow that the ∗ dual couple X has a dense intersection space, although this does follow if, for example, X0 and X1 are reﬂexive Banach spaces. A second natural assumption is that of Gagliardo completeness. This requires that the sets BXj = {x: xXj 1} are closed in the topology of Σ(X) for j = 1, 2. The weaker assumption that the closure of each BXj is contained in Xj allows us from the Closed Graph Theorem to obtain Gagliardo completeness for certain equivalent norms on each Xj . An easier example of a non-Gagliardo complete pair is the pair (C[0, 1], L1 [0, 1]) where it is easily seen that the closure of BX0 in the topology of the sum space is the ball of L∞ [0, 1]. Interpolation theory has its origins in the classical Riesz–Thorin and Marcinkiewicz theorems. Both these theorems lead to the idea of an interpolation method. This is a functor F which assigns to any Banach couple X an interpolation space F(X) in such a way that if X, Y are two interpolation couples and T ∈ L(X, Y ) then T maps F(X) to F(Y ) boundedly. The Riesz–Thorin theorem abstracts to the complex methods and the Marcinkiewicz theorem is abstracted in the real methods.

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One may isolate two basic types of questions implicit in interpolation theory. One can study a particular method in order to develop applications. In this case one needs to take speciﬁc practical examples of Banach couples and then calculate the effect of the corresponding method. The second type of question is to identify all interpolation spaces for a given couple. In Banach space theory, interpolation has played a pivotal role in several areas (particularly in the local theory and in problems related to Grothendieck’s theorem). It is probably fair to say that in Banach space theory one is mainly interested in knowing about the preservation of properties of spaces or operators under interpolation.

3. The K-functional and the (θ, p)-methods The fundamental notion of real interpolation theory is the K-functional. Suppose X is a Banach couple. We deﬁne the K-functional by

K(x, t) = K x, t; X = inf x0 X0 + tx1 X1 : x = x0 + x1 ,

x ∈ Σ(X).

Thus K(x, 1) is simply the usual norm of Σ(X) and each x → K(x, t) gives an equivalent norm on the sum space for which it becomes an exact interpolation space. It is easy to check that for ﬁxed x the function t → K(x, t) is increasing and concave for 0 < t < ∞. If x ∈ Δ(X) one has an estimate K(x, t) min xX0 , txX1 xΔ min(1, t). There is a dual construct known as the J -functional:

J (x, t) = J x, t, X = max xX0 , txX1 ,

x ∈Δ X .

These form a family of exact interpolation norms on Δ(X). If 0 < θ < 1 and 1 p < ∞ we deﬁne the real interpolation spaces Xθ,p = (X0 , X1 )θ,p by x ∈ Xθ,p if and only if

∞

xθ,p = 0

t −θp K(x, t)p

dt t

1/p < ∞.

(3.1)

If p = ∞ we deﬁne Xθ,∞ as the space of x such that xθ,∞ = sup t −θ K(x, t) < ∞. t >0

It is easily seen that both these deﬁnitions can be given in discrete form, e.g., xθ,p ≈

n∈Z

p 2−θpn K 2n , t

1/p .

(3.2)

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The functor which takes the couple X to Xθ,p is the (θ, p)-method; this clearly provides an example of an exact interpolation method. The theory of this method is well-developed and understood and we can refer to [5] and [8] for a full discussion of such topics as reiteration and duality. For our purposes it is useful to point out an equivalent deﬁnition in terms of the J -functional, ﬁrst obtained in the fundamental paper of Lions and Peetre [76]. Consider the case 1 p < ∞. Deﬁne for x ∈ X0 + X1 , x θ,p

p 1/p k θk = inf max xk 0 , 2 xk 1 : x= 2 xk , k∈Z

(3.3)

k∈Z

where the series converges in X0 + X1 . Then x ∈ Xθ,p if and only if x θ,p < ∞ and the norms x θ,p and xθ,p are equivalent. In (3.3) we have formulated the J -method discretely; it is more usual to use a continuous version. The equivalence of the J -method and the K-method of deﬁnition can be obtained from the Fundamental Lemma, which we discuss later (Theorem 6.1). Note that we must have that Δ(X) is dense in the spaces Xθ,p provided 1 p < ∞. Using this, one can show a duality theorem [74]: T HEOREM 3.1. Suppose Δ(X) is dense in both X0 and X1 . Then if 1 p < ∞ and 0 < θ < 1 the dual of (X0 , X1 )θ,p can be identiﬁed naturally with (X0∗ , X1∗ )θ,q where 1/p + 1/q = 1. The (θ, p)-methods have proved extremely useful in many branches of analysis including Banach spaces. We conclude this section by discussing the ﬁrst major application of interpolation in Banach spaces theory, the Davis–Figiel–Johnson–Pełczy´nski factorization theorem [35]. The basic idea of this theorem is to establish conditions under which certain interpolation spaces are reﬂexive, although in the initial paper the language of interpolation was not used. Later Beauzamy [3] established the general result. Consider the spaces Zn = (Δ(X), J2n ) where Jt (x) = J (x, t) is a norm on Δ(X). Now we can use (3.3) to deﬁne a quotient mapping Q : p (Zn )n∈Z → Xθ,p by

θk 2 ak . Q (ak )k∈Z = k∈Z

The following lemma is an easy gliding hump argument: L EMMA 3.2. Suppose 1 < p < ∞ and 0 < θ < 1. Suppose an = (ank )k∈Z is a bounded sequence in p (Zn ) such that for each k we have limn→∞ ank = 0 weakly in X0 + X1 . Then Qan converges to zero weakly in Xθ,p . P ROOF. It is enough to construct a sequence of convex combinations of (Q(ak ))kn which is weakly null. First by Mazur’s theorem we can take convex combinations and assume limn→∞ ank X0 +X1 = 0 for each k. It follows quickly that limn→∞ ank θ,p = 0 for each k. Indeed we can split ank = bnk + cnk where bnk is bounded in X0 and converges

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to zero in X1 while cnk is bounded in X1 and converges to zero in X0 . Then we use the estimate

1−θ θ θ ank θ,p C bnk 1−θ X0 bnk X1 + cnk X0 cnk X1 . It follows that we can ﬁnd a sequence Nn → ∞ so that θk Q(an ) − 2 ank |k|Nn

→ 0. θ,p

Let bnk = ank if |k| Nn and 0 otherwise. Standard gliding hump arguments show that bn is weakly null. It is then easy to conclude that an is also weakly null. An immediate consequence due to Beauzamy [3] is that: T HEOREM 3.3. Suppose 1 < p < ∞ and 0 < θ < 1. Then (X0 , X1 )θ,p is reﬂexive if and only if BΔ(X) is relatively weakly compact in Σ(X). This follows from the preceding lemma and the Eberlein–Smulian theorem. Now the Factorization Theorem of Davis, Figiel, Johnson and Pełczy´nski is given by: T HEOREM 3.4 ([35]). Suppose X and Y are Banach spaces and T : X → Y is weakly compact. Then there is a reﬂexive space R and a factorization of T = BA where A : X → R and B : R → Y are bounded. P ROOF. We use the following typical trick. Let K be the closure of T (BX ) and let Y0 be the Banach space generated by taking K as it’s unit ball. Let Y1 = Y and then take R = Yθ,p for some choice of 0 < θ < 1 and 1 < p < ∞. For A we treat T as an operator into R and for B we take the inclusion of R into Y . There is a sense in which the (θ, p)-methods give rise to Banach spaces with relatively simple structure. This is the content of a theorem of Levy [71]. T HEOREM 3.5. Suppose Δ(X) is not closed in Σ(X). Then for 0 < θ < 1, 1 p < ∞ the spaces (X0 , X1 )θ,p contain a complemented copy of p . In order to prove this theorem, we ﬁrst prove a preliminary lemma: L EMMA 3.6. Suppose Δ(X) is not closed in Σ(X). Then either (1) For every t > s > 0 and ε > 0 there exists x ∈ Δ(X) with τ −1 K(τ, x) (1 − ε)s −1 K(s, x), or

s < τ < t,

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(2) For every t > s > 0 and ε > 0 there exists x ∈ Δ(X) with K(τ, x) (1 − ε)K(t, x),

s < τ < t.

P ROOF. Note ﬁrst that t −1 K(t, x) is decreasing and K(t, x) is increasing. Assume (1) fails. Then there exists ε > 0 and t > s > 0 such that t −1 K(t, x) < (1 − ε)s −1 K(s, x). Then for any x ∈ Δ(X) we have K(t, x) (1 − ε)txX1 . Thus, putting δ = 1 − ε/2 we can ﬁnd u1 ∈ X0 , v1 ∈ X1 so that x = u1 + v1 and u1 X0 txX1 and v1 X1 δxX1 . Now we can iterate the argument as in the Open Mapping Theorem and write v1 = u2 + v2 where u2 X0 δtxX1 and v2 X1 δ 2 txX1 . Continuing in this way we construct ∞ (un )∞ n=1 in X0 and (vn )n=1 in X1 such that un X0 tδ n−1 xX1 ,

vn X1 tδ n xX1

and x = u1 + · · · + un + vn . Clearly ∞ n=1 un converges in X0 and its sum must be x (by computing in X0 + X1 ). Hence xX0 t (1 −δ)−1 xX1 . Similarly the failure of (1) implies that xX1 CxX0 whenever x ∈ Δ(X) for a suitable constant C. Thus if both (1) and (2) fail then the two norms · X0 and · X1 are equivalent on the intersection, and this implies the intersection is closed in X0 + X1 . We now turn to the proof of Theorem 3.5: P ROOF. Note that (3.2) the space Xθ,p can be regarded as a subspace of the p -sum of the space X0 + X1 with the norms 2−θn K(2n , x) for n ∈ Z. To show it has a complemented subspace isomorphic to p requires only the existence of a normalized sequence (xm )∞ m=1 in Xθ,p so that for each n we have limm→∞ K(2n , xm ) = 0 (i.e., xm converges to 0 in X0 + X1 ). This follows by standard gliding hump techniques. If such a sequence does not exist then there is a constant C so that

p 2−nθp K 2n , x

1/p CK(1, x).

n∈Z

That this is impossible follows from the preceding lemma.

4. The complex method We ﬁrst deﬁne the complex method for a Banach couple X which we now assume consists of complex Banach spaces. We introduce a Banach space F of analytic functions as follows. Let S = {z: 0 < !z < 1} and let F be the space of analytic functions F : S → Σ(X)

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such that F extends continuously to the closure S and the functions t → F (j + it) are continuous and bounded in Xj for j = 0, 1. We norm F by F F = max

sup

j =0,1 −∞
F (j + it) . X j

We then deﬁne the interpolation spaces Xθ = [X0 , X1 ]θ by x ∈ Xθ if and only if there exists F ∈ F with F (θ ) = x and then we set xXθ = inf F F : F (θ ) = x . This method is known as Calderón’s ﬁrst method or lower method, and is usually called simply the complex method. It was introduced independently by Lions [73] and Calderón [14]; most of the basic theory was developed by Calderón in [14]. Calderón’s second method or upper method ([14]) is described similarly but taking a of analytic functions. Suppose F : S → Σ(X) is a bounded analytic different family F function. Then any anti-derivative F # is Lipschitz and extends continuously to S. We say if t → F # (it) is Lipschitz into X0 and t → F # (1 + it) is Lipschitz into X1 . We then F ∈F put F F = max

j =1,2

F # (it) − F # (is)X0 , |t − s| s
F # (1 + it) − F # (1 + is)X1 sup . |t − s| s
Then we deﬁne the spaces X[θ] = [X0 , X1 ]θ by x ∈ X[θ] if and only if there exists F ∈ F with F (θ ) = x and we deﬁne xX[θ ] = inf F F: F (θ ) = x . It is clear that the complex interpolation spaces Xθ and X[θ] are further examples of exact interpolation spaces. It is also clear that in general Xθ ⊂ X[θ] and the injection is of norm one. The fundamental difference between the upper and lower methods is that Δ(X) is always dense in Xθ , but not necessarily in X[θ] . In fact it is shown in [7] that Xθ is simply the closure of Δ(X) in X[θ] . Let us discuss conditions under which the two methods coincide. We will need the Poisson kernel for the strip. These are maps P : ∂S × S → (0, ∞) such that if u is harmonic and bounded on S and extends continuously to S then u(z) = =

P (w, z)u(w)|dw|

∂S ∞

−∞

P (it, z)u(it) dt +

∞

−∞

P (1 + it, z)u(1 + it) dt.

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L EMMA 4.1. Suppose F ∈ F . Then F (θ ) exp P (w, θ ) logF (w)X X θ

∂S

!w

|dw| .

This lemma is proved very simply using the existence of appropriate outer functions (it is perhaps most easily seen by noting that the strip is conformally equivalent to the unit disk). Based on this we can quickly see the connection between the two methods described above: . If F # (j +it) is differentiable in Xj on a set of positive P ROPOSITION 4.2. Suppose F ∈ F measure for either j = 0 or j = 1 then F (θ ) ∈ Xθ for 0 < θ < 1 and xXθ F F . The following corollary was proved ﬁrst with the hypothesis that one space is reﬂexive in [14]; see [91]. C OROLLARY 4.3. If either X0 or X1 has the Radon–Nikodym property then the spaces [X0 , X1 ]θ and [X0 , X1 ][θ] coincide isometrically. P ROOF OF P ROPOSITION 4.2. For each h > 0 let Fh (z) = h−1 (F # (z + ih) − F # (z)). Then Fh ∈ F and Fh F F F . For ﬁxed θ we have that Fh (θ ) → F (θ ) in Σ(X). Assume F # (it) is differentiable on a set E of positive measure. Then Fh (it) converges in X0 a.e. on E. Note that Fh (w) − Fh (w) Fh (θ ) − Fh (θ ) exp P (w, θ ) log |dw| 1 2 1 2 X X θ

∂S

!w

and this inequality implies that Fh1 (θ ) − Fh2 (θ )Xθ converges to zero as h1 , h2 → 0, i.e., that limh→0 Fh (θ ) exists in Xθ . The proposition follows. The following is the duality theorem for complex interpolation due to Calderón [14]. T HEOREM 4.4. Suppose X is a Banach couple such that Δ(X) is dense in both X0 and X1 . Then the dual space of [X0 , X1 ]θ can be identiﬁed isometrically with [X0∗ , X1∗ ]θ . In particular if one of the spaces X0 or X1 is an Asplund space (i.e., either X0∗ or X1∗ has the Radon–Nikodym property) then the dual of [X0 , X1 ]θ can be identiﬁed with [X0∗ , X1∗ ]θ . There is a sense in which real interpolation scales can be regarded as special cases of complex scales. Let us quote the Re-iteration theorem (see [14,75,64,22]): T HEOREM 4.5. Suppose X is a complex Banach couple. Then: (1) If 0 φ1 < φ2 1 then [Xφ1 , Xφ2 ]θ coincides isometrically with [X0 , X1 ](1−θ)φ1 +θφ2 . (2) If 1 p < ∞ and 0 < φ1 < φ2 < 1 the space [Xφ1 ,p , Xφ2 ,p ]θ coincides with (up to equivalence of norm) Xφ1 (1−θ)+φ2 θ,p .

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See also [33] for some endpoint results and [30] for extensions to the quasi-Banach setting. One of the drawbacks of the complex method is that in general it seems relatively difﬁcult to calculate complex interpolation spaces. There is one exception to this rule, which is the case when one has a pair of Banach lattices. The following theorem follows from the work of Calderón [14]. T HEOREM 4.6. Suppose that X = [X0 , X1 ] is a couple of Banach function spaces on some measure space (K, μ). Assume that either X0 or X1 has the Radon–Nikodym property. Then the space [X0 , X1 ]θ is isometric to the space X01−θ X1θ , where θ 1−θ |h|θ . f X1−θ Xθ = inf g1−θ X0 hX1 : g ∈ X0 , h ∈ X1 , |f | = |g| 0

1

Let us note that this theorem can be applied when one can has a Banach couple X where X0 and X1 have a simultaneous unconditional basis; in particular it can be to study many types of function spaces (Besov spaces, Hardy spaces, Triebel–Lizorkin spaces, etc.) where one can ﬁnd such a basis using wavelets. Essentially this approach was used by Frazier and Jawerth [40] (using instead the essentially equivalent idea of the φ-transform). It seems however to be a general rule that the only cases where complex interpolation spaces are calculable are those when Theorem 4.6 can be used. For example, the interpolation of Schatten ideals by the complex method is possible only because it can be reduced to the interpolation of symmetric sequence spaces.

5. Properties preserved by interpolation There is vast literature on preservation of properties under interpolation. We consider properties of the underlying Banach spaces or of operators. Let us ﬁrst discuss the underlying spaces. Suppose P is a property of Banach spaces: we ask for conditions so that if one or both of the spaces X0 , X1 has the property P then the intermediate spaces (X0 , X1 )θ,p or [X0 , X1 ]θ obtained by real or complex interpolation inherit the property. It is also possible to discuss other methods of interpolation of course. Let us give an example. If X0 or X1 is reﬂexive so are the spaces (X0 , X1 )θ,p and [X0 , X1 ]θ for 0 < θ < 1 and 1 < p < ∞; the former is implied by Theorem 3.3 and the latter is due to Calderón [14]. In fact there is a very simple technique to see that certain types of properties interpolate for the real or complex methods. P ROPOSITION 5.1. Let X be a Banach couple. Then: (1) If 0 < θ < 1 and 1 < p < ∞ then (X0 , X1 )θ,p is isomorphic to a quotient of a subspace of the Banach space p (X0 ⊕ X1 ). (2) If 0 < θ < 1 then [X0 , X1 ]θ is isomorphic to a subspace of a quotient of Lp (X0 ⊕ X1 ) for any choice of 1 p < ∞.

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Case (i) of Proposition 5.1 is essentially proved in Section 3. Case (ii) follows from some alternative formulations of the complex method. From this proposition one can see immediately that if, say, X0 and X1 have non-trivial type then so do (X0 , X1 )θ,p and [X0 , X1 ]θ when 0 < θ < 1 and 1 < p < ∞. Suppose X is a Banach couple. We shall say that an interpolation space X is θ -exponential where 0 < θ < 1 if whenever T ∈ L(X, X) then θ T X→X CT 1−θ X0 →X0 T X1 →X1 .

The real interpolation spaces (X0 , X1 )θ,p and the complex interpolation spaces [X0 , X1 ]θ are examples of θ -exponential interpolation spaces. The following theorem is due to Garling and Montgomery-Smith [41]. It provides a strong converse to Proposition 5.1. T HEOREM 5.2. Let Z be the quotient of a subspace of a separable Banach space Y . Then there exists a Banach couple X such that both the end-point spaces X0 and Y0 are isomorphic to Y ⊕ Y ⊕ Y , but such for every 0 < θ < 1 and every θ -exponential interpolation space X we have that X contains a complemented subspace isomorphic to Z. So, for example, for any separable Banach space Z, there is a Banach couple X so that the end point spaces are isomorphic to 1 , but such that for every 0 < θ < 1 and every θ -exponential interpolation space X, we have that X contains a complemented subspace isomorphic to Z. Thus we see that many properties cannot be inherited by any interpolation method which is θ -exponential for some 0 < θ < 1. For example, the Radon–Nikodym property and non-trivial cotype can never be preserved by such a method. Indeed, Dilworth and Sobecki [36] showed that any Banach space property that is passed from the end point spaces to the spaces created by the real or complex method must also pass from any Banach space to any subspace of any quotient of that space. They remark that the only property of 1 preserved under either the real or complex methods is separability. Now suppose X and Y are two Banach couples and that T ∈ L(X, Y ). We now discuss properties of the operator T which can be interpolated. Let us ﬁrst note that if T : X0 → Y0 is weakly compact then T : (X0 , X1 )θ,p → (Y0 , Y1 )θ,p is weakly compact for any 0 < θ < 1 and 1 < p < ∞. This is due to Aizenstein (see [12] for a full discussion of interpolation of weak compactness by general real methods). In fact, in this case it can be seen to follow from the corresponding result for the property of reﬂexivity and the Factorization Theorem 3.4. The same argument establishes a similar result for complex interpolation. We shall now consider the question of interpolating compactness. If both T : X0 → Y0 and T : X1 → Y1 are compact then, as early as 1969, Hayakawa [45] showed that T : (X0 , X1 )θ,p → (Y0 , Y1 )θ,p is compact for 0 < θ < 1 and 1 p < ∞. The stronger one-sided result was proved, in full generality, only in 1992 by Cwikel [25]: T HEOREM 5.3. Suppose X and Y are two Banach couples and that T ∈ L(X, Y ) is such that T : X0 → Y0 is compact. Then T : (X0 , X1 )θ,p → (Y0 , Y1 )θ,p is compact for 0 < θ < 1 and 1 p < ∞.

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Curiously the same problem for the complex method is unsolved. It was ﬁrst considered by Calderón [14] in 1964. Surprisingly, it is not even known under two-sided conditions. P ROBLEM 5.4. Suppose X and Y are Banach couples and T ∈ L(X, Y ) is such that T : Xj → Xj is compact for j = 0, 1. Is T : [X0 , X1 ]θ → [X0 , X1 ]θ compact for 0 < θ < 1? This problem appears challenging for Banach space theorists, as the discussion in [29] shows. In [27] a partial result was given: T HEOREM 5.5. Suppose X and Y are complex Banach couples and that T ∈ L(X, Y ) is such that T : X0 → Y0 is compact. Suppose X0 is a UMD-space. Then T : [X0 , X1 ]θ → [Y0 , Y1 ]θ is compact for 0 < θ < 1. Some other conditions on X0 are also considered in [27]. For example, it sufﬁces (in place of assuming X0 is a UMD-space) that X0 is itself an interpolation space X0 = [E, X1 ]θ for some 0 < θ < 1 and some E. In this form the result can be regarded as an improvement (for complex interpolation) of the following earlier extrapolation result of [25]: T HEOREM 5.6. Suppose X and Y are Banach couples and that T ∈ L(X, Y ). Suppose 1 p < ∞ and that for some 0 < θ < 1 we have T : [X0 , X1 ]θ → [Y0 , Y1 ]θ (respectively T : (X0 , X1 )θ,p → (Y0 , Y1 )θ,p ) is compact. Then T : [X0 , X1 ]θ → [Y0 , Y1 ]θ (respectively T : (X0 , X1 )θ,p → (Y0 , Y1 )θ,p ) is compact for every 0 < θ < 1. Extrapolation theorems of this type can also be proved for certain properties of Banach spaces. Let us ﬁx our attention on complex interpolation. Suppose Xθ = [X0 , X1 ]θ . It is clear from Theorem 4.5, for example, that if Xθ is reﬂexive for some 0 < θ < 1 it is reﬂexive for every 0 < θ < 1. There is an abstract approach to these ideas via the notion of the Kadets distance (see [63]). Let X and Y be two subspaces of a Banach space Z. We deﬁne Λ(X, Y ) to be the Hausdorff distance between BX and BY , i.e., Λ(X, Y ) = max sup inf x − y, sup inf x − y . x∈BX y∈BY

y∈BY x∈BX

Now suppose X and Y are any two Banach spaces. We deﬁne the Kadets distance dK (X, Y ) Y ) over all spaces Z containing isometric copies of X, Y . It may to be the inﬁmum of Λ(X, be shown that dK is a pseudo-metric on any set of Banach spaces; unfortunately there are non-isomorphic Banach spaces X, Y so that dK (X, Y ) = 0. The following theorem now has the content that the map θ → [X0 , X1 ]θ is continuous for the Kadets metric. T HEOREM 5.7. Suppose X is a Banach couple and Xθ = [X0 , X1 ]θ for 0 < θ < 1. Then if 0 < θ < φ < 1, dK (Xθ , Xφ ) 2

sin(π(φ − θ ))/2 . sin(π(φ + θ ))/2

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Theorem 5.7 allows us to prove extrapolation theorems by showing that certain properties of Banach spaces deﬁne sets which are either open and closed or merely open. If the property is an isomorphic invariant then one can use Theorem 4.5 to give also the corresponding extrapolation result for the real method. Let us say a property P is stable if there exists α > 0 so that if X has P and dK (X, Y ) < α then Y has P. Stable properties deﬁne open and closed sets. Typical stable properties are separability, reﬂexivity, containing 1 , super-reﬂexivity and having non-trivial type. T HEOREM 5.8. Suppose X is a Banach couple and Xθ = [X0 , X1 ]θ for 0 < θ < 1. Then if P is a stable property and Xθ has P for some 0 < θ < 1 then Xθ has P for every 0 < θ < 1. Some other properties are merely open, i.e., they deﬁne open sets. For example (see [63]), X ≈ c0 and X ≈ 1 are both open for the Kadets metric. T HEOREM 5.9. Suppose X is a Banach couple and Xθ = [X0 , X1 ]θ for 0 < θ < 1. Then if P is an open property and Xθ has P for some 0 < θ < 1 then there exists δ > 0 so that Xφ has P for every |φ − θ | < δ. There are many unresolved questions about the Kadets metric. For example, we can consider the set of separable Banach spaces under the Kadets metric. This pseudo-metric space is not connected (for example, the super-reﬂexive spaces form an open and closed subset). It is not difﬁcult to show that the component of any Banach space X contains all isomorphic copies of X. One can ask to identify the component containing 2 . It is not clear if this contains all super-reﬂexive spaces. An old extrapolation result of Pisier [92] implies it contains all super-reﬂexive Banach lattices. Another intriguing question is to identify the component of c0 . It is tempting to conjecture this consists of all spaces isomorphic to a subspace of c0 . We refer to [63] for a fuller discussion of these ideas and of the relationship to the Gromov–Hausdorff distance.

6. Calderón couples We now turn to the interpolation theory question of determining all intermediate spaces for a given couple. Let us ﬁrst note that the construction of the (θ, p)-spaces in (3.1) and (3.3) can be generalized in an obvious way by replacing the a weighted Lp -space by an arbitrary Banach function space. To make this precise we deﬁne our notion of a Banach function space over a σ -ﬁnite measure space (Ω, μ). We say that a Banach space E, · E continuously embedded in the space M of all measurable functions (with the topology of convergence in measure on subsets of ﬁnite measure) is a Banach function space if whenever g ∈ E and |f | |g| a.e. then f ∈ E and f E gE .

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Let us suppose E is a Banach function space on the space (0, ∞) (with Lebesgue measure) with the property that min(1, t) ∈ E. Then given a Banach couple we can deﬁne an interpolation space XE as the space of all x such that K(t, x) ∈ E and we can put xXE = K(t, x)E . Each such function space E then induces an interpolation method. The interpolation method associated to E is called a K-method, with parameter E. We say that an interpolation space X for the Banach couple (X0 , X1 ) is K-monotone if there is a constant C so that whenever y ∈ X and x ∈ X0 + X1 with K(t, x) K(t, y),

0 < t < ∞,

then x ∈ X and xX CyX . It is clear that each K-method yields a K-monotone interpolation space. It is a deep result of Brudnyi and Kruglyak that for Gagliardo complete couples the converse is true, i.e., that every K-monotone interpolation space can be obtained by a K-method. The so-called Fundamental Lemma and K-divisibility Principle are the key ingredients of this result. The Fundamental Lemma appears in [24]. A forerunner appeared in Cwikel and Peetre [32]. T HEOREM 6.1. There is an absolute constant C with the following property. Let (X0 , X1 ) be a Gagliardo complete couple and suppose x ∈ Σ(X). Thenthere exists a sequence (uj )j ∈Z so that uj ∈ Δ(X) except for at most 2 values of j , x = j ∈Z uj in Σ(X) and: K(t, x)

min uj X0 , tuj X1 CK(t, x),

0 < t < ∞.

(6.1)

j ∈Z

R EMARK . Here if for some j , we have uj ∈ / X0 ∩ X1 we interpret

min uj X0 , tuj X1 < ∞ to imply that uj ∈ X0 ∪ X1 . The precise value of the constant C has been investigated further in [26]. The principle of K-divisibility of Brudnyi and Kruglyak [12] was announced in [11]. The Fundamental Lemma was used by Cwikel [24] to give an independent proof of Kdivisibility. T HEOREM 6.2. There is an absolute constant C so that if (X0 , X1 ) is a Gagliardo complete couple, x ∈ Σ(X) and ϕj is a sequence of concave functions such that ∞ j =1

ϕj (1) < ∞

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N. Kalton and S. Montgomery-Smith

and K(t, x)

∞

0 < t < ∞,

ϕj (t),

j =1

then there exist uj ∈ (X) such that x = 0 < t < ∞ and j ∈ N.

∞

j =1 uj

in Σ(X) and K(t, uj ) Cϕj (t) for

We now discuss the proofs of Theorems 6.1 and 6.2. Let us ﬁrst remark that they are equivalent. If we assume Theorem 6.2 then we can obtain Theorem 6.1 by observing that since t → K(t, x) is concave it may be represented (uniquely) in the form

∞

K(t, x) = a + bt +

min(s, t) dμ(s),

(6.2)

0

where μ is a Borel measure on (0, ∞) such that min(s, 1) dμ(s) < ∞. By approximation, if ε > 0 one can ﬁnd a sequences (sn )∞ n∈Z , (cn )n∈Z in (0, ∞) such that K(t, x) a + bt +

cn min(sn , t) (1 + ε)K(t, x),

0 < t < ∞.

(6.3)

n∈Z

Note that if K(t, u) a for all t then by Gagliardo completeness uX0 a and similarly if K(t, u) bt for all t then uX1 b. If K(t, u) c min(s, t) for all t then uX0 cs and uX1 c. Thus if apply Theorem 6.2 to (6.3) we obtain Theorem 6.1 with constant C(1 + ε). Next we consider the converse direction. In this case we ﬁnd (un ) as in Theorem 6.1 and let ψ(t) =

min un X0 , tun X1 .

n∈Z

Thus K(t, x) ψ(t) CK(t, x),

0 < t < ∞.

= (θj (t))j ∈N Now consider the set Γ of continuous maps θ : (0, ∞) → 1 of the form θ (t) where each θj is non-negative and concave and we have θj (t) ϕj (t) but n∈Z θj (t) C −1 ψ(t) for 0 < t < ∞). It is not difﬁcult to see by the Ascoli–Arzela theorem that Γ is compact for the topology of uniform convergence on compacta, and so has a minimal element σ (t) = (σj (t))n∈Z . It then follows without difﬁculty that in fact we have j ∈Z

σj (t) = C −1 ψ(t),

0 < t < ∞.

Interpolation of Banach spaces

1147

Indeed if σj > C −1 ψ(t) on some maximal open interval I then it is easy to see that σn must be afﬁne on I ; if I = (α, β) where 0 < α < β < ∞ then the fact that ψ is concave leads to a contradiction, while the other cases when α = 0 and/or β = ∞ can be treated similarly. Now using the deﬁnition of ψ and the fact that each σj is concave we see that σj (t) = C −1

δj k min uk X0 , tuk X1 ,

k∈Z

where k∈Z δj k = 1 and δj k 0. (One way to see this is to use the representation 6.2 for ψ and σj .) Let vj =

δ j k uk .

k∈Z

Then

∞

j =1 vj

converges absolutely in X0 + X1 to x and

K(t, vj ) Cσj (t) Cϕj (t). It is clear from the foregoing discussion that if we deﬁne γ1 as the inﬁmum of all constants C for which Theorem 6.1 holds and γ2 as the inﬁmum of all constants C for which Theorem 6.2 holds then γ1 = γ2 . Their common value, γ is called the K-divisibility constant. Its exact value is unknown and seems to be a challenging problem. The √ best estimate from above was obtained by Cwikel, Jawerth and Milman [26], γ 3 + 2 2. On the other hand, an example of Kruglyak [70] gives a lower estimate γ > 1.6. Let us now sketch the ideas in the proof of Theorem 6.1, but without attempting to give the most delicate estimates (following [5]); we refer the reader to [26] for these. For ﬁxed x ∈ X0 + X1 let us deﬁne t0 = 1 and then construct a sequence (tj )j ∈Z by two-sided induction such that for any j we have that one of the three mutually exclusive possibilities holds: K(tj , x) tj K(tj −1 , x) (1) , =2 min K(tj −1 , x) tj −1 K(tj , x) or (2) tj −1 = 0 and K(tj , x) tj K(t, x) , min < 2, K(t, x) tK(tj , x)

0 < t < tj ,

or (3) tj = ∞ and K(t, x) tK(tj −1 , x) , min < 2, K(tj −1 , x) tj −1 K(t, x)

tj −1 < t < ∞.

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For each j ∈ Z such that 0 < tj < ∞ pick vj ∈ X0 and wj ∈ X1 so that x = vj + wj and vj X0 + twj X1 < 2K(tj , x). If tj = 0 let vj = 0 and wj = x. If tj = ∞, let vj = x and wj = 0. Next let uj = vj − vj −1 . If 0 < tj −1 < tj < ∞ then uj X0 4K(tj , x),

uj X1 2tj−1 −1 K(tj −1 , X).

It follows that if tj −1 t tj then

min uj X0 , tuj X1 8K(t, x). If 0 = tj −1 < tj then uj = vj and uj X0 2K(tj , x). In this case either 1 lim K(t, x) K(tj , x) t →0 2 or tj−1 K(tj , x) lim t −1 K(t, x) 2tj−1 K(tj , x). t →0

In the latter case we use Gagliardo completeness to deduce that xX1 2tj−1 K(tj , x) and hence uj X1 xX1 + wj X1 4tj−1 K(tj , x). In either case we have

min uj X0 , tuj X1 8K(t, x),

0 < t tj .

Similarly if tj −1 < tj = ∞ we obtain

min uj X0 , tuj X1 8K(t, x),

tj −1 t < ∞.

Now if tj −1+r t tj +r where r ∈ Z \ {0} then we may see that K(t, x) 2r−1 K(tj , x),

r > 0,

and t −1 K(t, x) 21−r tj−1 −1 K(tj −1 , x),

r < 0.

Interpolation of Banach spaces

1149

Hence

min uj X0 , tuj X1 8 · 2−|r| K(t, x). Hence

min uj X0 , tuj X1 24K(t, x),

0 < t < ∞.

j ∈Z

It is clear that uj converges absolutely in X0 + X1 and it is not difﬁcult to check that its sum must be x in all possible cases. The above argument is clearly quite crude with regards to constants. In [26] a somewhat more delicate analysis is performed, keeping track of the intercepts on the axes of the tangents √ to the concave function t → K(t, x). With this analysis one can achieve the constant 3 + 2 2 + ε for any ε > 0. The main conclusion from the principle of K-divisibility is that K-monotone interpolation spaces coincide exactly with interpolation spaces obtained by the K-method: T HEOREM 6.3 (Brudnyi and Kruglyak [11]). If X is a Gagliardo complete couple then any K-monotone interpolation space is given by a K-method. Suppose Y is a K-monotone interpolation space. The idea of Theorem 6.3 is that one can deﬁne a Banach function space E on (0, ∞) by ∞ ∞ yj Y : f (t) K(t, yj ), 0 < t < ∞ . f E = inf j =1

j =1

Now if x ∈ X0 + X1 and K(t, x) ∈ E we can ﬁnd yj ∈ Y so that K(t, x)

∞

K(t, yj ),

0 < t < ∞,

j =1

and ∞ j =1

yj Y 2K(t, x)E .

Then by K-divisibility (Theorem 6.2) we can decompose x =

∞

j =1 uj

in X0 + X1 so that

K(t, uj ) CK(t, yj ), where C is a universal constant. If Y is K-monotone this implies that uj ∈ Y and that we have an estimate uj Y C1 yj Y for some constant C1 depending on Y . Hence x= ∞ j =1 uj ∈ Y and xY C2 K(t, x)E .

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We say that a Banach couple is a Calderón couple if every interpolation space is K-monotone (or, by Theorem 6.3, every interpolation space is given by a K-method). This terminology is based on the classical Calderón–Mitjagin theorem on interpolation spaces for (L1 , L∞ ). This theorem is in some sense already classical, but we will discuss it below as motivation. Let us ﬁrst make an equivalent formulation of the problem. Suppose 0 = x ∈ Σ(X). We can deﬁne an orbit space for x, Ox namely the space {T x: T ∈ L(X)} with the norm yOx = inf T X : T x = y . Then Ox is an interpolation space. If Ox is monotone then there is a constant C so that for any y satisfying K(t, y) K(t, x) we have T ∈ L(X) with T x = y and T X C. We thus have: P ROPOSITION 6.4. X is a Calderón couple if and only if for every x ∈ Σ(X) there is a constant C = C(x) so that if y ∈ Σ(X) and K(t, y) K(t, x) for 0 < t < ∞ then there exists T ∈ L(X) with T x = y. Somewhat surprisingly it appears to be unknown if the constant C can be chosen independent of x. If this is the case we call X a uniform Calderón couple. We now consider the special case of the pair L = (L1 (R), L∞ (R)) where R is equipped with standard Lebesgue measure. In this case the K-functional is computable and is given by the formula:

t

K(t, f ) =

f ∗ (s) ds,

0

where f ∗ is the decreasing rearrangement of |f |, i.e., the function on (0, ∞) given by f ∗ (s) = sup inf f (u). λ(F )=s u∈F

If we introduce f ∗∗ as usual by setting f ∗∗ (t) =

1 t

t

f ∗ (s) ds

0

then of course K(t, f ) = tf ∗∗ (t). We recall a Banach function space X is symmetric (or a symmetric lattice ideal) if it satisﬁes the condition that if f ∈ X and g is any measurable function with g ∗ f ∗ it follows that g ∈ X and gX f X . It is not difﬁcult to show any interpolation space for L is a symmetric space. Now for f, g ∈ L∞ + L1 let us say f ≺ g if f ∗∗ (t) g ∗∗ (t) for 0 < t < ∞. In many symmetric Banach function spaces X the property f ∈ X and g ≺ f implies g ∈ X and gX f X ; for example, this holds if X is separable. However it does not hold in

Interpolation of Banach spaces

1151

general, so we shall use the term rearrangement-invariant or r.i. space to mean a symmetric space with this additional property. The following theorem is due to Ryff [103] and Calderón [15]: P ROPOSITION 6.5. If f ≺ g then there is an operator T ∈ L(L) with T L = 1 and Tg = f. T HEOREM 6.6. The interpolation spaces for the couple (L1 , L∞ ) coincide with the r.i. spaces on R and (L1 , L∞ ) is a Calderón couple. Let us remark that this theorem is equally valid for the couple (L1 , L∞ [0, 1]). It is natural then to try to extend this Theorem 6.6 to other function spaces. A major advance was made in this direction by Sparr [106] and [107] (see also [1]): T HEOREM 6.7. The Banach couple (Lp0 (w0 ), Lp1 (w1 )) is a Calderón couple for any choice of 1 pj ∞ and any pair of weight functions wj . On the other hand, Ovchinnikov shows that the pair (L1 + L∞ , L1 ∩ L∞ ) is not a Calderon couple [88]. Indeed Maligranda and Ovchinnikov [78] showed that the Lp ∩ Lp and Lp + Lp are not K-monotone with respect to this couple when 1 < p < ∞, p = p and 1/p + 1/p = 1. However these spaces are complex interpolation spaces for this couple. This raises the general question of classifying pairs of r.i. spaces (X0 , X1 ) on either (0, 1) or (0, ∞) which form Calderón couples. For special examples (certain types of Lorentz spaces and Marcinkiewicz spaces) positive results were obtained by Cwikel [23] and Merucci [81]. In [56] a full study of this problem was undertaken and although the results are not complete, a good description was obtained for sufﬁciently “separated” pairs of spaces, in a sense to be described. Curiously enough some of the properties which surface in the characterization have a ﬂavor suggestive of Banach space theory. We ﬁrst need to introduce some standard ideas (see [72], for example). If X is an r.i. space on [0, 1] or [0, ∞) then the dilation operators Da on X are then deﬁned by Da f (t) = f (t/a) (where we regard f as vanishing outside [0, 1] in the former case). We can then deﬁne the Boyd indices pX and qX of X by log a a→∞ log Da X

pX = lim and qX = lim

a→0

log a . log Da X

In many texts, the reciprocals of pX and qX are used for the Boyd indices following the original convention of Boyd [10]. The Boyd indices are of course extremely useful in interpolation theory because of the classical Boyd interpolation theorem [10], which we will discuss in Section 7.

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For convenience we restrict our discussion to the case of r.i. spaces over (0, ∞). Let en = χ(2n ,2n+1 ) . Associated to each r.i. space X we can introduce a Banach sequence space SX modelled on Z deﬁned by ξ = (ξn )n∈Z if and only if n∈Z ξn en ∈ X and ξ e ξ SX = n n . X

n∈Z

Now it is clear from consideration of averaging projections that (X, Y ) is a Calderón couple if and only if (SX , SY ) is also a Calderón couple. It turns out we can answer this question under separation conditions on the Boyd indices in terms of some conditions with a Banach space ﬂavor. Let us suppose that E is a Banach sequence space modelled on Z. N We shall say that E has the right shift-property (RSP) if whenever (xn )N n=1 , (yn )n=1 are two sequences satisfying (1) supp x1 < supp y1 < supp x2 < · · · < supp xn < suppyn , (2) yn E xn E , n = 1, 2, . . . , N , then N N yn C xn . n=1

n=1

N Similarly we say E has the left-shift property (LSP) if whenever (xn )N n=1 , (yn )n=1 are two sequences satisfying (1) supp x1 > supp y1 > supp x2 > · · · > supp xn > supp yn , (2) yn E xn E , n = 1, 2, . . . , N , then

N N yn C xn . n=1

n=1

We then say that an r.i. space X on (0, ∞) is stretchable if SX has (RSP) and compressible if SX has (LSP). If X is both stretchable and compressible then X is elastic. The main theorems of [56] then assert the following: T HEOREM 6.8. Let (X, Y ) be a pair of r.i. spaces on (0, ∞) such that pY > qX . Then (X, Y ) is a (uniform) Calderón couple if and only if X is stretchable and Y is compressible. T HEOREM 6.9. Let X be an r.i. space on (0, ∞); then (X, L∞ ) is a Calderón couple if and only if X is stretchable. Of course the condition pY > qX is a quite strong separation condition on the Boyd indices; it asserts that the intervals [pX , qX ] and [pY , qY ] do not intersect. A remarkable feature of the conclusion of Theorem 6.8 is that the condition that X is stretchable (or Y is

Interpolation of Banach spaces

1153

compressible) is independent of the normalization of en . To illustrate this note that if X is a Lorentz space with norm

f ∗ (t)p w(t) dt

∞

f X =

1/p

0

for a suitable decreasing weight function w then X is always elastic because SX up to normalization is simply p . The above Theorems 6.8 and 6.9 have similar statements when (0, ∞) is replaced by [0, 1] and for sequence spaces. It is necessary simply to formulate the shift properties on sequence spaces modelled on N or Z \ N. We refer to [56] for details. It is possible to give a rather complicated characterization of stretchable and compressible Orlicz spaces. In fact for Orlicz spaces the conditions are equivalent and any such space is elastic. The following theorem is given in [56] (we specialize to [0, 1] for deﬁniteness). T HEOREM 6.10. Let F be an Orlicz function. Then the following conditions on F are equivalent: (1) LF [0, 1] is elastic (respectively, stretchable, respectively, compressible). (2) (L∞ [0, 1], LF [0, 1]) is a Calderón couple. (3) There is a bounded monotone increasing function w : [1, ∞) → R and a constant C so that if 1 s t and 0 < x 1 we have F (tx) F (sx) C + w(t) − w(s). F (t) F (s) (4) There is a bounded monotone increasing function w : [1, ∞) → R and a constant C so that if 1 s t and 0 < x 1 we have F (tx) F (sx) C + w(t) − w(s). F (s) F (t) These conditions are a little difﬁcult to check. They are related to somewhat similar criteria for Orlicz spaces to coincide with Lorentz spaces [84]. Perhaps the simplest practical condition which follows is the following. T HEOREM 6.11. Let X be an Orlicz space on [0, 1]. Then if (X, L∞ [0, 1]) is a Calderón couple we have pX = qX . This allows the construction of some very easy counter-examples to the conjecture that (LF [0, 1], L∞ [0, 1]) is a Calderón couple for every Orlicz function F . In spite of the difﬁculty in classifying Calderón couples, there is a form of converse to the theorem of Sparr, obtained by Cwikel and Nilsson [31]. Here we consider all possible changes of density. If X is a Banach function space we deﬁne X(w) = {f : f w ∈ X} with f X(w) = f wX , where w is a weight function (a strictly positive measurable function).

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T HEOREM 6.12. Let (X0 , X1 ) be a pair of Banach function spaces on [0, 1] or [0, ∞). Suppose that for every pair of weight functions the pair (X0 (w0 ), X1 (w1 )) is a Calderón couple. Then there exist 1 p0 , p1 ∞ and weight functions v0 , v1 so that X0 = Lp0 (v0 ) and X1 = Lp1 (v1 ) up to equivalence of norm. Finally let us note a problem raised by Cwikel which was solved in [80]. Cwikel asked if a pair (X, Y ) of complex Banach spaces is a Calderón couple if and only if every complex interpolation space is K-monotone. In [80] counter-examples are exhibited even for pairs of r.i. spaces. 7. Interpolation spaces for (Lp , Lq ) Throughout this section we will suppose that our rearrangement invariant spaces are over [0, 1] or [0, ∞). Let us start by noting that the K-functional for (Lp , Lq ) can be approximated in terms of the Hardy operators. To this end, we have the following formula of Holmstedt [47]: t

−1/p

K f, t 1/p−1/q ≈

t 1/p ∞ 1/q 1 1 ∗ p ∗ q f (s) ds + f (s) ds . t 0 t t

This formula, combined with the fact that (Lp , Lq ) is a Calderón couple, can be used to obtain useful results. For example, it is now easy to prove the following interpolation result [46]. Given an Orlicz function Φ, we will say that Φ is p-convex if the map t → Φ(t 1/p ) is convex, and q-concave if the map t → Φ(t 1/q ) is concave (we will say that all functions are ∞-concave). T HEOREM 7.1. Let 1 p < q ∞, and let X be an interpolation space for (Lp , Lq ). Then there is a positive constant c such that the following holds. If f, g are functions such that gΦ f Φ for every function Φ that is p-convex and q-concave, and if f ∈ X, then g ∈ X with gX c f X . In [61], the authors were able to obtain the following characterization of interpolation spaces for (Lp , Lq ). In order to state this result, we need the notion of conditional expectation. On [0, 1], this is standard. On [0, ∞) the same construction works, as long as the σ -ﬁeld’s atoms all have ﬁnite measure. T HEOREM 7.2. Let 1 p < q ∞. A rearrangement invariant space X is an interpolation space for (Lp , Lq ) if and only if there is a positive constant c such that for any function f , and any sub-σ -ﬁeld M whose atoms have ﬁnite measure, we have that p

1/p E |f | |M c f X X and if q < ∞

1/q . f X c E |f |q |M X

Interpolation of Banach spaces

1155

This gives the following collection of sufﬁcient conditions, the fourth of which is the classical Boyd interpolation theorem [10], which is also proved in [72] (see also [86]). T HEOREM 7.3. Let X be a rearrangement invariant space, and 1 p < q ∞ Suppose that any of the following hold: • X is p-convex and q-concave; • X is p-convex and has upper Boyd index less than q; • X is q-concave and has lower Boyd index greater than p; • X has Boyd indices strictly between p and q. Then X is an interpolation space for (Lp , Lq ). Finally we end this section with some results about the span of the Rademacher series in rearrangement invariant spaces. That is, given a rearrangement invariant space L on [0, 1], we can form a sequence space RL which is the space of sequences (an ) whose norm ∞ n=1 an rn L is ﬁnite. (Here (rn ) denotes the sequence of Rademacher functions on [0, 1].) It was shown by Rodin and Semenov [102] that RL is isomorphic to the space 2 if and only if L contains the space G, where G is the closure of the simple functions in the Orlicz space derived from an Orlicz function equivalent to exp(x 2 ). They went on to calculate the space RL for some lattices that do not contain G. More recently this work was extended by Astashkin [2]. One of the main results of this paper can be summarized as follows. T HEOREM 7.4. A symmetric sequence space S is naturally isomorphic to RL for some rearrangement invariant space L on [0, 1] if and only if S is an interpolation space for the couple (1 , 2 ).

8. Extensions Let us brieﬂy describe the elements of the theory of extensions (or twisted sums). This discussion overlaps the discussion in [60] but here our emphasis is slightly different. We note that a good reference for the general theory of twisted sums in the context of Banach space theory is [18]. Let X and Y be a Banach spaces (or more generally quasi-Banach spaces). An extension of X by Y is (formally) a short exact sequence 0 → Y → Z → X → 0, where Z is a quasi-Banach space. Less formally we regard the space Z is an extension of X by Y if Z ⊃ Y and Z/Y is isomorphic to X. One can of course restrict extensions to lie in the category of Banach spaces. There is a general construction of extensions via quasi-linear maps. Let V be any vector space containing Y (we may take Y = V but some ﬂexibility is useful here.) A map Ω : X → V is called quasi-linear if

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N. Kalton and S. Montgomery-Smith

• Ω(λx) = λΩ(x), x ∈ X, λ ∈ K. • There is a constant C so that if x1 , x2 ∈ X then Ω(x1 + x2 ) − Ω(x1) − Ω(x2) ∈ Y and

Ω(x1 + x2 ) − Ω(x1 ) − Ω(x2) C x1 + x2 .

(8.1)

We then can deﬁne an extension X ⊕Ω Y to be the subspace of X ⊕ V of all (x, v) such that v − Ωx ∈ Y , and equipped with the quasi-norm (x, v) = x + v − Ωx. In general this is not a norm, but it will be equivalent to a norm if it satisﬁes an estimate of the form n n n (xj , vj ), x1 , . . . , xn ∈ X, v1 , . . . , vn ∈ V . xj , vj C j =1

j =1

j =1

It follows that X ⊕Ω Y is isomorphic to a Banach space if (and only if) (8.1) is replaced by the stronger inequality: n n n Ωxk − Ω xk C xk . k=1

k=1

(8.2)

k=1

In fact, in the above construction, it is only necessary that Ω be deﬁned on a dense linear subspace; the construction above yields a space whose completion is an extension. Now it is a key fact that every extension can be represented in this form. Indeed if Z is an extension of X we can deﬁne two maps F : X → Z and L : X → Z such that qF = qL = IX where q is the quotient map. F is deﬁned to be homogeneous (not necessarily linear) and satisfy F (x) 2x, while L is required to be linear (but not necessarily bounded). If we set Ω(x) = F (x) − L(x) then Ω : X → Y is quasilinear and one can easily set up a natural isomorphism between Z and X ⊕Ω Y . Notice, however, that the choice of Ω depends heavily on certain arbitrary choices (e.g., of the linear map L). We refer to an extension Z of X as trivial if there is a bounded projection of Z onto X. In this case Z splits as a direct sum X ⊕ Y . It is easy to show that X ⊕Ω Y splits if and only if there is a linear map L : X → V so that Ωx − Lx ∈ Y for all x and Ωx − Lx Cx,

x ∈ X.

In [60] we discussed the case of minimal extensions. A minimal extension is an extension by the scalar ﬁeld K. In this case all Banach extensions are trivial, by the Hahn–Banach theorem. A Banach space X is called a K-space if all minimal extensions of X are trivial. The following proposition is then very useful: P ROPOSITION 8.1 ([51]). X is a K-space if and only if any extension of X by a Banach space is (isomorphic to) a Banach space.

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It is conjectured (see [60]) that X is a K-space if and only if X∗ has non-trivial cotype. It is known that any space with non-trivial type is a K-space. An extension of X by X is called a self-extension; in this case we introduce the notation dΩ X = X ⊕Ω X. We will see shortly that these are intimately related with interpolation theory, but ﬁrst let us discuss the historical origins of the study of self-extensions.

9. Self-extensions of Hilbert spaces A self-extension of a Hilbert space is called a twisted Hilbert space. The basic question of the existence of a non-trivial twisted Hilbert space was apparently ﬁrst raised by Palais. It was solved in 1975 by Enﬂo, Lindenstrauss and Pisier [39] who produced the ﬁrst nontrivial example of a twisted Hilbert space. A few years later in [62] an alternative example was constructed based on the ideas of Ribe’s construction of a non-trivial minimal extension of 1 (see [99] and [60]). We will discuss this example and some variants in this section. It is interesting that the link between minimal extensions of 1 and self-extensions of 2 is now much better understood than it was in 1979, and we will explain this connection later. Let us recall that Ribe’s space is associated to the quasilinear map Ω : c00 → R given by ∞ ∞ ∞ (9.1) ξn log |ξn | − ξn log ξn . Ωξ = n=1

n=1

n=1

We deﬁne a corresponding self-extension of 2 , denoted by Z2 by taking Ω : 2 → ω (ω is the space of all sequences) as |ξn | ∞ . Ωξ = ξn log ξ 2 n=1 Here we interpret 0 log 0 and 0 log 0/0 as 0. Thus Z2 is the space of pairs of sequences ∞ ((ξn )∞ n=1 , (ηn )n=1 ) such that (ξ, η) =

∞ n=1

1/2 |ξn |2

+

2 1/2 ∞ | |ξ n ηn − ξn log < ∞. ξ 2 n=1

This equation deﬁnes a quasi-norm; the fact that it is equivalent to a norm (and thus Z2 is a genuine Banach self-extension) follows from Proposition 8.1. The Banach space properties of the space Z2 are of some interest. It is immediate that there is a natural unconditional Schauder decomposition into two-dimensional spaces and it is shown in [62] that it has no unconditional basis; in [50] it is shown to fail local unconditional structure as well. In fact in [17] it is shown that any space with a two-dimensional UFDD (En ) (or even a UFDD with bounded dimensions) with local unconditional structure has an unconditional basis which can be chosen from the subspaces. The main unresolved problems concerning Z2 are:

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• Is Z2 prime? • Is Z2 isomorphic to its hyperplanes. It has been widely conjectured that Z2 is not isomorphic to its hyperplanes (in some sense, Z2 is even-dimensional and its hyperplanes are odd-dimensional!). Of course, since the celebrated example of Gowers [43], this problem is less pressing. Let us notice (as in [62]) that this construction can be generalized quite a bit. Let F : R → R be any Lipschitz map, and let E be any Banach sequence space and deﬁne ∞ |ξn | ΩF (ξ ) = ξn F log . ξ E n=1 Then ΩF induces a self-extension dΩF E of E. In fact one can go further and consider complex sequence spaces and then allow F : R → C. In [57] this idea was exploited taking E = 2 and F (t) = t 1+iα . This produces a complex Banach space Z2 (α) which is not isomorphic to its conjugate space. The conjugate space of a complex Banach space X is ¯ In this case it is not difﬁcult the space X on which multiplication is deﬁned by λ × x = λx. to see that the conjugate space of Z2 (α) is isomorphic to Z2 (−α) and then it can be shown without undue difﬁculty that Z2 (α) and Z2 (−α) are not isomorphic as complex Banach spaces; see [57] or [6]. Earlier examples had been constructed by probabilistic methods by Bourgain [9] and Szarek [108].

10. Analytic families of Banach spaces In this section we sketch the origins of the theory of non-linear commutators and analytic families and how it relates to the preceding examples. Let us introduce the idea of an analytic family of Banach spaces. To do this we will abstract the ideas of complex interpolation introduced in Section 4; this has the added convenience of incorporating the description of interpolating families of spaces by Coifman, Cwikel, Rochberg, Sagher and Weiss [20]. Let us suppose U is an open subset of the complex plane conformally equivalent to the open unit disk D; in fact we need only consider the case U = D and U = S := {z: 0 < !z < 1}. Next let W be some complex Banach space (the ambient space) and let F be a Banach space of analytic functions F : U → W . We assume that F is equipped with a norm · F such that: • The evaluation map F → F (z) (F → W ) is bounded for each z ∈ U . • If ϕ : U → D is a conformal equivalence then F ∈ F if and only if ϕF ∈ F and F F = ϕF F . We will call such a space F admissible. Then for z ∈ U and x ∈ W we deﬁne xz = inf F F : F (z) = x and let Xz = x ∈ W ; xz < ∞ .

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The family of spaces (Xz )z∈U is then called an analytic family of Banach spaces. If W0 is the linear span of the spaces {Xz : z ∈ U} then a linear map T : W0 → W0 will be called interpolating if F → T ◦ F is deﬁned and bounded on F . It then follows that T (Xz ) ⊂ Xz for each z ∈ U and T Xz →Xz T ◦ F F →F . If we take a Banach couple X = (X0 , X1 ) and deﬁne F as in Section 4 then one may see that {Xz : z ∈ U} is an analytic family and Xz = X!z where Xθ is the complex interpolation space between X0 and X1 . Thus our deﬁnition abstracts the ideas of complex interpolation. Under these assumptions any T ∈ L(X, Y ) is interpolating. The upper method also yields an analytic family at least when X is Gagliardo complete. Let us note that it is possible to describe the ideas of this section in much more generality, by relaxing our assumptions on F , so that real and other methods may be included; we refer to [28] for a fuller discussion, using an annulus in place of the disk. To keep our discussion reasonably crisp we will retain our much stronger conditions. We now invoke ideas of Rochberg and Weiss [101] (which in embryonic form appear in work of Schechter [104]). For each z we deﬁne a derived space dXz ⊂ W × W by dXz = {(x1 , x2 ): (x1 , x2 )dXz < ∞} where (x1 , x2 ) (10.1) = inf F F ; F (z) = x1 , F (z) = x2 . dXz Let Y be the subspace of dXz deﬁned by x1 = 0. We claim that Y is an isometric copy of Xz . Indeed let ϕ be a conformal map of U onto D with ϕ(z) = 0. Then if F (z) = 0 we can write F = ϕG where GF = F F . Then F (z) = ϕ (z)G(z) and so −1 (0, x2 ) = ϕ (z) x2 Xz . dXz On the other hand dXz /Y is trivially isometric to Xz so that we have a short exact sequence 0 → Xz → dXz → Xz → 0 and dXz is a self-extension of Xz . Thus we can use the ideas from Section 8 to give a representation of dXz in the form dΩ Xz where Ω : Xz → W is a quasilinear map. It is easy enough to see that an appropriate Ω is given by Ω(x) = F (z) where F is any choice of F ∈ F with F (z) = x and F F Cxz . In many circumstances there is a unique optimal choice of F with F F = xz and in this case one can deﬁne Ω in a very natural way. In general, there is some arbitrariness in the deﬁnition of Ω but any two such choices differ by a bounded function. Thus we have (x1 , x2 ) ≈ x1 z + x2 − Ωx1z . dXz Rochberg and Weiss used this construction to obtain commutator estimates. If T is an interpolating operator then (x1 , x2 ) → (T x1 , T x2 ) is bounded on dXz and this implies

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P ROPOSITION 10.1. If T is an interpolating operator then there is a constant C so that if x ∈ Xz then [T , Ω]x = T Ωx − ΩT x ∈ Xz and [T , Ω]x Cxz . z To conclude this section, let consider the case of interpolation of p -spaces. Suppose 1 p1 < p0 ∞. We will take W = ω, the space of all complex sequences, as our ambient space. Let us consider the space G of analytic functions on the strip S of the form F (z) = ∞ (fk (z))∞ k=1 where (fk )k=1 are bounded analytic functions. We can then extend each fk a.e. to the boundary of the strip by taking non-tangential limits, i.e., fk (j + it) = lim fk (x + it), x→j

j = 0, 1.

Now deﬁne F to be the space of all F ∈ G so that F (j + it) ∈ pj a.e. for j = 0, 1 and F F = max

ess sup F (j + it)p

j =0,1 −∞
j

< ∞.

It may be shown that this method coincides with the upper Calderón method, and hence by Corollary 4.3 yields the same interpolation spaces as complex interpolation, i.e., Xz = pz where if θ = !z we have 1 1−θ θ = + . pz p1 p0 The advantage of using this method, however, is that we can write down explicit extremals when computing norms in the interpolation spaces. Now suppose 0 < θ < 1 and x = (ξk )∞ k=1 ∈ p where p = pθ . In this case we can construct an optimal F so that F (θ ) = x and F F = xp . Let us suppose ﬁrst that xp = 1. Then F is given by fk (z) = xk |xk |

( pp − pp )(z−θ) 1

0

with fk (z) = 0 if xk = 0. It follows that we can deﬁne Ω : p → ω by the formula: Ω(x) =

∞ p p xk log |xk | − . p1 p0 k=1

In general by homogeneity we have ∞ p p |xk | Ω(x) = xk log − . p1 p0 xp k=1 If we take p0 = 1, p1 = ∞ and θ = 1/2 we have ∞ |xk | Ω(x) = 2 xk log x2 k=1

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1161

so that (except for a normalization factor of 2) one sees that Z2 is really nothing other than dX1/2 for this interpolation process. Of course exactly the same calculations are possible for function spaces and this was originally done in that context by Rochberg and Weiss [101]. In particular they noticed that if one applies this and Proposition 10.1 to the Hilbert transform H on the space Lp (T) where 1 < p < ∞ one obtains inequalities of the form

H f log |f | − Hf log |Hf | Cp f p . Lp Let us also notice that if one use the couple (L2 (p ), L2 (q )) where 1/p + 1/q = 1 it is not difﬁcult to use these ideas to see that Z2 is a (UMD)-space; this result was originally shown directly in [52]. Let us remark that it is also possible to develop a theory of commutator estimates for real interpolation spaces ([49]). However as remarked in Section 4 one can treat real interpolation scales as complex interpolation scales. For a uniﬁed treatment see [28]. We also remark that nothing prohibits us from generalizing our ideas to higher-order results by considering the map F → (F (z), F (z), F

(z), . . . , F (n) (z)) for n 1. There is quite a substantial literature on higher-order estimates (see, e.g., [16,82] and [100]).

11. Entropy functions and extensions In this section we will extend the ideas developed in the preceding sections to more general spaces. Our discussion will overlap with ideas in [60]. We will be discussing interpolation of lattices, but it will be convenient to discuss the special case of sequence spaces; function spaces can be treated almost identically but certain irritating (but fundamentally unimportant) complications arise. As in [60] we will use the term Banach sequence space to mean a Banach sequence space X equipped with a norm · X such that • The basis vectors en ∈ X. • If x ∈ X and |ηk | |xk | for every k then η ∈ X and ηX xX . • For every n ∈ N the linear functional η → ηn is continuous. • If x is a sequence such that n ∈ N Sn x = (x1 , . . . , xn , 0, . . .) ∈ X and sup Sn xX < ∞ then x ∈ X and xX = supn∈N Sn xX . Thus a key assumption is that a Banach sequence space will always be assumed to have the Fatou property. This is essentially the statement that BX = {x: xX 1} is closed for the topology of pointwise convergence. Of course if we consider only reﬂexive spaces this is immediate; the main signiﬁcance is that we consider ∞ and not c0 to be Banach sequence spaces. The analogous assumption for function spaces is that BX should be closed under convergence almost everywhere. We will use X∗ for the Köthe dual of X, i.e., x ∗ ∈ X∗ if and only if ∞ ∗ ∗ x |xk |: xX 1 < ∞. x ∗ := sup k X k=1

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We need a fundamental result of Lozanovsky [77]: P ROPOSITION 11.1. Suppose X is a Banach sequence space. Suppose u ∈ 1 with u 0 and u1 = 1. Then there exist x ∈ BX , x ∗ ∈ BX∗ with x 0, x ∗ 0 and u = xx ∗ , i.e., uk = xk xk∗ for 1 k < ∞. Furthermore x, x ∗ are uniquely determined if we insist that xk∗ = xk = 0 when uk = 0. This result was originally obtained by Lozanovsky by interpolation techniques; it is also valid in function spaces. Several subsequent proofs has appeared (e.g., [42]). The factorization u = xx ∗ is called the Lozanovsky factorization of u. Let us also introduce the notion of the entropy function of X: ΦX (u) = sup

∞

xX 1 k=1

uk log |xk |.

(11.1)

The study of this functional goes back to Gillespie [42] in 1981. The term indicator function of X was used in [55] and the name entropy function of X in [87] where it plays an important role in the solution of the distortion problem. As explained in [60] for any such Banach sequence space X the entropy function ΦX extends to a quasi-linear map ΦX : c00 → R which induces a minimal extension of 1 . The entropy function was used by Gillespie [42] to give a simple proof of the Lozanovsky factorization (Proposition 11.1). We treat the case of sequence spaces; with some extra technical work, the argument can be extended to function spaces. It sufﬁces to prove the proposition if u ∈ c00 (as then standard limiting arguments can be used, exploiting ∞the Fatou property). For ﬁxed u pick x ∈ BX with x 0 to maximize the expression k=1 uk log |xk |. Now if ξ ∈ BX with ξ 0 we have ∞

∞

uk log xk + t (ξk − xk ) uk log xk ,

k=1

0 t 1.

k=1

Note that if uk > 0 then we must have xk > 0. Hence differentiating uk =0

uk

ξk − xk 0. xk

Let xk∗ = uk /xk if uk = 0 and 0 otherwise. Then ∞

ξk xk∗

k=1

so that x ∗ X∗ 1.

∞ k=1

uk = 1

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1163

Uniqueness is immediate since if u = xx ∗ = yy ∗ where xX = yX = x ∗ X∗ = y ∗ Y ∗ and x, x ∗ , y, y ∗ 0 then ∞ 1 k=1

4

(xk + yk ) xk∗ + yk∗ 1

and this can only happen if xk = yk and xk∗ = yk∗ whenever uk = 0. + Note that this implies that if u ∈ c00 , ΦX (u) =

∞

uk log |xk |,

k=1

where x ∈ BX is determined by the Lozanovsky factorization of u/u1 . We can exploit to canonically extend ΦX to c00 by deﬁning ΦX (u) =

∞

uk log |xk |,

k=1

where x is given by the Lozanovsky factorization of |u|/u1 . This deﬁnition of ΦX is also quasi-linear on c00 . Next we turn to complex interpolation of Banach sequence spaces, using the upper method which may be formulated as described at the end of Section 10. If X0 and X1 are two Banach sequence spaces we deﬁne F to be the subset of G of functions such that F F := max ess sup F (j + it)X < ∞ j =0,1 −∞
j

and then we obtain an analytic family Xz for z ∈ S. In this case Xθ for 0 < θ < 1 is obtained by the Calderón formula Xθ = X01−θ X1θ , i.e., x ∈ Xθ if and only if θ 1−θ |x1 |θ < ∞. xXθ := inf x0 1−θ X0 x1 X1 : |x| = |x0 | As we have seen in Theorem 4.6 above the spaces Xθ are simply the usual interpolation spaces for complex interpolation when either X0 or X1 has the Radon–Nikodym property (essentially this means one is separable under our restrictions). Note that the entropy functions linearize this interpolation method, i.e., ΦXθ = (1 − θ )ΦX0 + θ ΦX1 . The Lozanovsky factorization theorem gives the formula ΦX + ΦX∗ = Λ,

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N. Kalton and S. Montgomery-Smith

where Λ = Φ1 is given by Λ(x) =

∞

xn log

n=1

|xn | . x1

Thus Λ coincides on the positive cone with Ribe functional given by (9.1). Note all also that Φp = p1 Λ for 1 p < ∞ and in particular that Φ∞ = 0. The following result is the specialization to our situation of Theorem 5.2 of [55], giving a full characterization of entropy functions: + T HEOREM 11.2 ([55]). Let Φ : c00 → R be any functional. Then in order that there exists a Banach sequence space X such that ΦX = Φ it is necessary and sufﬁcient that: (1) Φ is positive homogeneous, i.e., Φ(αu) = αΦ(u) if α 0. (2) Φ is convex. (3) Λ − Φ is convex.

Note that (X, · X ) is unique: in fact we can characterize BX by BX = x:

∞

uk log |xk | Φ(u) ∀u 0 .

k=1

Moreover, the inequality ΦY (u) − ΦX (u) Cu1 ,

u 0,

is equivalent to the statement that Y = X and e−C xY xX eC xY ,

x ∈ X.

It is amusing to point out that this identiﬁcation gives a “calculus” for Banach sequence spaces which can be used to prove an old extrapolation result of Pisier [92]: C OROLLARY 11.3 ([92]). Let X be a Banach sequence space which is p-convex and q-concave where 1 < p < 2 and 1/p + 1/q = 1. Then there is a Banach sequence space Y such that X = Y 1−θ θ2 where θ = 2/p − 1. To prove Corollary 11.3 it sufﬁces to note that X is p-convex (with constant one) if and only if there is a sequence space Y with X = Y 1/p so that ΦX satisﬁes that p1 Λ − ΦX is convex. Similarly q-concavity (with constant one) means that p1 Λ − ΦX∗ is convex. If we now solve the equation ΦX = (1 − θ )Φ + 12 θ Λ it can be shown that Φ and Λ − Φ are convex so that we can determine Y with Φ = ΦY . Now ﬁxing X0 and X1 let us describe the derived spaces dXθ . Assume x ∈ Xθ and xXθ = 1. Then there is an optimal factorization |x| = |x0 |1−θ |x1 |θ where xj ∈ BXj . It

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1165

can be assumed that x0 , x1 have the same support as X. The optimal choice of F ∈ F with F (θ ) = x is given by F (z) = x|x1 |z−θ |x0 |θ−z . Thus dXθ can be identiﬁed with dΩ Xθ where

Ωx = x log |x1| − log |x0 | . However if we similarly interpolate X0∗ and X1∗ then it follows from the calculus developed above (or in most cases from the basic duality theorem of [14]) that the corresponding intermediate spaces are given by Xθ∗ . Now assume x can be normed by some x ∗ ∈ Xθ∗ (e.g., if x ∗ is in the closure of c00 ). Then x ∗ has a similar factorization |x ∗ | = |x0∗ ||x1∗ | and the corresponding optimal F ∗ is given by F ∗ (z) = x ∗ |x1∗ |z−θ |x0∗ |θ−z . Let us suppose F (z) = (fk (z)) and G(z) = (gk (z)). Then ∞ k=1 fk (z)gk (z) assumes its maximum value 1 at the interior point z = θ and hence is constant. It then follows that for each k the functions fk (z)gk (z) can only take real values and thus individually constant. Thus |x||x ∗| = |x0∗ ||x1∗ | = u say. Thus x1 and x0 are determined by the Lozanovsky factorization of |x||x ∗ | where x ∗ norms x. Note also that we have !

"

Ωx, x ∗ = ΦX1 xx ∗ − ΦX0 xx ∗

if x ∈ c00 . This last formula can be extended to a larger domain but not to all x ∈ X. It may also be shown that arbitrary x ∈ X and x ∗ ∈ X∗ (not necessarily norming x we have ! "

Ω(x), x ∗ − Φ xx ∗ CxX x ∗

X∗

,

x ∈ X, x ∗ ∈ X∗ ,

(11.2)

where Φ = ΦX1 −ΦX0 , for a suitable constant C. There is thus an intimate relation between entropy functions and the derived spaces dXθ . To illustrate the meaning of this statement suppose X1 is obtain from X0 by a change of weight, i.e., X1 = {x: xw ∈ X0 } for some positive weight sequence w = (wn ) and xX1 = xwX0 . Then xXθ = xwθ X0 . In this case Φ is linear and Φ(u) = −

∞

un log wn .

n=1

The spaces dXθ which arise in this way are special self-extensions of Xθ , in the sense that the multiplication operators Ma x = ax for a ∈ ∞ must naturally extend to dXθ , since they are interpolating operators. In terms of Ω this means an estimate of the form Ω(ax) − aΩ(x) Ca∞ xX . θ X θ

We may say that the spaces dXθ are lattice self-extensions (in [53] the term lattice twisted sum was used). This notion can be made precise but we will not do so here for lack of space. Let us now consider the problem from the opposite direction. Suppose we ﬁx a Banach sequence space X; we wish to characterize all lattice self-extensions. It can be shown that this reduces to looking at the notion of a centralizer (cf. [53]). A map Ω : X → ω is called

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a centralizer if it is homogeneous (i.e., Ω(αx) = αΩ(x) for α ∈ C and x ∈ X) and there is a constant C so that: aΩx − Ω(ax) Ca∞ xX , a ∈ ∞ , x ∈ X. (11.3) X It is easy to show that a centralizer is quasilinear and so induces a self-extension dΩ X which is a lattice self-extension. Let us say that Ω is real if Ω(x) is real-valued whenever x is real-valued (in particular, this holds if we form a lattice self-extension of the real space X and then complexify). In general Ω is equivalent to Ω1 + iΩ2 for suitable real centralizer Ω1 , Ω2 . Now a key idea (and a rather simple calculation) is that every centralizer on X lifts to a centralizer on 1 : P ROPOSITION 11.4 ([53]). Let Ω be a real centralizer on X. Then there is real centralizer Ω on 1 such that for a suitable constant C we have ∗

Ω xx − x ∗ Ω(x) CxX x ∗ ∗ , x ∈ X, x ∗ ∈ X∗ . 1 X ∞

For u ∈ c00 we deﬁne Φ(u) = k=1 (Ω (u))k . From the fact that Ω is a centralizer it follows that the series must converge and that Φ : c00 → C is quasilinear on 1 and takes real sequences to R. Furthermore we have: ∗ ! " Φ xx − Ω(x), x ∗ Cxx ∗ , x ∈ X, x ∗ ∈ X∗ . (11.4) We thus are led back to the problem of characterizing all quasi-linear maps on 1 or alternatively all minimal extensions of 1 (see [60].) If we compare (11.4) and (11.2) it is apparent that we can represent dΩ X in the form dXθ for suitable X0 , X1 and 0 < θ < 1 if we show that Φ is equivalent to ΦX1 − ΦX0 . The main result we need is as follows: + → R is T HEOREM 11.5. Suppose 0 < ε < 1; then there is a constant C so that if Φ : c00 a map satisfying: + (1) Φ(αu) = αΦ(u), α 0, u ∈ c00 . + (2) |Φ(u + v) − Φ(u) − Φ(v)| (1 − ε) log 2(u1 + v1 ), u, v ∈ c00 , then there exists a Banach sequence space X so that

Φ(u) − ΦX (u) − ΦX∗ (u) Cu1 , u ∈ c+ . 00

Now if X is super-reﬂexive Theorem 11.5 can be applied to show (see [55]): T HEOREM 11.6. Let X be a super-reﬂexive Banach sequence space and suppose Ω is a real centralizer on X. Then for some c > 0 there exist super-reﬂexive Banach sequence 1/2 1/2 spaces X0 , X1 so that X = X1/2 = X0 X1 and the derived space dX1/2 is induced by a centralizer Ω equivalent to cΩ, i.e., Ω (x) − cΩ(x) CxX , x ∈ X. X In particular the lattice self-extension dΩ X is isomorphic to the derived space dX1/2 .

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1167

We stress that this entire program can also be carried for function spaces and it is in this form that it is developed in [53] and [55]. 12. Commutator estimates and their applications We now revisit Proposition 10.1, which so far we have not exploited. Initially we continue to treat sequence spaces for technical reasons, but in fact our main interest lies in similar calculations for function spaces. Let us suppose X0 and X1 are two reﬂexive Banach sequence spaces, and that T : Xj → Xj is a bounded operator for j = 0, 1. In this case the interpolation procedure described in the preceding section coincides with complex interpolation. Let 0 < θ < 1 we have the estimate for Xθ = X01−θ X1θ . [T , Ω]x CxX . θ X θ

Suppose x, x ∗ ∈ c00 ; then we have ! " ! " Ω(x), T ∗ x ∗ − Ω(T x), x ∗ CxX x ∗ ∗ . θ X θ

Noting that Ωx ∈ c00 as well, we can apply 11.2 to obtain

Φ x.T ∗ x ∗ − Φ T x.x ∗ CxX x ∗ ∗ , x ∈ X, x ∗ ∈ X∗ , θ X θ

where Φ = ΦX1 − ΦX0 . Noting that Φ is quasilinear we obtain the following estimate:

Φ x.T ∗ x ∗ − T x.x ∗ CxX x ∗ ∗ . θ X θ

We can do slightly better than this by exploiting the Krivine theorem (see [72]) that T actually maps Xj (2 ) to Xj (2 ). We ﬁnally obtain: L EMMA 12.1. Suppose X0 , X1 are reﬂexive Banach sequence spaces and T ∈ L(X, X). If 0 < θ < 1 then there is a constant C so that if x1 , . . . , xn , x1∗ , . . . , xn∗ ∈ c00 then if Φ = ΦX1 − ΦX0 . N N ∗ ∗ ∗ xn .T xn − T xn .xn C xn Xθ xn∗ X . Φ θ n=1

(12.1)

n=1

These calculations are carried out in [53] for function spaces, although the strengthening by using Krivine’s theorem has not been observed before. In order to push these estimates further we now suppose T is bounded on p0 and p1 where p0 < p1 . If we ﬁx p0 < p < p1 we can obtain (12.1) with Φ(u) =

p0−1

− p1−1 Λ(u) = p0−1

− p1−1

∞ n=1

un log |un |.

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However more can be achieved if we apply Boyd’s theorem for sequence spaces (Theorem 7.3) and note that T is bounded on any r.i. sequence space whose Boyd indices obey p0 < pX < qX < p1 . Thus we can consider any pair X1 , X0 of r.i. sequence spaces satisfying these conditions and X01−θ X1θ = p . We then get a family of estimates. In fact it sufﬁces to consider Orlicz sequence spaces and we must consider the family of quasilinear maps given by Φ(F ; u) =

∞

un F log |un | ,

F ∈ Lip1 ,

n=1

where Lip1 is the family of Lipschitz maps F : R → R with Lipschitz constant one. We then get a uniform estimate. Let us replace Lip1 by the space of all Lip1,c of all F ∈ Lip1 so that F is compactly supported (i.e., F is constant on (−∞, −a] and on [a, ∞) for some a > 0. For such F we have the advantage that Φ(F ; u) is deﬁned for all u ∈ 1 . Then we can deﬁne a quasisym Banach space h1 (the symmetric Hardy sequence space) as the space of all sequence ∞ ξ = (ξn )n=1 such that ξ h1sym =

∞

|ξk | + sup Φ(F ; ξ ) < ∞.

n=1

F ∈Lip1,c

It is clear that ξ ∈ h1sym implies ξ ∈ 1 and that ∞ n=1 ξn = 0. A convenient description of 1 ∞ hsym is as follows. Suppose ξ ∈ 1 . Let (λn )n=1 be a sequence such that |λn | is monotonically decreasing and such that if α = 0 the sets {n: λn = α} and {n: ξn = α} have the same cardinality. Then ξ ∈ h1sym if and only if ∞ |λ1 + · · · + λn | n=1

n

< ∞.

(12.2)

We refer to [53] and [54] for details. This equivalence actually hinges on replacing Orlicz spaces by Lorentz-type spaces in the above argument. One can then show that the above ideas yield the following theorem: T HEOREM 12.2. Suppose 1 < p0 < p < p1 < ∞ and 1/p + 1/q = 1. Suppose that T : pj → pj is bounded for j = 0, 1. Then the bilinear form BT (x, x ∗ ) = x.T ∗ x ∗ − T x.x ∗ is bounded from p × q to h1sym . Although we have given our exposition in terms of sequence spaces, the most interesting applications of these ideas are found with function spaces, where essentially the same steps can be made (with some annoying technicalities). Let us consider a measure space (K, μ) where K is a Polish space and μ is a σ -ﬁnite non-atomic Borel measure with either

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1 (K, μ) to μ(K) = 1 or μ(K) = ∞. We deﬁne the symmetric Hardy function space Hsym be the space of f ∈ L1 (K, μ) such that

f Hsym = 1

|f | dμ + sup K

F ∈Lip1,c K

|f |F log |f | dμ < ∞.

1 is a quasi-Banach space of functions. A description such as (12.2) can be As before Hsym 1 if and only if !f, 'f ∈ H 1 . If f ∈ L (K , μ ) and obtained. Let us note that f ∈ Hsym 0 1 1 sym g ∈ L0 (K2 , μ2 ) are two measurable functions we write f ∼ g if whenever B is any Borel 1 if and subset of C \ {0} we have μ1 (f1 ∈ B) = μ2 (f2 ∈ B). Let us note that f ∈ Hsym 1 only if !f, 'f ∈ Hsym . Therefore it sufﬁces to consider real functions. If f ∈ L1 (K, μ) is real then we may ﬁnd a function fd : R → R so that fd is decreasing and non-positive on (−∞, 0), decreasing and non-negative on (0, ∞), and such fd ∼ f . That is, if B is any Borel subset of R \ {0} we have μ(f ∈ B) = m(fd ∈ B) where m denotes either Lebesgue measure on R. Now let

M(t) =

t −t

fd (s) ds,

t > 0.

1 is equivalent to: Then f ∈ Hsym

0

∞

|M(t)| dt < ∞. t

(12.3)

The continuous analogue of Theorem 12.2 is given by: T HEOREM 12.3. Suppose 1 < p0 < p < p1 < ∞ and 1/p + 1/q = 1. Suppose that T : Lpj (K, μ) → Lpj (K, μ) is bounded for j = 0, 1. Then the bilinear form BT (f, g) = 1 (K, μ). f.T ∗ g − Tf.g is bounded from Lp × Lq to Hsym Let us interpret this result by considering K = T and dμ = (2π)−1 dθ . It is natural to consider the Riesz projection (or equivalently the Hilbert transform) given by Rf (θ ) ∼

fˆ(n)einθ ,

n0

where fˆ(n) =

dθ . f eiθ e−inθ 2π T

Since R is bounded on Lp for 1 < p < ∞ we can apply Theorem 12.3 when p = q = 2. Note that R ∗ is the Banach space adjoint of R (not the Hilbert space adjoint). Suppose

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f ∈ H2 and g ∈ H2,0 (i.e., g ∈ H2 with g(0) = 0). Then R ∗ g = 0 and Rf = f . Hence we obtain an estimate fgHsym Cf 2 g2 . But by standard factorization this reduces to 1 f Hsym Cf 1 , 1

f ∈ H1,0.

In particular if f ∈ H1,0 then !f satisﬁes (12.3). This result is, in fact, the main part of a theorem of Ceretelli [19], proved independently somewhat later by Davis [34]. T HEOREM 12.4. Let f ∈ L1 (T) be real-valued. Then there exists g ∈ H1,0 (T) with 1 . !g ∼ f if and only if f ∈ Hsym Next we consider non-commutative analogues of the ideas developed above. Suppose H is a separable Hilbert space. If X is a symmetric Banach sequence then we denote by CX the space of all operators T whose singular numbers sn (T ) satisfy

∞ T CX = sn (T ) n=1 X < ∞. In the case X = 1 we obtain the trace-class C1 . Much of the foregoing theory can be generalized using the fact that [CX0 , CX1 ] = CX1−θ Xθ if X0 , X1 are reﬂexive Banach sequence spaces. Probably the most interesting ap0 1 plication is to theory of commutators (or traces). A trace τ on a two-sided ideal of compact operators J is any linear map such that τ (AB) = τ (BA) for all a ∈ J and B ∈ B(H ). Let us deﬁne the commutator subspace Comm J to be the space of all A ∈ J so that τ (A) = 0 for all traces τ on J . Then Comm J is the linear span of all commutators [A, B] = AB − BA for a ∈ J and B ∈ B(H ). The problem of identifying Comm C1 goes back to [89]. It was shown by Gary Weiss in 1980 [109] (see also [110]) that Comm C1 does not coincide with {T ∈ C1 : tr T = 0} or, equivalently, that there exist discontinuous traces on C1 . The precise identiﬁcation, however, requires the non-commutative analogue of h1sym and this was done by interpolation-style arguments in [54]. We deﬁne the eigenvalue sequence (λn (T ))∞ n=1 for a compact operator T to be the sequence of non-zero eigenvalues of T repeated according to algebraic multiplicity, completed by zeros if there are only ﬁnitely many, and arranged so that (|λn (T )|)∞ n=1 is decreasing. There is some possible ambiguity here if T has two different eigenvalues with the same absolute value, but this does not cause problems. T HEOREM 12.5. Suppose T ∈ C1 . Then T ∈ Comm C1 if and only its eigenvalue sequence 1 (λn (T ))∞ n=1 ∈ hsym or equivalently ∞ |λ1 (T ) + · · · + λn (T )| n=1

n

< ∞.

It was shown in [54] that if T ∈ Comm J then T is the sum of at most six commutators. This theorem has recently been improved dramatically following work of Dykema, Figiel,

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Wodzicki and Weiss [37] (see also [59] and [38]) where the case of general ideals is treated; in particular it is shown in [37] that three commutators sufﬁce in Theorem 12.5. One can introduce a quasi-normed analogue of h1sym denoted by Ch1 := Comm C1 . It can be shown that this is made into a quasi-Banach space by the quasi-norm:

∞ T C h1 := T C1 + λn (T ) n=1 hsym . 1

1 and Ch are all examples of logconvex We remark that the quasi-Banach spaces h1sym , Hsym 1 spaces. A quasi-Banach space X is logconvex if it satisﬁes an estimate of the type n n 1 xk C xk log xk k=1

k=1

n

k=1 xk = 1.

whenever

One can apply this to prove results of the following type:

T HEOREM 12.6 ([54]). Suppose (An ) is a sequence of trace-class operators and Bn is a sequence of bounded operators. Suppose ∞

An C1 Bn 1 + log+

n=1

Then

1 An C1 Bn

< ∞.

∞

n=1 [An , Bn ] ∈ Comm C1 .

Now suppose X is any symmetric sequence space. Then the entropy function ΦX obeys an estimate ΦX (u) Cu

h1sym ,

u ∈ h1sym .

We can then deﬁne ΦX (T ) := ΦX ((λn (T )∞ n=1 ) for T ∈ C1 . Suppose S, T ∈ C1 . Consider the operator: A=

S+T 0 0

0 −S 0

0 0 −T

.

Then A is the sum of two commutators and it may shown that we have an estimate:

AC h1 C SC1 + T C1 . This leads quickly to an estimate:

ΦX (S + T ) − ΦX (S) − ΦX (T ) C SC + T C . 1 1 Thus ΦX is quasi-linear on C1 and induces a minimal extension of C1 .

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We close by remarking that it would be natural to attempt to classify all minimal extensions of C1 in a similar way to Theorem 11.5 (for 1 ). However it is not clear how to do this.

Acknowledgement We would like to thank Michael Cwikel for some helpful comments.

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CHAPTER 27

Probabilistic Limit Theorems in the Setting of Banach Spaces M. Ledoux Institut de Mathématiques, Université Paul-Sabatier, 31062 Toulouse, France E-mail: [email protected]

J. Zinn Department of Mathematics, Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . 2. Almost sure limit theorems . . . . . . . . . . . . 3. The central limit theorem and weak convergence 4. Bootstrap and empirical processes . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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. 1179 . 1180 . 1188 . 1192 . 1198

Abstract We survey the developments of Banach space techniques in the context of classical limit theorems in probability theory. Weak convergence and the relation to type and cotype, symmetrization techniques as well as the exponential bounds parts of the concentration phenomenon for product measures are the highlights of the story. Applications to empirical measures and the bootstrap complete the exposition.

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2 Edited by William B. Johnson and Joram Lindenstrauss © 2003 Elsevier Science B.V. All rights reserved 1177

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1. Introduction Throughout this chapter, B will denote a real separable Banach space with norm · , and topological dual space B . The separability assumption is most convenient to avoid a number of measurability questions. Complex Banach spaces are treated similarly. The basic object of investigation is a Borel random variable X on some probability space (Ω, A, P) with values in B, and a sequence (Xn )n∈N of independent copies of X. For each n 1, set Sn = X1 + · · · + Xn . Classical probability theory on R or Rk is mostly concerned with the limiting behaviour of the partial sum sequence (Sn )n1 . The most important and famous results are the (strong) law of large numbers (LLN), the central limit theorem (CLT) and the law of the iterated logarithmic (LIL) which, for real-valued random variables, may be summarized in the following way. (We refer to [32] for the history of these results.) Let X be a real-valued random variable. – The sequence (Sn /n)n1 converges almost surely to E(X) if and only if E(|X|) < ∞ (we then say that X√satisﬁes the LLN, or strong LLN). – The sequence (Sn / n)n1 converges in distribution (to a normal random variable G) if and only if E(X) = 0 and σ 2 = E(X2 ) < ∞ (and in this case, G is centered with variance σ 2 ) (we then say that X satisﬁes the CLT). – Deﬁne, on R+ , the function LLx = log log x if x e, and LLx = 1 if x < e, and set an = (2nLLn)1/2 for every n 1. The sequence (Sn /an )n1 is almost surely bounded if and only if E(X) = 0 and σ 2 = E(X2 ) < ∞, and in this case, lim inf n→∞

Sn = −σ an

and

lim sup n→∞

Sn = +σ an

with probability 1. Moreover, the set of limit points of the sequence (Sn /an )n1 is almost surely equal to the interval [−σ, +σ ] (we then say that X satisﬁes the LIL). With the exception of the last statement on the LIL these statements may be shown to easily extend to ﬁnite-dimensional random variables, with the obvious modiﬁcations. The deﬁnitions of these basic limit theorems extend to random variables taking values in a inﬁnite-dimensional real separable Banach space B. For example, weak convergence in the central limit theorem has to be understood as weak convergence in the space of Borel probability measures on the complete separable metric space B. For the LIL, one has to distinguish between a bounded form (the sequence (Sn /an )n1 is almost surely bounded in B), and a compact form (the sequence (Sn /an )n1 is almost surely relatively compact in B). In the latter case, it may be shown, completely generally [21], that the set of limit points of the sequence (Sn /an )n1 is a compact convex symmetric set in B (the unit ball of the reproducing kernel Hilbert space associated to the covariance structure of the random variable X). Moment conditions on the law of X fully characterize the preceding limit theorems in ﬁnite dimension. However, as it was soon realized, this is no longer true in inﬁnite dimension. At this point emphasis was put on understanding what kind on conditions on the space can ensure an extension of the ﬁnite-dimensional statements, and what new descriptions are available in this setting. In the ﬁrst part of this survey, we describe the almost sure limit

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theorems (LLN and LIL). As a main observation, it was established, as a consequence of deep exponential bounds, which are parts of the concentration of measure phenomenon for products measures, that the almost sure statements actually reduce to the corresponding ones in probability or in distribution under necessary moment conditions. It is a main contribution of the Banach space approach to realize that moment conditions are actually used to handle convergence in distribution. This fact is further illustrated in Section 3 in the investigation of the classical central limit theorem using type and cotype. In the last paragraph, we describe applications of these ideas and techniques to empirical processes and bootstrap in statistics. For convenience, we mostly refer to the monograph [32] for a complete account on the subject of probability in Banach spaces, and for further references and historical developments. We also refer to [32] for the complete proofs that are only outlined here.

2. Almost sure limit theorems In the early ﬁfties, emphasis was made in trying to understand the strong limit theorems (LLN and LIL) for inﬁnite-dimensional random variables, following early indications by Kolmogorov. In this direction, Mourier and Fortet (cf. [32]) extended the LLN in a statement completely analogous to the ﬁnite-dimensional setting. T HEOREM 1. Let X be a random variable with values in a Banach space B. Then the sequence (Sn /n)n1 converges almost surely to E(X) if and only if E(X) < ∞. Here, when E(X) < ∞, the expectation E(X) has to be understood as the element of B such that ξ, E(X) = E(ξ, X) for every ξ ∈ B . The modern proof (see [14]) of Theorem 1 is rather straightforward. P ROOF. The necessity of the moment condition E(X) < ∞ is proved as in the real case with the Borel–Cantelli lemma. Assume thus that E(X) < ∞. Without loss of generality, we can then assume that E(X) = 0. For each ε > 0, let then Y be a centered step random variable (taking ﬁnitely many values only) in B such that E(X − Y ) ε. Consider independent copies (Yn )n∈N of Y , and for every n 1, set Tn = Y1 + · · · + Yn . By the ﬁnite-dimensional LLN, 1 Tn = 0 n→∞ n lim

(1)

almost surely. On the other hand, by the triangle inequality, 1 1 Sn − Tn Xi − Yi , n n n

i=1

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and by the LLN on the line applied to X − Y , with probability 1, n

1 Xi − Yi = E X − Y ε. n→∞ n

lim

(2)

i=1

Summarizing (1) and (2), lim sup n→∞

1 Sn ε n

almost surely. Since ε > 0 is arbitrary, the conclusion follows.

Soon after Theorem 1, research concentrated for some time on related forms of the strong LLN in Banach spaces, in particular the so-called Kolmogorov LLN that states that if (Yi )i∈N are independent, but not necessarily identically distributed, mean-zero real random variables such that 1

E Yi2 < ∞, i2 i

then the sequence n1 ni=1 Yi converges almost surely to 0. Beck’s fundamental discovery in 1962 [4] was that the extension of Kolmogorov’s LLN could not take place in any inﬁnite-dimensional Banach space. More importantly, he characterized, through a geometric condition known as Beck’s convexity, those Banach spaces B for which every sequence (Yi )i∈N of independent mean-zero B-valued random variables with supi Yi ∞ < ∞ satisﬁes the LLN, i.e., 1 Yi → 0 almost surely. n n

i=1

In particular, spaces such as 1 , L1 , c0 , C(K) do not satisfy Beck’s condition, and therefore Kolmogorov’s LLN. Beck’s convexity condition was then recognized in the early seventies through the basic work of Pisier [38] as equivalent to a probabilistic type condition. This condition, as well as a related “dual” condition, were introduced independently by Maurey [35] and HoffmanJørgensen [12] (see [18,36] for the deﬁnitions). Lp -spaces are of type min(p, 2) and of cotype max(p, 2). By Kwapie´n’s theorem [24], Banach spaces of both type 2 and cotype 2 are isomorphic to Hilbert spaces. Pisier’s discovery in 1973 was that Beck’s convexity condition for a Banach space B is equivalent to the fact that B is of type p for some p > 1. Now, the probabilistic type condition is easily seen to be well-adapted to the investigation of the LLN and resulted in the following theorem due to Hoffman-Jørgensen and Pisier [15] that extends Beck’s result. It should be noted that together with [20] this last paper brought to the forefront the usefulness of symmetrization by Rademacher variables (εi )i∈N (extensively developed in [9], e.g.). This still has a great impact.

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T HEOREM 2. A Banach space B is of type p > 1 if and only if for every sequence (Yi )i∈N of independent mean-zero B-valued random variables such that 1

E Yi p < ∞, p i i

one has 1 Yi → 0 n n

i=1

almost surely. Up to this point, the results developed as natural extensions, in spaces with some type, of the classical theorems in ﬁnite dimension. An important step was performed with the contribution of Yurinskii [49,50] (whose interests were in exponential inequalities), and applications of his ideas by Kuelbs [22] (for the LIL) and in [23] (for the LLN). With these results, the Banach space conceptualizations started to have an important impact on the probabilistic analysis. Given Y1 , . . . , Yn independent integrable Banach space valued random variables, Yurinskii’s observation was that the norm of the sum S = ni=1 Yi centered at its mean may be written as a sum of martingale differences n

S − E S = di i=1

with respect to the ﬁltration Fi = σ (Y1 , . . . , Yi ), i = 1, . . . , n (i.e., E(di |Fi−1 ) = 0), with the property that, for every i = 1, . . . , n,

|di | Yi + E Yi .

(3)

In a sense, S − E(S) is as good as the sum ni=1 Yi , so that, provided E(S) is under control, the classical one-dimensional results should apply similarly. Using this representation the following was proved in [23]. T HEOREM 3. Let (Yi )i∈N be a sequence of independent B-valued random variables such that, for some p, 1 p 2, 1

E Yi p < ∞. p i i

Then 1 Yi → 0 n n

i=1

almost surely

Probabilistic limit theorems in the setting of Banach spaces

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if and only if 1 Yi → 0 in probability. n n

i=1

P ROOF. Under the hypothesis, Yi /i → 0 almost surely, so that we may assume that Yi ∞ i for every i 1. We then symmetrize so that blocking is easily managed. Here is one way to symmetrize (see immediately after the proof for symmetrization in Lp ). Set Sn = Y1 + · · · + Yn , n 1. Clearly, Sn /n → 0 almost surely if (Sn − Sn )/n → 0 almost surely, where Sn /n is formed from an independent copy of the original (Yi ), and Sn /n → 0 in probability. Indeed, if (Sn − Sn )/n → 0 almost surely, then, by Fubini’s theorem, we may ﬁnd ω so that Sn /n − Sn (ω )/n → 0 almost surely which, in particular, implies this last quantity goes to zero in probability. But, then since Sn /n → 0 in probability, Sn (ω )/n → 0. Hence, Sn /n → 0 with probability 1. We may thus reduce ourselves to the case of independent symmetric random variables Yi . Assume thus that Sn /n → 0 in probability. Then, as a consequence of the Hoffman-Jørgensen inequalities ([14], [32], Chapter 6),

1 E Sn → 0 n as n → ∞. Moreover, by the maximal inequalities for sums of independent symmetric random variables, it is enough to show that n

1 2n

2

Yi → 0

i=2n−1 +1

with probability 1. By Yurinskii’s result, for every ε > 0, and every n, 2n P n−1 i=2

+1

Yi − E n

4 ε2 22n

2 i=2n−1 +1

n

2 i=2n−1 +1

Yi εn

4 E Yi 2 2 ε

n

2 i=2n−1 +1

1 E Yi p . p i

The conclusion then immediately follows from the Borel–Cantelli lemma.

The important feature of Theorem 3 is that, under the convergence in probability of the partial sum sequence, no assumption has to be imposed on the Banach space. With respect to Theorem 2, the type condition is actually only used in order to achieve this convergence in probability. The argument relies on a basic symmetrization procedure, classical in the area. Assume indeed the Yi ’s to be centered and denote by (Yi )i∈N an independent copy of

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M. Ledoux and J. Zinn

the sequence (Yi )i∈N . Then, by Jensen’s inequality and the triangle inequality, for every n, and p 1, p n n p

E Yi − Yi Yi E i=1 i=1 p p n n

p =E εi Yi − Yi 2 E εi Yi , i=1

(4)

i=1

where the Rademacher sequence (εi )i∈N is independent from the previous sequences. Using the type inequality conditionally on the Yi ’s, for some constant C depending only on B, n n p

E Yi 2p C E Yi p . i=1

i=1

Therefore, if 1

E Yi p < ∞, p i i

then n1 ni=1 Yi → 0 in Lp by Kronecker’s lemma, and thus in probability. Another aspect of the preceding proof is the use of the Hoffman-Jørgensen inequalities. Again, these may be considered as a consequence of the Banach space conceptualization. In its early formulation, the Hoffman-Jørgensen inequality [12,13] indicates that whenever Y1 , . . . , Yn are independent symmetric random variables with values in B, for every s and t > 0, 2 P S s + 2t P max Yi s + 4 P S t , 1in

(5)

where S = ni=1 Yi . Inequality (5) is one amongst a variety of similar inequalities. Typically, it may be used to show that if (an )n∈N is a sequence of positive numbers increasing to inﬁnity and if (Yi )i∈N is a sequence of independent symmetric random variables, then, whenever the sequence n 1 Yi , an

n 1,

i=1

is bounded or converges to 0 in probability in B, the sequence n 1 Yi I{Yi Can } , an i=1

n 1,

Probabilistic limit theorems in the setting of Banach spaces

1185

is bounded or converges to 0 in Lp (B) for any p > 0 (cf. [14,32] and the references therein). The main consequence of Theorem 3 is that the classical probabilistic limit theorems have to be investigated, in a Banach space setting, in two distinct steps. Namely, under the classical moment conditions, prove convergence in probability or in distribution with the help of the type (or cotype) conditions. The resulting statements thus only hold for classes of Banach spaces with the appropriate geometric conditions. One typical and fundamental example of this situation is the central limit theorem which we investigate in the next section. Once the convergence in probability is achieved, or simply assumed, prove, in any Banach space, the corresponding almost sure statement. The lesson learned for limit theorems in Banach spaces is that moment conditions are needed to ensure convergence in probability and that, more or less, convergence in probability then always implies almost sure convergence. Provided with these fundamental observations, we turn to some more reﬁned almost sure statements, such as the law of the iterated logarithm (LIL). As in Theorem 1, let X be a B-valued random variable, and let (Xn )n∈N be a sequence of independent copies of X. For each n 1, Sn = X1 + · · · + Xn . Recall also an = (2nLLn)1/2 . As expected from the preceding conclusions, and using exponential bounds on Yurinskii’s martingale, Kuelbs showed in 1977 [22] that the sequence (Sn /an )n1 is relatively compact in B as soon as E(X2 ) < ∞ and Sn /an → 0 in probability. Although this result was a powerful extension of the classical LIL, it was not entirely satisfactory since the moment condition E(X2 ) < ∞ was known not to be necessary for the LIL. The necessary moment condition on the law of X to satisfy the LIL in an inﬁnitedimensional Banach space B actually splits into two parts: ﬁrst, for every linear functional ξ ∈ B , the scalar random variable ξ, X satisﬁes the LIL, and thus E(ξ, X) = 0 and E(ξ, X2 ) < ∞. Secondly, if the sequence (Sn /an )n1 is almost surely bounded, so is the sequence (Xn /an )n1 , and thus, by the Borel–Cantelli lemma, E(X2 /LLX) < ∞. The occurrence of weak moments with respect to the usual norm conditions made this investigation signiﬁcantly harder than most of the previous results and showed that it lays at a much deeper level. It was thus open for some time to know whether these necessary moment conditions, together with the control of the sequence (Sn /an )n1 in probability, were also sufﬁcient for the LIL to hold. This conjecture was ﬁrst settled in Hilbert spaces [11] using the scalar product structure, and then further extended in smooth spaces (uniformly convex spaces [27]). The ﬁnal breakthrough was accomplished with the help of the isoperimetric and concentration ideas. The Gaussian isoperimetric inequality may be considered at the origin of this development (cf. [32], Chapter 3). This inequality in particular implies that if G is a Gaussian random vector with values in B, for every t 0,

2 2 P G − E G t e−t /2σ , where σ 2 = supξ 1 E(ξ, G2 ). In particular,

E exp αG2 < ∞

(6)

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M. Ledoux and J. Zinn

if and only if α < 1/2σ 2 . This fundamental Gaussian property led to a ﬁrst result [30] together with a Gaussian randomization argument put forward in [9] (and close in spirit to the proof of Lemma 14 below). This was actually the starting point of the deep investigation by Talagrand of isoperimetric and concentration inequalities in product spaces, with applications to a number of various areas in both Banach space and probability theory [45,46] (cf. [43]). In particular, this approach yields optimal extensions of the classical exponential bounds on sums of independent random variables in the spirit of the Gaussian inequality (6). Let, for example, Y1 , . . . , Yn be independent mean-zero B-valued random variables such that Yi ∞ C for every i = 1, . . . , n. Set S = Y1 + · · · + Yn and deﬁne σ 2 = supξ 1 ni=1 E(ξ, Yi 2 ). Then, for every t 0,

Ct t P S − E S t K exp − , log 1 + 2 KC σ + CE(S)

(7)

where K > 0 is some numerical constant. This type of inequality is, using the centering factor E(S), the complete analogue of the classical Bennett inequalities for sums of independent real random variables, as well as the natural extension of (6). It describes the Gaussian behavior of sums of independent random variables for the small values of t, and the Poissonian behavior for the large values. A weaker form of (7) goes back to the early paper [44] by Talagrand, which led the author to a complete study of the concentration phenomenon for product measures [45]. The precise form of (7) is taken from the more recent work [47]. An alternate simpler approach based on logarithmic Sobolev inequalities has been recently proposed in [28] (see also [29] and the references therein for further developments on sharper numerical constants). Together with such an estimate, it is easy to characterize the LIL in Banach spaces (see [30]). T HEOREM 4. Let X be a random variable with values in a Banach space B. The sequence (Sn /an )n1 is almost surely bounded if and only if it is bounded in probability, E(X2 /LLX) < ∞, and for every linear functional ξ ∈ B , E(ξ, X) = 0 and E(ξ, X2 ) < ∞. The sequence (Sn /an )n1 is almost surely relatively compact if and only Sn /an → 0 in probability, E(X2 /LLX) < ∞, and the family of random variables ξ, X2 , ξ 1, is uniformly integrable. P ROOF. Let us concentrate only on the bounded form of the LIL. The conditions have been seen to be necessary. As in the LLN, by a classical blocking argument, it is enough to show that 2n 1 Xi < ∞ sup n a2n i=1

(8)

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1187

almost surely. One can show that, under the integrability E(X2 /LLX) < ∞, condition −k there exists a sequence of integers (kn )n∈N such that n 2 n < ∞ and

k n (r) X n a2n < ∞, P 2

n

r=1

(r)

where X2n is the r-th largest element of the sample (Xi )1i2n . In particular, it is enough to prove (8) with the Xi ’s replaced by Yi = Xi I{Xi a2n / kn } , i = 1, . . . , 2n . Since the sequence (Sn /an )n1 is bounded in probability, by the Hoffman-Jørgensen’s inequalities, for some ﬁnite constant M, 2n 1 sup E Yi M. n a n 2 i=1

On the other hand, by the closed graph theorem,

σ 2 = sup E ξ, X2 < ∞. ξ 1

Apply then (6) to S = Y1 + · · · + Y2n to get, for t = T a2n , T > 0, 2n 2n P Yi − E Yi T a2n i=1 i=1 M Mkn log 1 + 2 K exp − . K (σ kn /LL2n ) + 1 Since n 2−kn < ∞, the preceding exponential bound is the general term of a summable sequence for any T large enough. The conclusion follows. As for the LLN, it is an easy exercise to see that in a type 2 Banach space, whenever X has mean zero and E(X2 /LLX) < ∞, then Sn /an → 0 in probability. As a corollary to Theorem 4, one thus obtains quite a nice characterization of the LIL in type 2 Banach spaces only relying on moment conditions on the law of X. C OROLLARY 5. Let X be a mean-zero random variable with values in a type 2 Banach space B. Then the sequence (Sn /an )n1 is almost surely relatively compact if and only if E(X2 /LLX) < ∞ and the family of random variables ξ, X2 , ξ 1, is uniformly integrable. Again, the type property on the Banach space B is only use to ensure the convergence in probability, or weak convergence. As we will now develop it, type and cotype are actually intimately connected with weak convergence, and in particular the central limit theorem.

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3. The central limit theorem and weak convergence It was soon realized, at the beginning of probability in Banach spaces, that the classical moment conditions E(X) = 0 and E(X2 ) < ∞ are neither sufﬁcient, nor necessary for a random variable X to satisfy the central limit theorem in an arbitrary Banach space B. They are actually sufﬁcient (only) in type 2 spaces, and necessary (only) in cotype 2 spaces. However, in these ﬁrst considerations of the CLT in Banach spaces, the interplay, which we now understand well, between the geometry of the space and the conditions on the random variable was still unclear. Results analogous to the classical results were proved in special spaces such as Hilbert space and for specialized random variables with values in spaces with a basis. Results were also proved in the form of limit theorems for stochastic processes. Some of the early highlights were Varadhan’s central limit theorem in Hilbert space [48], the results of Dudley and Strassen [6] on processes in C[0, 1] and Dudley’s work on empirical processes [5]. For more details on the historical development, see [32], p. 296. In a conference in Durham, England, 1975, Dudley posed the question: in which (separable) Banach spaces do the classical conditions of mean zero and ﬁnite variance (E(X2 ) < ∞) always imply that the CLT holds? The question was answered at this conference and resulted in the theorem of Hoffman-Jørgensen and Pisier [15] below. T HEOREM 6. A random variable X with values in a type 2 Banach space B satisﬁes the CLT as soon as E(X) = 0 and E(X2 ) < ∞. Conversely, if in a Banach space B, all random variables X such that E(X) = 0 and E(X2 ) < ∞ satisfy the CLT, then B is of type 2. The companion theorem for cotype soon followed [16]. T HEOREM 7. A random variable X with values in a cotype 2 Banach space B satisfying the CLT is such that E(X) = 0 and E(X2 ) < ∞. Conversely, if in a Banach space B, all random variables X satisfying the CLT are such that E(X) = 0 and E(X2 ) < ∞, then B is of cotype 2. To illustrate the idea of these statements, let us outline the proof of Theorem 6. P ROOF OF T HEOREM 6. Assume B is of type 2. By the symmetrization argument (4), for every n 1, n 2

E Xi 4Cn E X2 . i=1

(9)

Probabilistic limit theorems in the setting of Banach spaces

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For ε > 0, let Y be a mean-zero step random variable such that E(X − Y 2 ) ε/4C. Therefore, if (Yi )i∈N is a sequence of independent copies of Y , (9) applied to X − Y yields 2 n 1 sup E √ (Xi − Yi ) ε. n n1 i=1

√ Since Y is ﬁnite-dimensional, it satisﬁes the CLT. The sequence (Sn / n)n1 is thus uniformly close in L2 to a weakly convergent sequence, and thus is tight. Since, by the ﬁnitedimensional CLT, the only possible limit is Gaussian with the same covariance structure as X, X satisﬁes the CLT. Conversely, assuming that every mean-zero random variable X with values in B such that E(X2 ) < ∞ satisﬁes the CLT, we deduce from a closed graph argument that there is a ﬁnite constant C such that 2 n 1

Xi C E X2 . sup E √ n n1 i=1

Applying this inequality to a random variable X taking ﬁnitely many values x1 , . . . , xn , then shows that B is of type 2. Together with Kwapie´n’s isomorphic characterization of Hilbert spaces, one can deduce from Theorems 6 and 7 the following “probabilistic” characterization of Hilbert spaces to which several authors contributed. C OROLLARY 8. A Banach space B is isomorphic to a Hilbert space if and only if the classical moment conditions E(X) = 0 and E(X2 ) < ∞ are necessary and sufﬁcient for a random variable X with values in B to satisfy the CLT. As for the LIL, the strong second moment E(X2 ) < ∞ is not always well-adapted, and one has rather to consider weak moments. In the context of the CLT, an additional necessary condition for a random variable X to satisfy the CLT is the existence of a (centered) Gaussian random vector G with values in B (the limiting distribution) with the same covariance structure as X, that is E(ξ, X2 ) = E(ξ, G2 ) for every ξ in B . In an inﬁnitedimensional context, it is not always true that there exists a Gaussian distribution with a given covariance structure. Actually, Theorems 6 and 7 may be rephrased replacing the central limit property by only the existence of a limiting Gaussian distribution. The proofs are rather similar. In the presence of a limiting Gaussian distribution, the natural necessary moment condition on the norm, in order for a random variable X with values in a Banach space B to satisfy the CLT is that (10) lim t 2 P X t = 0. t →∞

In particular, E(Xp ) < ∞ only for 0 < p < 2. It is then a challenging question to characterize those Banach spaces B in which the preceding condition (10) together with

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M. Ledoux and J. Zinn

the existence of a limiting Gaussian distribution are (necessary and) sufﬁcient for a Bvalued mean-zero random variable X to satisfy the CLT. As shown in [42], Lp -spaces with 2 p < ∞ share this property. If, in a Banach space a (certain) version of an inequality of Rosenthal holds, then [11] the CLT holds if and only if (10) holds and the limiting Gaussian distribution exists (in the space). An attempt is made in [26] to characterize such spaces. These ideas lead to necessary and sufﬁcient conditions for the CLT in spaces of the form Lp (Lq ) [8]. In a general Banach space, one cannot hope for any reasonable description of random variables satisfying the CLT. However, one can give sufﬁcient conditions for some classes of random variables. Let, for example, C(K) be the Banach space of continuous functions on a compact metric space (K, d) equipped with the uniform norm · ∞ . Consider the class of Lipschitz random variables X on C(K), that is such that for some non-negative random variable M, X(s, ω) − X(t, ω) M(ω)d(s, t),

s, t ∈ K,

for all (or almost all) ω. There were many partial steps in the direction of the “ﬁnal” theorems in this direction, with contributions in [6,5] and [7]. A big step was taken in [17] with the following result. T HEOREM 9. Let X be a Lipschitz mean-zero random variable such that E(M 2 ) < ∞. Then, whenever d dG where dG (s, t) = G(s) − G(t)2 , s, t ∈ K, is the L2 -metric of a Gaussian random vector G in C(K), then X satisﬁes the CLT. P ROOF. We follow the approach in [51] which illustrates the type 2 ideas. Consider indeed the canonical injection map j : Lip(K) → C(K). The space Lip(T ) is equipped with the norm xLip = x∞ + sup s=t

|x(s) − x(t)| . d(s, t)

Although C(K) is of no type p > 1, under the hypothesis of the theorem, the linear operator j is actually of type 2. Let indeed x1 , . . . , xn be elements in Lip(K) such that, by homogeneity, ni=1 xi 2Lip = 1. By an elementary comparison (cf. [32]), 2 2 n n 2E , E εi j (xi ) gi j (xi ) i=1

∞

i=1

∞

where gi are independent standard normal variables. Since the xi ’s are element of Lip(K), the Gaussian process = G(t)

n i=1

gi xi (t),

t ∈ K,

Probabilistic limit theorems in the setting of Banach spaces

1191

is such that, for every s, t ∈ K, n xi (s) − xi (t)2 dG (s, t)2 . 2 = G(s) − G(t) E i=1

Classical Gaussian comparison theorems (cf. [32], Chapter 3) then show that 2 n E C, εi j (xi ) ∞

i=1

where C > 0 only depends on G. Therefore, by homogeneity, j is an operator of type 2 in the sense that, whenever x1 , . . . , xn are elements in Lip(K), then 2 n n C E εi j (xi ) xi 2Lip . ∞

i=1

i=1

Mimicking the proof of Theorem 6, we get that 2 n 1

1 √ E √ j (Xi ) CE X2Lip . n n i=1

(11)

∞

One then concludes as in Theorem 6. There is however a little difﬁculty due to the fact that Lip(K) need not be separable. To handle this problem, show that, for every ε > 0, there exists a ﬁnite-dimensional subspace F of C(K) such that if QF is the quotient map C(K) → C(K)/F , the type 2 constant of the operator j ◦ QF is less than ε. Applying (11) to j ◦ QF then easily yields the result. With some further work, the integrability condition on M may be weakened into limt →∞ t 2 P{M t} = 0, provided it is assumed in addition that there is a continuous Gaussian process with the same covariance structure as X. To some extent, the preceding analysis of the classical central limit theorem with Gaussian limits may also be developed similarly in the case of the general central limit theorem for triangular arrays with inﬁnitely divisible limits [1]. Under the assumptions of tightness of the partial sum sequence, limits are identiﬁed more or less as in the scalar case. Type or cotype assumptions are sufﬁcient or necessary for tightness. In particular, stable limits may be characterized through stable type. We refer to the book [2] for an account on the general central limit theorem. There are several other concepts and results from Banach space theory which were crucial to the investigation of stable limit theorems and the CLT for triangular arrays. In particular, Banach spaces in which ∞ is not ﬁnitely representable [37] (of ﬁnite cotype) may be used to investigate Poissonnization and the accompanying laws in inﬁnite dimension [3]. Banach spaces B not containing c0 were characterized by Hoffman-Jørgensen [13] and Kwapie´n [25] as those in which almost sure bounded partial sum sequences (Sn ) of independent symmetric B-valued random variables are almost surely convergent. There was a

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nice interplay between the Banach space and probability communities around stable type. In [34] a representation theorem for stable random variables and vectors (obtained in [33] for the Hilbert case) was extended and used to analyze stable random Fourier series. Let X be a p-stable (0 < p < 2) random vector with values in B with spectral measure m symmetrically distributed on the unit sphere of B (cf. [32], Chapter 5). Denote by (Yj )j ∈N independent random variables distributed as m/|m|. Let furthermore Γj = λ1 + · · · + λj , j 1, where the λi ’s are independent standard exponential random variables and assume the sequences (Yj )j ∈N and (Γj )j ∈N to be independent. Then, the series ∞

−1/p

Γj

Yj

j =1

converges almost surely and is distributed as cp |m|−1/p X for some cp > 0 only depending on p. This representation later allowed Pisier in [39] to obtain an extension of results of Johnson and Schechtman [19] on spaces in which p , 1 p < 2, is not ﬁnitely representable. Since Γj ∼ j by the LLN, Pisier’s idea was to use this representation together with Yurinskii’s inequality to produce a proof similar to the Gaussian proof of Dvoretzky’s theorem based on (5) (cf. [41,43]). 4. Bootstrap and empirical processes Exponential bounds on empirical processes have been proved extremely useful very recently in selection of models in Statistics. To describe one inequality more precisely, let, on some probability space (Ω, A, P), (Xi )i∈N be independent identically distributed random variables with values in some measurable space (S, S) and with common law P . For every n 1, consider the empirical measures 1 δ Xi . n n

Pn =

i=1

Now, let F be a class of measurable functions on S with real values. The theory of empirical processes runs into various measurability questions in which we do not want to enter here. So let us assume for simplicity the class F to be countable. The statistical treatment of empirical measures shows that the unknown law P can be recovered from the observations Pn on some class F , the larger the class, the more accurate the result. In particular, a class F is said to be a Glivenko–Cantelli class if supPn (f ) − P (f ) → 0 F

with probability 1. F is said to be a Donsker class if the sequence

√ n Pn (f ) − P (f ) , f ∈ F , converges in distribution (in a sense to be made precise) to a centered Gaussian process GP indexed by F with covariance P (fg) − P (f )P (g), f, g ∈ F . These deﬁnitions extend the

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classical results of Glivenko and Cantelli, and Donsker for the class of the characteristic functions of the intervals (−∞, x], x ∈ R. In statistical applications however, one is interested in estimates at ﬁnite range, that is on Pn for ﬁxed n. To this end, the exponential bound (7) is of fundamental importance. Assume that |f | C for every f ∈ F and set Zn = sup Pn (f ) − P (f ). f ∈F

Then, for every t 0, Ct nt P Zn − E(Zn ) t K exp − log 1 + 2 , KC σ + CE(Zn )

(12)

where σ 2 = supf ∈F (P (f 2 ) − P (f )2 ) (cf. [47,29]). A similar inequality also holds for E(Zn ) − Zn which thus yields a concentration property. It is very important in statistical applications that (12) holds with E(Zn ) and not a multiple of E(Zn ) as was the case in the earlier bounds [32]. A special class F is given by the example of the family of characteristic functions of Vapnik–Cervonenkis classes of sets. Let S be a set and C be a class of subsets of S. Let A be a subset of S of cardinality k. Say that C shatters A if each subset of A is the trace of an element of C. C is said to be a Vapnik–Cervonenkis class (VC class in short) if there is an integer k 1 such that no subset A of S of cardinality k is shattered by C. Denote by v(C) the smallest k with this property. The class of all interval (−∞, x], x ∈ R, is a VC class with v(C) = 2. The most striking fact about VC classes is that whenever C is a VC class in S and v = v(C), any subset A of S with Card(A) = n v satisﬁes Card(C ∩ A)

en v

v .

(13)

Let now Q be a probability measure on a measurable space (S, S) and consider a VC class C of subsets of S. For any measurable subsets A, B in S, set dQ (A, B) = IA − IB 2 (where the norm is understood with respect to Q). For any ε > 0, let N(ε) = N(C, dQ ; ε) be the minimal number of balls of radius ε in the metric dQ which are necessary to cover C. As a consequence of (13), it was shown by Dudley [5] that the growth of N(ε) as ε goes to zero is controlled by v(C). Indeed, for any ε > 0, 1 . log N(ε) Kv(C) 1 + log ε

(14)

From this result, it is not so difﬁcult to deduce that C is a Donsker class for every probability measure P on (S, S). Rather than to directly prove such a property, let us relate, as in Section 3, the nice limit properties of VC classes to the type type 2 property of a

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certain operator between Banach spaces. Denote by M = M(S, S) the Banach space of all bounded measures μ on (S, S) equipped with the norm μ = |μ|(S). Consider the operator j : M → ∞ (C) deﬁned by j (μ) = (μ(C))C∈C . Denote by T2 (j ) the type 2 constant of j , that is the smallest constant C such that for all μ1 , . . . , μn in M, 2 n n E εi j (μi ) C μi 2 . i=1

C

i=1

The next theorem has been observed by Pisier [40]. T HEOREM 10. For some numerical constant K > 0, # # K −1 v(C) T2 (j ) K v(C). P ROOF. We only show the right-hand side inequality. Let μ1 , . . . , μn in M. To prove the type 2 inequality, we may assume that the measures μi are positive and, by homogeneity, that ni=1 μi 2 = 1. Set Q = ni=1 μi μi . Then Q is a probability measure on (S, S), and we clearly have that

n μi (A) − μi (B)2

1/2 dQ (A, B)

i=1

for all subsets A and B. A classical entropic bound on the Rademacher process n i=1 εi μi (C), C ∈ C (cf. [32], Chapter 11), then shows that 2 2 n n E εi μi = E sup εi μi (C) C∈C i=1 i=1 C ∞ #

1/2 1+K dε 1 + K v(C), log N(ε) 0

where we used (14) in the last step. Since v(C) 1, the claim is established.

One can then deduce that VC classes are uniformly Donsker by the type theory of Section 3. Further and reﬁned limit theorems for empirical processes have been developed, especially in [5] and [9], under various conditions (entropy with bracketing, random entropy, etc.). The following is the Banach space formulation of Theorem 10. T HEOREM 11. Let x1 , . . . , xn be functions on some set T taking the values ±1. Let n r(T ) = E sup εi xi (t) . t ∈T i=1

Probabilistic limit theorems in the setting of Banach spaces

1195

There exists a numerical constant K > 0 such that for every k r(T )2 /Kn, one can ﬁnd m1 < m2 < · · · < mk in {1, . . . , n} such that the set of values {xm1 (t), . . . , xmk (t)}, t ∈ T , is exactly {−1, +1}k . In other words, the subsequence {xm1 , . . . , xmk } generates a subspace isometric to k1 in ∞ (T ). For the proof, apply Theorem 10 to the class C of subsets of {1, 2, . . . , n} of the form i ∈ {1, . . . , n}; xi (t) = 1 ,

t ∈ T.

The functional calculus of probability has also proved useful in other statistical problems, and, in the ﬁnal part of this exposition, we present applications to the bootstrap in statistics. The “bootstrap”, introduced by Efron in 1979, is a resampling method for approximating ω , i = 1, . . . , n, be independent the distribution of various statistics. For every ω ∈ Ω, let X ni and identically distributed random variables with common distribution Pn (ω). Denote, for ω , i = 1, . . . , n, that is each n 1, Pn (ω) the empirical distribution based on X ni 1 δX Pn (ω) = ω . ni n n

i=1

It is then expected that the distribution of statistics ω

ω ,...,X nn n (ω) = Hn X ; Pn (ω) H n1 is ω-almost surely asymptotically close to that of Hn (X1 , . . . , Xn ; P ). This suggestive method has been validated with limit theorems for many particular statistics Hn , improving rates with respect to the classical central limit theorem approximations. Let F = supf ∈F |f |. The following has been obtained in [10]. T HEOREM 12. The conditions sufﬁcient for

F 2 dP < ∞ and P is a Donsker class are necessary and

√ n Pn (ω) − Pn (ω) → G

weakly

ω-almost surely for a centered Gaussian process G independent of ω. Furthermore G = GP . This theorem completely settles (modulo measurability) the question of the validity of the bootstrap for the CLT for empirical processes. The proof of Theorem 12 is based on an almost sure version of the CLT which we state in the setting of Banach space valued random variables for simplicity. In the statement, X is a Banach space valued random variable and g a standard normal random variable independent of X. Accordingly, if (Xi )i∈N , resp. (gi )i∈N , is a sequence of independent copies of X, resp. g, the two sequences are understood to be independent (i.e., constructed on different probability spaces). The following result is due to Talagrand and the authors [31].

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T HEOREM 13. Let X be a random variable with values in a Banach space B and let g be a standard normal random variable independent of X. The following are equivalent. (i) E(X2 ) < ∞ and X satisﬁes the CLT with limiting Gaussian distribution G. (ii) For almost every√ω of the probability space supporting the Xi ’s, the sequence ( ni=1 gi Xi (ω)/ n)n1 distribution. converges in √ In either case, the limit of ( ni=1 gi Xi (ω)/ n)n1 does not depend on ω and is distributed as G. P ROOF. We concentrate on the implication (i) ⇒ (ii). It is not difﬁcult to see that it is enough to establish the following lemma. An alternate proof may be found in [32]. L EMMA 14. If E(X2 ) < ∞, almost surely 1 lim sup √ Eg n n→∞

n n 1 gi Xi 4 lim sup √ E gi Xi , n n→∞ i=1

i=1

where we denote by Eg partial integration with respect to the sequence (gi )i∈N . P ROOF. We only outline one argument. Let n 1 M = lim sup √ E gi Xi n n→∞ i=1

assumed to be ﬁnite. By the Borel–Cantelli lemma, it sufﬁces to show that for every ε > 0, P n

sup

2n−1
1 √ Eg k

k gi Xi 2(2M + 5ε) < ∞. i=1

By Lévy’s inequalities for symmetric random vectors, and a simple truncation argument under the hypothesis E(X2 ) < ∞, it is enough to show that n

2n P Eg gi Ui 2M + 5ε < ∞,

i=1

where, for every i = 1, . . . , 2n , Ui = Ui (n) = 2−n/2 Xi I{Xi ε√2n } . By Hoffman-Jørgensen’s inequality (5), for every n, 2n 2 2n gi Ui 2M + 5ε P Eg gi Ui M + 2ε , P Eg

i=1

i=1

Probabilistic limit theorems in the setting of Banach spaces

1197

so that, by deﬁnition of M, it ﬁnally sufﬁces to show that 2n 2 2n gi Ui − E gi Ui ε < ∞. P Eg n

i=1

i=1

By Yurinskii’s martingale representation and Chebyshev’s inequality, 2n 2n 2n 4 2

P Eg gi Ui − E gi Ui ε 2 E di . ε

i=1

i=1

i=1

Now, di = E fi |Fi ) − E(fi |Fi−1 ), where 2n 2n gj Uj − Eg gj Uj , fi = Eg j =1

j =1,j =i

i = 1, . . . , 2n . For every i, 0 fi gi Ui . Moreover, by identical distribution, 2n M +ε 1 E(fi ) n E gj Uj 2 2n j =1

for every n large enough. Therefore,

E di2 E fi2 1/2

E gi Ui 3/2 fi

1/2

1/2 M +ε 3 1/2 3 E(fi ) E gi Ui . E gi Ui 2n

One is thus left to show that the series

2−n/2 E X3 I{Xε√2n }

n

is summable which is an easy consequence of the second moment assumption.

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We brieﬂy conclude the proof of Theorem 13. Since X satisﬁes the CLT, gX also satisﬁes the CLT. Choose then, for every k 1, a ﬁnite-dimensional subspace Fk of B such that if Tk = TFk : B → B/Fk is the quotient map, n 1 1 sup √ E gi Tk (Xi ) . k n n1 i=1

Apply Lemma 14 to Tk (X) for every k. There exists Ωk with P(Ωk ) = 1 such that for every ω ∈ Ωk , n

4 1 gi Tk Xi (ω) . lim sup √ Eg k n n→∞ i=1

Let also Ω 0 be the set of full probability obtained when Lemma 14 is applied to X itself. Let Ω 0 = k0 Ωk . If ω ∈ Ω 0 , for each ε > 0, there exists a ﬁnite-dimensional subspace F of B such that if T = TF is the quotient map, 1 lim sup √ Eg n n→∞

n

gi T Xi (ω) ε2 . i=1

Hence, if n n(ε), n √ Pg T gi Xi (ω)/ n ε ε. i=1

√ It follows that the sequence ni=1 gi Xi (ω)/ n, n 1, is tight. The proof is easily completed by identifying the limit.

References [1] A. de Acosta, A. Araujo and E. Giné, On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces, Adv. Probab. 4 (1978), 1–68. [2] A. Araujo and E. Giné, The Central Limit Theorem for Real and Banach Space Valued Random Variables, Wiley (1980). [3] A. Araujo, E. Giné, V. Mandrekar and J. Zinn, On the accompanying laws in Banach spaces, Ann. Probab. 9 (1981), 202–210. [4] A. Beck, A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13 (1962), 329–334. [5] R.M. Dudley, Central limit theorems for empirical measures, Ann. Probab. 6 (1978), 899–929. [6] R.M. Dudley and V. Strassen, The central limit theorem and ε-entropy, Probab. Inform. Theory, Lecture Notes in Math. 89, Springer (1969), 224–231. [7] E. Giné, On the central limit theorem for sample continuous processes, Ann. Probab. 2 (1974), 629–641. [8] E. Giné and J. Zinn, Central limit theorems and weak laws of large numbers in certain Banach spaces, Z. Wahrsch. Verw. Gebiete 62 (1983), 323–354.

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[9] E. Giné and J. Zinn, Some limit theorems for empirical processes, Ann. Probab. 12 (1984), 929–989. [10] E. Giné and J. Zinn, Bootstrapping general empirical measures, Ann. Probab. 18 (1990), 851–869. [11] V. Goodman, J. Kuelbs and J. Zinn, Some results on the LIL in Banach space with applications to weighted empirical processes, Ann. Probab. 9 (1981), 713–752. [12] J. Hoffman-Jørgensen, Sums of independent Banach space valued random variables, Aarhus Univ. Preprint Series 1972/73, no. 15 (1973). [13] J. Hoffman-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159–186. [14] J. Hoffman-Jørgensen, Probability in Banach spaces, Ecole d’Eté de Probabilités de St-Flour 1976, Lecture Notes in Math. 598, Springer (1976), 1–186. [15] J. Hoffman-Jørgensen and G. Pisier, The law of large numbers and the central limit theorem in Banach spaces, Ann. Probab. 4 (1976), 587–599. [16] N.C. Jain, Central limit theorem in a Banach space, Probability in Banach Spaces (Proc. First Internat. Conf., Oberwolfach, 1975), Lecture Notes in Math. 526, Springer, Berlin (1976), 113–130. [17] N.C. Jain and M.B. Marcus, Central limit theorem for C(S)-valued random variables, J. Funct. Anal. 19 (1975), 216–231. [18] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. 1 [19] W.B. Johnson and G. Schechtman, Embedding m p into p , Acta Math. 149 (1982), 71–85. [20] J.-P. Kahane, Some Random Series of Functions, D. C. Heath and Co., Raytheon Education Co., Lexington, MA (1968), viii+184 pp. [21] J. Kuelbs, A strong convergence theorem for Banach space valued random variables, Ann. Probab. 4 (1976), 744–771. [22] J. Kuelbs, Kolmogorov’s law of the iterated logarithm for Banach space valued random variables, Illinois J. Math. 21 (1977), 784–800. [23] J. Kuelbs and J. Zinn, Some stability results for vector valued random variables, Ann. Probab. 7 (1979), 75–84. [24] S. Kwapie´n, Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefﬁcients, Studia Math. 44 (1972), 583–595. [25] S. Kwapie´n, On Banach spaces containing c0 , Studia Math. 52 (1974), 187–190. [26] M. Ledoux, Sur une inégalité de H. P. Rosenthal et le théorème limite central dans les espaces de Banach, Israel J. Math. 50 (1985), 290–318. [27] M. Ledoux, The law of the iterated logarithm in uniformly convex Banach spaces, Trans. Amer. Math. Soc. 294 (1986), 351–365. [28] M. Ledoux, On Talagrand’s deviation inequalities for product measures, ESAIM Probab. Statist. 1 (1996), 63–87. [29] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monographs 89, Amer. Math. Soc., Providence (2001). [30] M. Ledoux and M. Talagrand, Characterization of the law of the iterated logarithm in Banach spaces, Ann. Probab. 16 (1988), 1242–1264. [31] M. Ledoux and M. Talagrand, Un critère sur les petites boules dans le théorème limite central, Probab. Theory Related Fields 77 (1988), 29–47. [32] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer (1991). [33] R. LePage, M. Woodroofe and J. Zinn, Convergence to a stable distribution via order statistics, Ann. Probab. 9 (1981), 624–632. [34] M.B. Marcus and G. Pisier, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta. Math. 152 (1984), 245–301. [35] B. Maurey, Espaces de cotype p, Séminaire Maurey–Schwartz 1972–73, Ecole Polytechnique, Paris (1973). [36] B. Maurey, Type, cotype and K-convexity, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1299–1332 (this Handbook). [37] B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et géométrie des espaces de Banach, Studia Math. 58 (1976), 45–90.

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[38] G. Pisier, Sur les espaces qui ne contiennent pas de 1n uniformément, Séminaire Maurey–Schartz 1973–74, Ecole Polytechnique, Paris (1974). [39] G. Pisier, On the dimension of the np -subspaces of Banach spaces, for 1 p < 2, Trans. Amer. Math. Soc. 276 (1983), 201–211. [40] G. Pisier, Remarques sur les classes de Vapnik–Cervonenkis, Ann. Inst. H. Poincaré 20 (1984), 287–298. [41] G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and Analysis, Varenna (Italy) 1985, Lecture Notes in Math. 1206, Springer (1986), 167–241. [42] G. Pisier and J. Zinn, On the limit theorems for random variables with values in the spaces Lp (2 p < ∞), Z. Wahrsch. Verw. Gebiete 41 (1978), 289–304. [43] G. Schechtman, Concentration, results and applications, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1603–1634 (this Handbook). [44] M. Talagrand, Isoperimetry and integrability of the sum of independent Banach space valued random variables, Ann. Probab. 17 (1989), 1546–1570. [45] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Publ. Math. I.H.E.S. 81 (1995), 73–205. [46] M. Talagrand, A new look at independence, Ann. Probab. 24 (1996), 1–34. [47] M. Talagrand, New concentration inequalities in product spaces, Invent. Math. 126 (1996), 505–563. [48] S.R.S. Varadhan, Limit theorems for sums of independent random variables with values in a Hilbert space, Sankh`ya A 24 (1962), 213–238. [49] V.V. Yurinskii, Exponential bounds for large deviations, Theory Probab. Appl. 19 (1974), 154–155. [50] V.V. Yurinskii, Exponential integrability for sums of random vectors, J. Multivariate Anal. 6 (1976), 476– 499. [51] J. Zinn, A note on the central limit theorem in Banach spaces, Ann. Probab. 5 (1977), 283–286.

CHAPTER 28

Quotients of Finite-Dimensional Banach Spaces; Random Phenomena Piotr Mankiewicz∗ ´ Institute of Mathematics, PAN, Sniadeckich 8, 00-956 Warsaw, Poland E-mail: [email protected]

Nicole Tomczak-Jaegermann† Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries, Gaussian quotients . . . . . . . . . . . . . . . . . 3. The asymptotic growth of the diameter of Minkowski compacta 4. Spaces with large basis constant . . . . . . . . . . . . . . . . . . 5. Mixing operators and quotients of N 1 . . . . . . . . . . . . . . . 6. Other results on random quotients of N . . . . . . . . . . . . . 1 7. Random quotients of convex bodies in special position . . . . . . 8. Spaces with large Euclidean subspaces and quotients . . . . . . . 9. An inﬁnite-dimensional construction . . . . . . . . . . . . . . . . 10. Constructions in a general setting . . . . . . . . . . . . . . . . . 11. Updates and recent results . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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∗ Partially supported by KBN Grants No. 2 P03A 022 15 and 2 P03A 013 19. † When the work on this paper began, this author held Canada Council Killam Fellowship in 1997-99. Since 2001

the author holds Canada Research Chair in Geometric Analysis. HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2 Edited by William B. Johnson and Joram Lindenstrauss © 2003 Elsevier Science B.V. All rights reserved 1201

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1. Introduction There are several directions in which close connections can be found between the geometry and linear-metric structure of ﬁnite- and inﬁnite-dimensional Banach spaces. In the ﬁnitedimensional context, the geometry of convex bodies in Rn has a fundamental inﬂuence on structural phenomena in random subspaces and quotients of the corresponding normed spaces. Random phenomena that occur with probability close to 1 determine our understanding of what the governing rules are, notwithstanding the fact that such rules may have a very different character. To illustrate, consider the following two examples. It is well known by now that typical n-dimensional sections of the unit ball of 2n 1 are nearly Euclidean, for n ∈ N. In fact, they are uniformly close to a suitable multiple of corresponding sections of the Euclidean ball. Consequently, they admit lots of symmetries. In contrast, the same procedure for the cube, i.e., the unit ball in 2n ∞ , results in typical sections that admit virtually no symmetries at all. For a general convex body it turns out that these two possibilities are the only alternatives to each other. Which one actually occurs is determined by the geometry of the body, more precisely, by volumetric relationships between the body and the Euclidean ball. Studies of each of these directions developed independently. The investigation of Euclidean sections has a long tradition, dating from Dvoretzky’s theorem in 1961 and its random treatment by Milman at the end of that decade. Over the years, a rich structural theory has been developed. It includes new geometric inequalities of an isomorphic type such as the inverse Santaló inequality and the inverse Brunn–Minkowski inequality, furthermore, the concepts of volume ratio and of Milman’s M-ellipsoid, to name just results of fundamental importance in our paper. For detailed account of this theory we refer the reader to the article by Giannopoulos and Milman [8] in this Handbook and the references therein. In this paper we present the other alternative, namely, the study of random phenomena of non-Euclidean nature. Family of random sections of the cube plays an important role in this study. We shall also brieﬂy discuss some inﬁnite-dimensional applications. The theory began with Gluskin’s result (1981) that established the asymptotic order of the diameter of the Minkowski compactum by showing that two n-dimensional sections of the 2n-dimensional cube have typically as large Banach–Mazur distance as possible. Soon after, further results were discovered, which indicated, in various ways, a complete lack of symmetries for typical proportional-dimensional sections of the cube. It is interesting that, as it will be seen in this paper, all invariants which measure norms of “non-trivial” operators acting on such sections of the cube, have an asymptotic growth of the largest possible order, perhaps up to a logarithmic factor. It is also noteworthy that there are no examples known, other than random constructions of type presented here, of ﬁnite-dimensional normed spaces lacking symmetries in such a strong quantitative sense. For general convex bodies, it turns out that random sections displaying similar lack of symmetries can be always found, provided that the volume of the body is not too close to the volume of the ellipsoid of maximal volume contained in the body. In particular, this led to a solution of a ﬁnite-dimensional version of the homogeneous space problem. The above phenomena can be used in the constructions of inﬁnite-dimensional Banach spaces that do not admit continuous operators with natural (algebraic) properties, or sequences of such operators. In fact, such constructions require just a few canonical oper-

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ations, starting from an arbitrary Banach space non-isomorphic to 2 . This in particular leads to unexpected characterizations of Banach spaces isomorphic to Hilbert spaces. Before we pass to describing the details of the paper, let us ﬁrst make a technical remark of a general nature. Since we are interested in estimating norms of operators, it is easier to work with quotients of m 1 (mainly proportional-dimensional), which have an easier description of extreme points than sections of m ∞ . By standard duality, these two situations are equivalent. After the preliminary section, we provide in Section 3 a complete proof of Gluskin’s theorem on the diameter of Minkowski compactum, we present this in the Gaussian setting in order to unify the exposition. In Section 4 the notion of mixing operators is introduced, and the Szarek and Gluskin result on basis constant of quotients of n1 ’s is proved. Section 5 starts with a structural theorem concerning the distance of an operator T to the line {λId | λ ∈ R}. Further, various results on quotients of n1 of proportional dimension are discussed. Section 6 is devoted to sub-proportional quotients of n1 , the Gordon–Lewis property of proportional-dimensional sections of the cube, and other related facts, including Szarek’s examples of spaces without well bounded Dvoretzky–Rogers factorization. In Section 7 we turn our attention to general convex bodies. We discuss relevant geometric invariants and introduce a notion of a special position of a convex body. For bodies in such a position we prove results parallel to those from Sections 3 and 4, which are then used in the next section to present a solution of the ﬁnite-dimensional homogeneous space problem. Furthermore, in Section 8 we also present several results concerning structural properties of ﬁnite- and inﬁnite-dimensional Banach spaces. The next two sections are devoted to inﬁnite-dimensional constructions, obtained by gluing together ﬁnite-dimensional random spaces. In the former section we concentrate on speciﬁc examples, while in the latter section we discuss constructions inside an arbitrary Banach space as mentioned above. The theory presented here is relatively new and still evolving quite fast. In particular new results which are still appearing are partially changing the picture; this is especially true for random quotients of general convex bodies. This article presents the development of the theory as seen in the Fall 1999, with some additions made in the Spring 2000. The last Section 11, written in September 2002, contains information on some developments which took place after April 2000.

2. Preliminaries, Gaussian quotients We shall mainly deal with convex bodies and operators on Rn . The Euclidean norm on Rn is denoted by · 2 , and the Euclidean unit ball by B2n . For a linear subspace E ⊂ Rn , by PE we denote the orthogonal projection onto E. For a Borel subset B of Rn , by vol(B) = voln (B) we denote the standard Lebesgue measure of B. Let (Ω, P) be a probability space. We call a random vector g : Ω → Rn a normalized n −1/2 Gaussian vector if g = n i=1 hi ei , where hi are standard N(0, 1) distributed independent Gaussian variables, and {ei } is the usual unit vector basis in Rn . It is well known that this notion in fact does not depend on the choice of an orthonormal basis in Rn and the distribution of a normalized Gaussian vector is N(0, 1/nIn ), where In denotes the identity matrix on Rn . Therefore this concept can be naturally extended to the notion of normalized

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Gaussian vectors in an arbitrary n-dimensional Hilbert space H = (H, · 2 ). The density of a normalized Gaussian vector in H is equal to (n/2π)n/2 exp (−nx22 /2). Clearly, for a normalized Gaussian vector g the second moment of its norm satisﬁes Eg22 = 1, and this explains our use of the term “normalized” in the present context. Further basic properties we shall need in this article are listed in the following fact. FACT 1. Let g be a normalized Gaussian vector in an n-dimensional Hilbert space H . Then √ (i) For every k-dimensional subspace E ⊂ H , n/kPE g is a normalized Gaussian vector in E. (ii) For every orthogonal pair of subspaces E1 , E2 ⊂ H the random vectors PE1 g and PE2 g are independent. (iii) For any a, b > 0 we have the estimates √ 2 n P ω ∈ Ω | g(ω)2 a 1 − 2e−a /4

(1)

and √ n P ω ∈ Ω | g(ω)2 1/b 1 − e/b .

(2)

In particular, P ω ∈ Ω | 1/2 g(ω)2 2 1 − e−cn , where c > 0 is a universal constant. (iv) For every Borel set B ⊂ H we have P ω ∈ Ω | g(ω) ∈ B en/2 vol B/ vol B2n . O UTLINE OF THE PROOF. Properties (i) and (ii) are direct consequences of the density formula. To prove (iv) observe that P ω ∈ Ω | g(ω) ∈ B = (n/2π)n/2

B

exp −nx22 /2 dx

(n/2π)n/2 vol B2n

B

dx/ vol B2n .

(3)

Also recall that

n/2 , vol B2n = π n/2 /Γ (1 + n/2) C /n for some numerical constant C . This immediately implies the required estimate with a factor C n , for some numerical constant C; which is sufﬁcient for our purpose. The factor en/2 can be obtained by using the formula for the volume of B2n and the Stirling formula.

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Estimate (2) in (iii) immediately follows from (3). To get (1) write a normalized Gaussian vector g in Rn in the form g = n−1/2 ni=1 hi ei , where hi are standard N(0, 1) distributed independent Gaussian variables. Fix an arbitrary λ ∈ (0, 1/2). Setting y = √ 1 − 2λt we have ∞ ∞ 2 −1/2 (λ−1/2)t 2 −1/2 −1/2 (2π) e dt = (1 − 2λ) (2π) e−y /2 dy −∞

= (1 − 2λ)

−1/2

−∞

.

Hence for every a > 0 we have P ω ∈ Ω | g(ω)2 a n 2 2 =P ω∈Ω | hi (ω) a n e

−a 2 λn

i=1

e

λ

n

2 i=1 hi (ω)

Ω

e

−a 2 λ

(1 − 2λ)

−1/2 n

dP(ω) = e

−a 2 λn

n $ i=1

.

Letting λ = 1/4, we get (1), completing the proof of (iii).

(2π)

−1/2

∞ −∞

2

e(λ−1/2)xi dxi (4)

By a random n-dimensional space we mean a (measurable) map from (Ω, P) into the set of all n-dimensional Banach spaces. In particular, we shall use this notion and its consequences when dealing with families of n-dimensional spaces equipped with a probability measure. (This is consistent with the usage generally adopted in the asymptotic theory of normed spaces, which refers to elements of a subset of such a family with measure close to 1.) On the other hand, a random space deﬁnes in a standard way a probability measure on its range, which will be also denoted by P, without leading to any confusion. To present important examples, for every integer n 1 consider a sequence gn,1 , gn,2 , . . . of independent normalized Gaussian vectors in Rn . For every m ∈ N and ω ∈ Ω, deﬁne BXn,m (ω) = abs conv e1 , . . . , en , gn,1 (ω), . . . , gn,m (ω) , where abs conv A denotes the convex hull of A ∪−A. The Banach space Xn,m (ω) is deﬁned to be Rn equipped with the norm for which BXn,m (ω) is the unit ball. In this way, for any integers n, m 1, we deﬁne a random space ω → Xn,m (ω). The range of this random space, i.e., the set of all spaces Xn,m (ω) for ω ∈ Ω, is denoted by Xn,m . Every space X(ω) ∈ Xn,m can be identiﬁed with a quotient of n+m , via the quotient 1 map qn,m (ω) : Rn+m → Rn deﬁned by the formula for 1 i n, ei qn,m (ω)(ei ) = (5) gn,i−n (ω) for n < i n + m. Indeed, BXn,m (ω) = qn,m (ω)(B1n+m ).

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To describe another related example, for every m n, set BYn,m (ω) = abs conv gn,1 (ω), . . . , gn,m (ω) , and by Yn,m (ω) we denote Rn with BYn,m (ω) as the unit ball. The set of all spaces Yn,m (ω) is denoted by Yn,m . As above, the spaces Yn,m (ω) can be identiﬁed with suitable ndimensional quotients of m 1 in a manner similar to (5). This identiﬁcation has a distinct geometric ﬂavor that will be discussed at the beginning of Section 6. In technical arguments concerning classes Xn,m and Yn,m , it is convenient to restrict considerations to a natural smaller class of random spaces, on which there is an additional 0 as the control of Euclidean norms of the Gaussian vectors involved. Thus we deﬁne Xn,m n set of all X ∈ Xn,m such that BX ⊂ 2B2 . Fact 1 yields 0

1 − m exp(−cn), P Xn,m

(6)

where c > 0 is a universal constant. Similar notation and estimates can be used for Yn,m . More generally, for every n, m 1 and an arbitrary (n + m)-dimensional space W = (Rn+m , · ) we deﬁne a random n-dimensional (quotient) space of W , ω → Zn,m (ω) by Zn,m (ω) = qn,m (ω)(W ). That is, Zn,m (ω) is Rn with the unit ball equal to qn,m (ω)(BW ). The set of all spaces Zn,m (ω) is denoted by Zn,m , or by Zn,m,BW , if W may be not clear from the context. Given a sequence of random n-dimensional spaces Xn (·) and a certain property P of ﬁnite-dimensional spaces, we say that P is satisﬁed for a majority of the spaces Xn , if for every n, the set of ω ∈ Ω such that Xn (ω) does not satisfy P has probability less than or equal to exp(−cn), where c > 0 is some universal constant. For a random variable f deﬁned on a probability space Ω and a subset A of the range of f , we use the shorthand notation ω ∈ Ω | f (ω) ∈ A = f (ω) ∈ A = {f ∈ A}. In general, we follow the notation from [14]. For Banach spaces X and Y , and a (bounded) operator T from X to Y , the norm of T will be denoted by T : X → Y . If both X and Y are Hilbert spaces (not necessarily of the same dimension, ﬁnite or not), then T 2→2 stands for the norm of T . If B is a symmetric convex body in Rn , by XB we shall denote Rn equipped with the norm for which B is the unit ball. We shall frequently use the polar decomposition of operators in L(Rn ). To ﬁx the notation recall that FACT 2. Everylinear operator T on Rn can be represented in the polar decomposi tion form T = si (T )u¯ i (T ) ⊗ ui (T ), i.e., T (x) = si (x, u¯ i )ui with s1 (T ) s2 (T ) · · · sn (T ) 0, where {u¯ i }ni=1 and {ui }ni=1 are orthonormal systems in Rn . Moreover, while such a representation (in general ) is not unique, the sequence of s-numbers si (T ) is unique. By γ2 (T : X → Y ) we denote the 2 -factorable norm of an operator T : X → Y ; that is, γ2 (T : X → Y ) = inf S1 S2 where the inﬁmum runs over all operators S1 : X → 2

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and S2 : 2 → Y satisfying T = S2 S1 . We write γ2 (T : X → Y ) = ∞ if no operators S1 , S2 exist. In most of this paper we study the behaviour of numerical invariants of ﬁnitedimensional Banach spaces. In fact we are only interested in their asymptotic behaviour, up to universal constants, when the dimension n tends to inﬁnity. Therefore, all inequalities and estimates are required to hold for sufﬁciently large n only: it is obvious that by a suitable change of universal constants involved, we may force such an inequality to hold for every n 1. 3. The asymptotic growth of the diameter of Minkowski compacta For an integer n 1 the Minkowski compactum Mn is the set of all n-dimensional Banach spaces equipped√with the Banach–Mazur distance (cf. [14], Section 8). From John’s theorem, d(X, n2 ) n, and hence d(X, Y ) n, for √ all X, Y ∈ Mn . On the other hand one can easily check that d(n1 , n2 ) is of the order n. Despite several attempts, √ no examples of sequences of Banach spaces Xn , Yn ∈ Mn such that sup d(Xn , Yn )/ n = ∞ were known until 1980. The breakthrough was made by Gluskin who introduced random ﬁnite-dimensional spaces in order to show in [9] that the diameter of the Minkowski compactum Mn is asymptotically of order n. We present the complete proof of this result in the Gaussian setting. In the sequel we shall frequently be concerned with inequalities for some classes of operators acting on random spaces. A typical proof consists of few steps. First we prove that a slightly stronger inequality holds for an arbitrary ﬁxed operator in the class, on random n-dimensional spaces, except for a set of such spaces with small probability. This is done by reducing the problem to estimating probability that some normalized Gaussian vectors hit a suitably chosen subset in Rn and then applying Fact 1(iv). The next step consists of constructing sufﬁciently dense net in the considered class of operators. Then a uniform continuity argument (more precisely, a Lipschitz condition) allows us to conclude that the starting inequality holds for all operators and for random spaces except for those for which the stronger inequality was violated for a single operator from the net. T HEOREM 3 ([9]). For every δ > 0 there exists c > 0 such that for every integer n 1 there exist random spaces Xn , Yn in Xn,[δn] such that d(Xn , Yn ) cn. In fact, the estimate is valid for a majority of the pairs (Xn , Yn ) ∈ Xn,[δn] × Xn,[δn] . For the rest of this section we ﬁx an integer n 1. For m 1, α > 0, an n-dimensional space Yn = (Rn , · Yn ), and an operator T ∈ L(Rn ), we let √ (7) A(m, α, Yn , T ) = Xn ∈ Xn,m | T : Xn → Yn α n . L EMMA 4. Let Yn = (Rn , · Yn ) be an n-dimensional Banach space and let n/4 k n. For every operator T ∈ L(Rn ) with sk (T ) 1 and every α > 0 and m 1 one has √ k vol(PE BYn ) m P A(m, α, Yn , T ) 2e1/2α n , vol B2k

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where E = span[ui (T ) | 1 i k] and the vectors ui (T ) are given by the polar decomposition of T , as in Fact 2. P ROOF. Let F = span[u¯ i (T ) | 1 i k], where the vectors u¯ i (T ) are given by the polar decomposition of T . Since (T |F )−1 2→2 1, by Fact 1 we have

√ P A(m, α, Yn , T ) = P T BXn,m (ω) ⊂ α nBYn

√ P PE T BXn,m (ω) ⊂ α nPE BYn =

m $ √ P PE T gi ∈ α nPE BYn i=1

m % $ # = P n/k T PF gi ∈ α n2 /kPE BYn i=1

m $ # √ P n/kPF gi ∈ 2α n(T |F )−1 PE BYn i=1

m $ 1/2 √ k vol((T |F )−1 PE BYn ) 2e α n vol B2k i=1 √ k vol(PE BYn ) m 2e1/2α n . vol B2k

L EMMA 5. For every κ 0 there exists c2 > 0 such that for every integer k 1, and for every convex body B ⊂ Rk of the form B = abs conv{xi | 1 i (1 + κ)k}, we have vol B

c2 K k

k ,

where K = max{xi 2 | 1 i (1 + κ)k}. P ROOF. For every k-element subset σ of {1, 2, . . . , [(1 + κ)k]} set Bσ = abs conv{xi | i ∈ σ }. By the Caratheodory theorem, B = σ Bσ . Therefore, by the Hadamard inequality and the Stirling formula (recall that vol B1k = 2k /k!) vol B

σ

vol Bσ

[(1 + κ)k] (2K)k c2 K k . k k! k

0 For every operator T ∈ L(Rn ) with s[n/2] (T ) 1 and every random space Yn ∈ Xn,[δn] we have, by Lemmas 4 and 5,

P A [δn], α, Yn , T (c3 α)[n/2][δn]

(8)

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for sufﬁciently small α > 0. (Recall that vol B2n ∼ n−n/2 .) In fact, it is sufﬁcient to consider the sets A([δn], α, Yn , T ) for a sufﬁciently dense net in a suitable set of operators. Recall that if · is a norm on Rn , A ⊂ Rn and δ > 0 then N is a δ-net in A with respect to · if N ⊂ A and for every x ∈ A there is z ∈ N such that x − z < δ. The following lemma is a standard ingredient. L EMMA 6. Let B1 ⊂ B2 be two symmetric convex bodies in Rn . For every subset A ⊂ B2 there exists a 1-net N in A with respect to the norm · B1 with cardinality card(N ) 3n vol B2 / vol B1 . P ROOF. Let N = {x1, . . . , xN } be a maximal subset of A satisfying xi − xj B1 1 for all i = j . The maximality condition implies that it is a 1-net in A. Consider the set N i=1 (xi + 1 1 1 2 B1 ) ⊂ (1 + 2 )B2 . Observing that this set is a union of translates of 2 B1 with mutually disjoint interiors, we get N( 12 )n vol B1 ( 32 )n vol B2 . We shall identify, in the canonical way, operators from L(Rn ) with n × n matrices, 2 considered as elements of Rn . In particular, this deﬁnes an n2 -dimensional volume on any Borel set of operators. We shall consider two sets of operators n = T ∈ L Rn T : n2 → n2 1 , Bop and, for a space Z = (Rn , · Z ), n Bop,Z = T ∈ L Rn T : n1 → Z 1 . n n (c /n)n L EMMA 7. We have vol Bop,Z = (vol B(Z))n and vol Bop universal constant.

2 /2

, where c > 0 is a

The lemma is based on the well-known fundamental upper estimate for the norm of a Gaussian matrix. For sake of completeness we brieﬂy describe it here and give a short elementary proof. Let G(ω) be an n × n Gaussian matrix whose columns are independent normalized Gaussian vectors in Rn . Then for any a > 0 we have

n √ 2 P ω ∈ Ω | G(ω) : n2 → n2 a 1 − 6 2e−a /16 .

(9)

Fix an arbitrary a > 0. Since matrices G and GU have the same distribution for any ﬁxed orthogonal matrix U , then by (1) we have, for every x ∈ S n−1 , √

2 n P G(ω)x 2 > a = P G(ω)e1 2 > a 2e−a /4 . By Lemma 6, the unit sphere S n−1 admits an 1/2-net N with respect to · 2 with cardinality not greater than 6n . (In fact a direct argument analogous to Lemma 6 shows a standard

Quotients of ﬁnite-dimensional Banach spaces

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estimate that card(N ) (1 + 2/δ)n , for a δ-net N .) Set A = {ω ∈ Ω | G(ω)x2 a for √

2 all x ∈ N }. Then P(A) 1 − (6 2e−a /4 )n . An easy approximation argument shows that G(ω) : n2 → n2 2a for every ω ∈ A. Thus the conclusion follows from the estimate for P(A), setting a = a/2. Now we can return to Lemma 7. n P ROOF. To prove the former statement, ﬁrst observe that the set Bop,Z consists of all operators T such that T (ei ) ∈ B(Z) for all 1 i n; and then identify this set with the Cartesian product (B(Z))n = (B(Z)) × · · · × (B(Z)). n , consider again a Gaussian matrix G(ω) as in To prove the volume estimate for Bop

(9). Observe that n−1/2 G is a normalized Gaussian vector in Rn . Therefore, applying n ⊂ Rn2 we get Fact 1(iv) for the set B = 8n−1/2 Bop 2

n2 /2 2

vol B/ vol B2n n/82 e P G : n2 → n2 8 . By (9), probability of the set above is larger than or equal to 1/2, hence, using the formula 2 for the volume vol B2n we get

n2 /2

n2 /2 n vol Bop (1/2) n/82 e vol B2n c /n , where c > 0 is a universal constant.

For future use, we state the next corollary in a slightly more general form than needed here. C OROLLARY 8. Let B ⊂ Rn be a symmetric convex body and let t > 0 be such that tB2n ⊂ B. Denote by A the set of all operators T ∈ L(Rn ), such that T : l1n → XB 1. Every subset A ⊂ A admits a t-net N , with respect to the operator norm on n2 with 2 card(N ) (C/t)n (vol B/ vol B2n )n , where C is an absolute constant. Clearly, the set N forms also a 1-net in the operator norm from n2 to XB . 0 0 For any α > 0 and Yn ∈ Xn,[δn] , consider the set A(α, Yn ) of all spaces Xn ∈ Xn,[δn] such √ n that there exists T ∈ L(R ), with s[n/2] (T ) 1 and T : Xn → Yn α n. That is,

A(α, Yn ) =

A [δn], α, Yn , T , T

where the union runs over all T ∈ L(Rn ) with s[n/2] (T ) 1. L EMMA 9. For sufﬁciently small α and sufﬁciently large n one has the estimate 2 P(A(α, Yn )) 2−n .

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n P ROOF. Let N be an α-net with minimal cardinality √ in the set of all operators T ∈ L(R ), n with s[n/2] (T ) 1, such that T : l1 → Yn α n, with respect to the operator norm on 2 n2 . By Corollary 8, card(N ) C1n , where C1 > 0 is a universal constant. By (8), for sufﬁciently small α > 0 and sufﬁciently large n we get

2 2 A [δn], 3α, Yn , T C1n (3c3 α)kδn (1/2)n . P T ∈N

So it remains to prove that A(α, Yn ) ⊂ T ∈N A([δn], 3α, Yn , T ). To this end,√ ﬁx Xn ∈ A(α, Yn ). Then there exists T with s[n/2] (T ) 1, such that T : Xn → Yn α n. Pick T0 ∈ N with T − T0 : n2 → n2 α. Since n−1/2 B2n ⊂ B(Xn ) and B(Yn ) ⊂ 2B2n , we get T0 : Xn → Yn T : Xn → Yn + T0 − T : Xn → Yn √ √ √ α n + 2 nT0 − T : n2 → n2 3α n. Thus Xn ∈ A([δn], 3α, Yn , T0 ).

P ROOF OF T HEOREM 3. Fix α > 0 and n ∈ N satisfying Lemma 9. Denote by An the 0 0 × Xn,[δn] such that for all T ∈ L(Rn ) with s[n/2] (T ) 1 set of all pairs (Xn , Yn ) ∈ Xn,[δn] √ one has T : Xn → Yn > α n. Set Bn = {(Xn , Yn ) | (Yn , Xn ) ∈ An }. Using the Fubini theorem for the complement of An , by (6) and Lemma 9, we get P × P(An ) = P × P(Bn ) 1 − δn exp(−cn) − 2−n . 2

Hence P × P(An ∩ Bn ) 1 − 2(δn exp(−cn) + 2−n ). To complete the proof note that d(Xn , Yn ) > α 2 n for (Xn , Yn ) ∈ An ∩ Bn . Indeed, let T ∈ L(Rn ) be an arbitrary isomorphism. By multiplying T by a suitable constant we may assume that s[n/2] (T ) 1 and s[n/2] (T√−1 ) 1. Hence each of the norms T : Xn → Yn and T −1 : Yn → Xn is larger than α n. 2

4. Spaces with large basis constant Questions related to the existence of Schauder bases are among the most classical ones in the Banach space theory and have been studied from the earliest days (see [14], Section 3). The problem whether every inﬁnite-dimensional separable Banach space has a basis was solved in the negative by Enﬂo in 1972, but his argument yielded no quantitative ﬁnitedimensional estimates. The problem whether there exists a sequence of n-dimensional Banach spaces Xn such that sup bc(Xn ) = ∞ was solved only in the early 80’s, independently by Gluskin [10] and Szarek [47]. The basis constant of an n-dimensional space X, denoted by bc(X), is deﬁned as the inﬁmum of the basis constants of {xi }ni=1 , taken over all bases √ {xi }ni=1 of X. From John’s theorem one obviously has bc(X) dim X. In this section we present the approach by Szarek, which in particular shows that the obvious upper estimate is asymptotically sharp. The result of Gluskin which goes in a slightly different direction will be stated in Theorem 25.

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T HEOREM 10 ([47]). For every δ > 0 there exists c > 0 such that for every √ integer n 1 there exists a random space Xn ∈ Xn,[δn] with basis constant bc(Xn ) c n. In fact, the estimate is valid for a majority of the spaces Xn ∈ Xn,[δn] . In contrast to Theorem 3, where operators were acting between two independent random spaces, here we deal with operators acting from a space into itself. This causes additional conceptual difﬁculties. They can be overcome by the crucial notion of mixing operators, explicitly introduced in [48], which technically speaking allows us to split Gaussian vectors into two independent components. D EFINITION 11. Let n, k ∈ N, with 1 k n/2, and β > 0. An operator T ∈ L(Rn ) is said to be (k, β)-mixing if there exists a subspace E ⊂ Rn , with dim E k, such that dist(T x, E) = PE ⊥ T x2 βx2

for all x ∈ E.

(10)

The set of all (k, β)-mixing operators on Rn will be denoted by Mixn (k, β). The connection between mixing operators and the basis constant is given by the following observation. Let P ∈ L(Rn ) be an arbitrary projection of rank k n/2. Then P is (k, 1/2) mixing, and Id − 2P is (k, 1) mixing. Indeed, if P1 is a rank 1 projection in R2 , one picks vectors u ∈ P1 (R2 ) and v ∈ ker(P1 )√of norm 1 such that (u, v) 0, and let E = span[u + v]. Then PE ⊥ P1 (u + v)2 (1/ 2), and hence P1 ∈ Mix2 (1, 1/2). In the n-dimensional case, one represents a rank k projection as a sum of k projections of rank 1 acting in orthogonal two-dimensional subspaces. The second statement easily follows from the ﬁrst one. This leads us to the conclusion that despite the fact that the mixing properties of an operator in Rn depend on the choice of a scalar product on Rn , some operators satisfying a simple algebraic condition preserve these properties for all scalar products. In our case the condition is P 2 = P . Another example, which can be shown in a similar manner, are operators T ∈ L(Rn ) satisfying T 2 = −IdRn . Theorem 10 is now a consequence of the following more general result. T HEOREM 12 ([48]). For every 0 < κ 1/2 and δ > 0 there exists α > 0 such that√for every integer n 1 a majority of the spaces Xn ∈ Xn,[δn] satisfy T : Xn → Xn α n, for every operator T ∈ Mixn (κn, 1). These spaces Xn satisfy also B1n ⊂ BXn ⊂ 2B2n . Another technical trick is the use of a projection annihilating some random vectors followed by a lifting argument to essentially decrease the set of operators that have to be considered. This allows more efﬁcient control of the cardinality of nets. The annihilating projections can be deﬁned for an arbitrary subset of random vectors in a similar fashion as below. Our choice depends on their interplay with mixing operators. Given 0 < κ 1/2 and δ > 0, for ω ∈ Ω let Qω ∈ L(Rn ) be the orthogonal projection with ker Qω = span[gi (ω) | 1 i min(δn, κn/8)]. To avoid additional technicalities, we shall present the proof of Theorem 12 assuming that δ κ/8. Note that for every Xn (ω) = Xn ∈ Xn,[δn] we have

(11) Qω (BXn (ω) ) = Qω B1n for all ω ∈ Ω.

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We shall actually show that for κ, δ as above and T ∈ Mixn (κn, 1) we have √ Qω T : Xn → Qω B n α n, 1

(12)

for a majority of ω ∈ Ω. (Clearly, by (11), T : Xn → Xn Qω T : Xn → Qω (B1n ).) The proof of (12) is based on two lemmas. L EMMA 13. Let S : R → Rm be an operator with at least k0 s-numbers larger than or equal to 1, and let g be a normalized Gaussian vector in R . Then for every y ∈ Rm and every measurable set B ⊂ Rm one has k

P Sg(ω) ∈ y + B C k0 (/k0 )k0 /2 sup vol(PE B)/ vol B2 0 , E

where the supremum is taken over all subspaces E ⊂ Rm with dim E = k0 , and C is a universal constant. The proof is an easy consequence of the polar decomposition formula and Fact 1 (cf. the proof of Lemma 4), and is left to the reader. For a ﬁxed T ∈ L(Rn ) and α, δ > 0, set √ 0 A(α, δ, T ) = Xn ∈ Xn,[δn] | Qω T : Xn → Qω B1n α n . ¯ α > 0 and for every T ∈ Mixn (κn, ¯ 1), L EMMA 14. For every 0 < κ¯ 1/2, 0 < δ 3κ/4, one has

k [δn] ¯ δ)n/k0 0 , P Xn ∈ A(α, δ, T ) αC(κ, where k0 = [(κ¯ − δ)n]. P ROOF. Let T ∈ Mixn (κn, ¯ 1). Pick E ⊂ Rn with dim E κn, ¯ satisfying (10), with β = 1. Consider independent variables PE gi and PE ⊥ gi , for 1 i δn. Let Qω,E be the orthogonal projection with ker Qω,E = span[E, PE ⊥ gi | 1 i δn]. Thus Qω,E depends on PE ⊥ gi only (1 i δn), hence it is independent of PE gi for 1 i δn. Furthermore, ker Qω ⊂ ker Qω,E , for every ω, and hence there is the orthogonal projection Rω such that Qω,E = Rω Qω . Thus we have

√ Xn ∈ A(α, δ, T ) ⊂ Qω T gi (ω) ∈ α nQω B1n for 1 i δn ⊂

[δn]

√ Rω Qω T gi (ω) ∈ α nRω Qω B1n .

i=1

Since gi = PE gi + PE ⊥ gi , for all i, the latter set is equal to [δn] i=1

√ Qω,E T PE gi (ω) ∈ yi + α nQω,E B1n ,

(13)

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where yi = Qω,E T PE ⊥ gi (ω), for 1 i δn. For each ﬁxed values of PE ⊥ gi , with 1 i δn, the operator S = Qω,E T PE : Rn → Rn has at least dim E − δn k0 s-numbers larger than or equal to 1. So, by Lemmas 13 and 5 and Fact 1, the probability of each individual set in (13) is less than or equal to (αC(κ, ¯ δ)n/k0 )k0 , for 1 i δn. The conclusion follows from the independence of gi ’s. P ROOF OF T HEOREM 12. Consider the set of operators

√ T ∈ L Rn | T ∈ Mixn (3κn/4, 1) and T : n1 → n1 α n and let N be an α-net in it with respect to the operator norm on n2 , of minimal cardinality. 2 By Corollary 8, card(N ) C1n . T ). By a simple Let T ∈ Mixn (κn, 1) and pick ω0 ∈ Ω such that Xn (ω0 ) ∈ A(α, δ,√ ∈ L(Rn ) such that T : n → n α n and Qω0 T = lifting argument, there exists T 1 1 . Since dim ker Qω0 δn, we have T ∈ Mixn (3κn/4, 1). Pick T0 ∈ N such that Qω0 T T − T0 : n2 → n2 α. By the deﬁnidenote Qω0 (B1n ) by B. We claim that Xn (ω0 ) ∈ A(3α, √ δ, T0n). To nsee this, n tion of T , and the inclusion (1/ n)B2 ⊂ B1 ⊂ B2 , we get, for all 1 i δn,

gi gi + Qω0 T0 − T Qω0 T0 gi B Qω0 T B B

√ Qω0 T gi B + n T0 − T gi 2 √ √ √ α n + 2α n = 3α n. √ Moreover, Qω0 T0 ej B T0 ej B1n α n, for all 1 j n. Set A(α, δ, κ) = T ∈Mixn (κn,1) A(α, δ, T ). We have proved that for every α > 0 A(α, δ, κ) ⊂

A(3α, δ, T ).

T ∈N

By Lemma 14, applied for κ¯ = 3κ/4, the probability of the set on the right-hand side is 2 2 less than or equal to (3αC(3κ/4)n/k0 )k0 [δn] C1n . Hence we have P(A(α, δ, κ) 2−n , for sufﬁciently small α > 0. The proof is completed by observing that for α > 0 sufﬁciently small P(Xn,[δn] \ 2 A(α, δ, κ)) 1 − e−cn − 2−n and that, by (11), we have T : Xn (ω) → Xn (ω) Qω T : Xn (ω) → Qω Xn (ω) √ Qω T : Xn (ω) → Qω B1n α n, for ω such that Xn (ω) ∈ Xn,[δn] \ A(α, δ, κ) and T ∈ Mixn (κn, 1).

P ROOF OF T HEOREM 10. This is an immediate consequence of the remark following the deﬁnition of mixing operators combined with Theorem 12. Indeed, if bc(Xn ) K, for

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some K, then from the deﬁnition of the basis constant, there exists a rank [n/2] projection P in Xn with P K.

5. Mixing operators and quotients of N 1 In this section we shall discuss several geometric invariants of a ﬁnite-dimensional Banach space √ X. For all these invariants, the trivial upper estimates yielded by John’s ellipsoids are dim X. It turns out that these estimates are asymptotically sharp, even when considered for quotients of N 1 of proportional dimension. Our arguments depend on the random approach presented in the previous sections. The argument behind Theorem 10 clearly shows that mixing properties of operators deserve a closer look. For information on different methods of establishing mixing properties of a given operator, the reader is referred to [23]. In a sense, mixing properties of an operator T are directly related to the distance of T to the line {λId | λ ∈ R}. The precise description of this relation is contained in the next lemma which is crucial for further studies. Its proof is straightforward linear algebra [24]. L EMMA 15. Let n 2 and 0 < ε < 1. Let T ∈ L(Rn ) be an operator such that for every subspace E ⊂ Rn with dim E (1 − ε)n and every λ ∈ R we have (T − λId)|E2→2 1. Then T is (εn/4, 1/8)-mixing. Given an n-dimensional space X = (Rn , · ) and 0 < κ 1/2, we let m(X, κ) = inf S : X → X | S ∈ Mixn (κn, 1) .

(14)

(It should be noted that this invariant, that will be referred to as the mixing invariant, depends on a speciﬁc representation of X as Rn . If κ is understood from the context, we write m(X) instead of m(X, κ).) As a formal consequence of Lemma 15 we get a structural theorem describing properties of operators acting on an arbitrary ﬁnite-dimensional Banach space X in terms of the mixing invariant of X ([24,25,32]). Its main point is that if m(X, κ) is large for some suitable κ then every operator acting on such a space is close to a multiple of the identity. This fact, which is stated in full generality for future references, will be fundamental for several results throughout the rest of this survey. Intuitively, the function T → λT deﬁned by the theorem below behaves as an “approximative” character (see the proof of Theorem 43, cf. [25,45]). T HEOREM 16. Let n 8 and 1/n κ < 1/8. For every operator T ∈ L(Rn ) there exists λT ∈ R and subspaces ET , FT ⊂ Rn with dim ET (1 − 4κ)n and dim FT (1 − 8κ)n satisfying the following: if T = 0 then λT = 0 and ET = FT = Rn . For every ndimensional Banach space X = (Rn , · ), set RT = T − λT Id and m(X) = m(X, κ). Then we have (i) RT |ET 2→2 (8/ m(X))T : X → X,

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(ii) for every pair of operators S, T ∈ L(Rn ) with rank(S − T ) < (1 − 8κ)n,

|λS − λT | 8/ m(X) S : X → X + T : X → X , Moreover, if S = 0 then the inequality is satisﬁed for all T with rank T < (1 − 4κ)n. (iii) T |FT 2→2 (32/ m(X))γ2 (T : X → X). P ROOF. Let T ∈ L(Rn ). The proof of (i) is a direct application of Lemma 15. (ii) follows from (i) and the inequality

0 = (λS Id + RS ) − (λT Id + RT ) x 2

|λS − λT | − RS |ES 2→2 + RT |ET 2→2

|λS − λT | − 8/ m(X) S : X → X + T : X → X , which holds for every x ∈ ker(T − S) ∩ ES ∩ ET with x2 = 1. If S = 0 then ker T ∩ ET = {0} whenever rank T < (1 − 4κ)n, and the moreover part follows. Now, if rank T < (1 − 4κ)n, condition (iii), with the constant 16/ m(X), trivially follows from the moreover part of (ii) and (i), letting FT = ET . Indeed,

T |ET 2→2 RT |ET 2→2 + |λT | 16/ m(X) T : X → X

16/ m(X) γ2 (T : X → X).

(15)

In the general case, let T = S2 S1 , where S1 : X → l2 , S2 : l2 → X and γ2 (T ) = S1 : X → l2 S2 : l2 → X and pick an orthonormal projection P in l2 such that each operator T1 = S2 P S1 and T2 = S2 (Id − P )S1 has rank less than (1 − 4κ)n. Since T = T1 + T2 and γ2 (Ti ) γ2 (T ) for i = 1, 2, condition (iii), with FT = ET1 ∩ ET2 , follows by the triangle inequality and (15) applied to T1 and T2 separately. We formulated Theorem 16 in terms of existence of subspaces ET and FT . However it may be viewed as lower estimates for T − λT Id : X → X and γ2 (T : X → X) in terms of s4κn (T − λT Id) and s8κn (T ) respectively. The ﬁrst invariant we shall discuss is the minimal distance of an even-dimensional real Banach space to a complex one; in other words, the minimal norm of a complex structure the space can admit. Identify R2n with Cn in a standard way, that is, the standard unit vector basis (e1 , . . . , en ) ∈ Cn is identiﬁed with (e1 , ie1 , . . . , en , ien ) ∈ R2n . Then for a complex ndimensional space Y = (Cn , · ), by Y R denote (R2n , · (R) ) with the norm · (R) transported to R2n from Y by this identiﬁcation. Given a 2n-dimensional real√Banach space X, there exists an n-dimensional complex space Y , such that d(X, Y R ) 2n. Indeed, one can take as Y a complex n-dimensional Euclidean space. The next theorem shows that this estimate is asymptotically the best possible.

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T HEOREM 17 ([48]). For every δ > 0 there is c > 0 such that for every integer n 1, a majority of the spaces X2n ∈ X2n,2[δn] satisfy

√ inf d X2n , YnR c n, where the inﬁmum is taken over all n-dimensional complex spaces Yn . If T : X2n → YnR is a real isomorphism, then the operator A = T −1 iT : X2n → X2n satisﬁes A2 = −Id, where i : YnR → YnR is the real operator of multiplication by i. To prove the theorem it is enough to observe that every such A is (n, 1)-mixing. This can be done by decomposing R2n into n two-dimensional mutually orthogonal A-invariant subspaces; and checking the two-dimensional case by hand. The Mazur–Ulam theorem states that every isometry F (i.e., a mapping preserving the distance) from a real normed space X onto a real normed space Y with F (0) = 0 is linear. For ﬁnite-dimensional complex Banach spaces this raises a question: if X, Y are complex n-dimensional (metrically) isometric Banach spaces, are they “nicely” isomorphic by means of a complex linear operator? The identiﬁcation of R2n with Cn allows us to associate with every X ∈ X2n,2[δn] the ndimensional complex space XC by taking as its unit ball the set BXC which is the convex hull of the set |z|=1 zBX . For a complex Banach space X (resp., Cn ) we denote by X (resp., Cn ) the same underlying space under the new multiplication by scalars given by λ ( x = λ¯ x, for λ ∈ C and x ∈ X. Note that the identity map from X onto X is an isometry which is clearly not complex-linear. T HEOREM 18 ([48]). For every δ > 0 there is c > 0 such that for every integer n 1, a majority of the spaces X2n ∈ X2n,2[δn] satisfy C

C cn. d X2n , X2n The speciﬁc trick the theorem depends upon establishes mixing properties of complexlinear operators acting between complex-linear spaces Cn and Cn . L EMMA 19. Let T : Cn → Cn be a C-linear operator with at least n/2 of s-numbers larger than or equal to 1. Then T as an operator in L(R2n ) is (n/12, 1/16)-mixing. The same holds for C-linear operators T : Cn → Cn . P ROOF. Let E0 ⊂ Cn be a complex subspace with dimC (E0 ) n/2 such that T x2 x2 for all x ∈ E0 . We shall prove that for every real subspace E ⊂ R2n with dimR (E) 5n/3 and every λ ∈ R we have (T − λId)|E2→2 1/2. The conclusion will follow by using Lemma 15 for the operator 2T . Pick E ⊂ R2n with dimR (E) 5n/3 and λ ∈ R, and assume to the contrary that (T − λId)|E2→2 < 1/2. Since E0 is a complex subspace a simple dimension consideration shows that there exists x = 0 such that both x and ix belong to E ∩ E0 . We may clearly

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assume that x2 = 1. The deﬁnitions of E0 and E easily imply that |λ| > 1/2. Since T (ix) = −iT (x), then (T − λId)(ix) = −i(T x − λx) − 2λix 2 2 2|λ| x2 − (T − λId)x 2 > 1/2, a contradiction. The other case follows by the same token.

To prove the theorem one needs an adaption of Theorem 12 to the present context. It sufﬁces to establish an analogue of (12) for orthogonal projections Qω with the kernel span[ker Qω , i ker Qω ], that is,

√ Qω T : XC → Qω B n,C c n. 2n 1 This can be done by exactly the same argument as in Theorem 12. The symmetry constant of an n-dimensional Banach space X is deﬁned by sym(X) = inf sup T , G T ∈G

(16)

where the inﬁmum is taken over all compact groups G ⊂ L(X) such that whenever T g = gT for all g ∈ G then T is a scalar multiple of the identity. With some abuse of terminology, in such a case we shall speak about an irreducible group. T HEOREM 20 ([22]). For every δ > 0 there exists c > 0 such that for every integer n √ 1, for a majority of the spaces Xn ∈ Xn,[δn] , the symmetry constants satisfy sym(Xn ) c n. It is not difﬁcult to prove that if an irreducible group G consists of isometries of Rn then G contains a (n/20, 1/4)-mixing operator. A modiﬁcation of the argument shows the same for a general irreducible group. Hence the estimate for sym(Xn ) follows from Theorem 12. One can consider a notion more subtle than the basis constant. Namely, for a given ndimensional Banach space X and m n, we deﬁne the m-th basis factorization constant of X by bf m (X) = inf γ (X, Y ) bc(Y ), Y

(17)

where the inﬁmum is taken over all m-dimensional Banach spaces Y , and where γ (X, Y ) is the constant of factorization of the identity on X through Y , that is, γ (X, Y ) = inf T S | S : X → Y, T : Y → X, with T S = IdX . That is, bf m (X) C means that X is C1 -isomorphic to a C2 -complemented subspace of an m-dimensional space √Y with the basis constant bc(Y ) C3 , where C1 C2 C3 C. Note that bfn (X) = bc(X) n for every n-dimensional Banach space. Pełczy´nski proved in [42] that for an n-dimensional space X, bfm (X) 2 for m n3/2 . His argument also shows

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that bfm (X) Cn3/2 /m, for all n m n3/2 . In fact, this estimate is optimal, up to a constant, as shown by the following theorem. T HEOREM 21 ([24]). For every δ > 0 there exists c > 0 such that for every integer n 1, a majority of the spaces Xn ∈ Xn,[δn] satisfy bfm (Xn ) cn3/2/m. P ROOF. To avoid technicalities, we shall assume that n is even and that m is a multiple of n. If X is an arbitrary n-dimensional space, then consider a space Y for which the inﬁmum in (17) is achieved. Using a basis in Y write IdY as a sum of 2m/n projections Sj on Y of rank n/2, with Sk 2 bc(Y ) and k1 Sj bc(Y ), for all 1 k 2m/n. 2m/n Using the factorization condition, one can then construct a representation IdX = 1 Sj k where rank Sk n/2, and 1 Sj bfm (X), for all 1 k 2m/n. Now, let Xn be one of spaces given by Theorem 12 for some κ < 1/20, and assume without loss of generality that m(Xn , κ) 1/16 (for small n the theorem follows by a 2m/n proper adjustment of the constant c > 0). By the previous remark write IdXn = 1 Sj k as above. Set Tk = 1 Sj for k = 1, 2, . . . , 2m/n. By Theorem 16(ii) we have that |λT1 | and |λTk+1 − λTk | for 1 k 2m/n are less than or equal to C n−1/2 bfm (Xn ) where C is a suitable numerical constant. Since T2m/n = IdXn we also have, by the assumption on m(Xn , κ), that λT2m/n > 1/2. Thus C n−1/2 bfm (Xn )2m/n > 1/2 which yields the required estimate.

6. Other results on random quotients of N 1 To see the importance of the family Yn,m , ﬁrst identify its elements Yn,m (ω) with (random) quotients of m 1 . Then the distribution of Yn,m (ω) coincides with the distribution of quotients m /E, where E is a random (m − n)-dimensional subspace of Rm , with respect to 1 the Haar measure on the Grassman manifold of all (m − n)-dimensional subspaces of Rm . Therefore this random space carries more information on the structure and geometry of m quotients of m 1 , and hence, by duality, on subspaces of ∞ . The equality of the distributions easily follows from two facts: a quotient space is uniquely determined by the kernel of the corresponding quotient map; due to the rotation invariance of the normalized Gaussian vectors in Rn , the distribution of the kernels of quotient maps corresponding to Yn,m (·) is rotation invariant, hence is equal to the Haar measure on the Grassman manifold. To stress the difference between Xn,m and Yn,m , note that the next two theorems √ 0 we have d(X, n1 ) 2 n and fail miserably for X ∈ Xn,m : indeed, for every X ∈ Xn,m IdRn : n1 → X IdRn : X → n2 2. T HEOREM 22 ([51]). For every θ > 0 sufﬁciently small there exists c > 0 such that for n 1+θ every √ integer n 1 and for m = [n ], a majority of Yn ∈ Yn,m satisfy d(Yn , 1 ) c n log n.

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The proof in [51] (see the top of p. 925) shows that the restriction 0 < θ < 2/7 is sufﬁcient. Recall that an n-dimensional Banach space has a Dvoretzky–Rogers factorization if an afﬁne image of its unit ball can be “squeezed” between the inscribed n1 -ball and a bounded from above multiple of the n2 -ball. The next theorem shows that this need not be the case in general. T HEOREM 23 ([51]). For every integer n 1 and m = 2n, a majority of Yn ∈ Yn,m have the property that for all operators S, T ∈ L(Rn ) such that ST x = x for x ∈ Rn we have T : n1 → Yn S : Yn → n2 c(n/ log n)1/10 , where c > 0 is a universal constant. Let us mention that, in contrast, for any n and 0 < δ < 1, the formal identity operator I : [δn] → [δn] admits a bounded factorization through any n-dimensional space X, that 1 2 is, I = ST and T : [δn] → X S : X → [δn] 1 2 C, where C = C(δ) depends on δ only. This result which is called the “proportional Dvoretzky–Rogers factorization”, was proved in [4], and then in [53] with a better dependence on δ in C(δ) (for recent improvements see [8]). The previous two theorems require much more delicate methods. To obtain appropriate probabilistic estimates one needs a description of the distribution of s-numbers of random Gaussian matrices. A qualitative description of this distribution is given by the classical Wigner Semicircle Law [57]. Here the following quantitative distributional inequality is needed ([51,52]). T HEOREM 24. For an integer n 1, let G = Gn (ω) be a n × n matrix with independent Gaussian entries with N(0, 1/n) distribution. Then for every 1 k n/2 one has

P ω | c1 k/n sn−k (G) c2 k/n > 1 − C exp −ck 2 , where {sj (G)} is the sequence of s-numbers of the matrix G considered as an operator on Rn , and c1 , c2 , c, C are positive universal constants. Let us also mention that since the unit vectors of Rn need not belong to the unit balls of Yn , and invariants considered here depend on speciﬁc ﬁxed norms on Rn (coming from n1 , n2 ), arguments concerning the cardinality of nets in suitable sets of operators require much more sophisticated calculations. The family Yn,m allows control of the norm of an arbitrary operator in terms of its mixing properties. T HEOREM 25. For every θ > 0 there exists c > 0 such that for every integer n 1 and for m = [n1+θ ], a majority of Yn ∈ Yn,m satisfy, for every 0 < κ 1/2, # T : Yn → Yn cκβ n/ log n, whenever T ∈ Mixn (κn, β).

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This theorem has been proved for θ = 1 and projections (of an arbitrary rank) by Gluskin, in his solution of the ﬁnite-dimensional basis problem [10]. For a general operator, the theorem was formulated by Szarek in [48]. A deeper study of mixing properties of a compact group of operators leads to an interesting description of any such a group G acting boundedly on a space Yn satisfying Theorem 25. Namely ([23]), there exists a subspace EG of Rn with dim EG n − cAn1/2 log3/2 n on which G acts trivially, i.e., T |EG = ±IdEG for every T ∈ G. Here A = supT ∈G T : Yn → Yn , and c > 0 is an universal constant. The Gordon–Lewis property, which we shall discuss now, provides a striking example of the interplay of the theory of operator ideals with the structural theory and local theory of Banach spaces. Gordon and Lewis [12] related the existence of an unconditional basis, or more generally, of (local) unconditional structure in a Banach space X, to the ratio of certain operator ideal norms of operators from X to 2 . The GL-property so introduced is weaker than the existence of an unconditional basis in a Banach space, but in many circumstances provides already satisfactory information and it is much easier to handle, due to its connections with a wide variety of tools developed in operator ideal theory. A detailed account of this theory can be found in books [43,5] and [56]; this latter book also describes many applications to Banach spaces. Here we shall restrict ourselves to one aspect only, that is, to constructing spaces with “large” GL-constant, in Theorems 26 and 39. A Banach space X is said to have the GL-property if every 1-summing operator T : X → 2 is L1 -factorable. For such a space, the GL-constant, gl(X), is deﬁned as gl(X) = supT γ1 (T )/π1 (T ), where the supremum is taken over all non-zero operators T : X → 2 . (Recall that for 1 p < ∞, the deﬁnition of p-summing operators is given in [14], and Lp -factorable operators are deﬁned in Section 2 of this article.) It is easy to see that gl(X) unc(X) ([12], see also [56], √ Section 34). Clearly, for an m-dimensional ) m. The natural question whether this estiBanach space X one has gl(X) d(X, m 2 mate is asymptotically sharp was solved in [7]. Related properties of subspaces of n∞ of proportional dimension were studied in depth in [6]. n T HEOREM √ 26 ([6]). Let 0 < a, α < 1. Let E ⊂ R be an [αn]-dimensional subspace such that (a n)x2 x1 for√all x ∈ E. Let E∞ denote the space E endowed with the norm from n∞ . Then gl(E∞ ) c n, where c = c(a, α).

Take, for example, α = 1/2, and recall ([18,46] cf. also [44], Chapter 6) that there exists an absolute constant 0 < a < 1 such that the subset of the Grassman manifold consisting of all [n/2]-dimensional subspaces E satisfying the above hypothesis is of measure close to 1 (exponentially in n) (see comments in Section 7). (The same holds for an arbitrary α√with a = a(α).) Thus for all such [n/2]-dimensional subspaces E, one has gl(E∞ ) c n, where c is an absolute constant. The proof of Theorem 26 can be also found in [56], Section 34. Recall that by the deﬁnition of the family Xn,[δn] , the cardinality of the set Ex(BX ) of extreme points of the unit ball of a space X ∈ Xn,[δn] , is less than or equal to (1 + δ)n. Let X ∈ Xn,[δn] be one of the spaces given by Theorem 12. Clearly,√for every T ∈ Mixn (κn, 1) there exists at least one vector x ∈ Ex(BX ) such that T x c n. Due to the probabilistic

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nature of the argument a stronger statement is in fact true: the set of such points x has the cardinality comparable to the cardinality of all of Ex(BX ), up to a logarithmic factor ([26], cf. also [31]). T HEOREM 27 ([26]). For every 0 < κ 1/2 and δ > 0 there exists c > 0 such that for every integer n 1, a majority of Xn ∈ Xn,[δn] satisfy √ card x ∈ Ex(BXn ) | T x c n κn/32 log n, for every operator T ∈ Mixn (κn, 1). Theorem 27 suggest that a still stronger property may hold, namely, given a sufﬁciently rich group of operators acting on Rn , one can ﬁnd many points in the ball BX on which many operators from the group are simultaneously large. Such a result was proved in [26], where also several applications were discussed. To formalize, for a compact group G of operators acting on Rn , and for 0 < κ < 1/2, β > 0 and 0 < p 1, we write G ∈ M(κn, β, p) if the set of all T ∈ G such that T ∈ Mix(κn, β) has Haar measure hG larger than or equal to p. By a straightforward use of Fubini theorem we get ([26]) T HEOREM 28. If X ∈ Xn,[δn] satisﬁes the conclusion of Theorem 27 for some 0 < κ < 1/2, and if G is a compact group from M(κn, β, p), for some β > 0 and 0 < p 1, then sup

x=1 G

T xdhG (T )

1 N

T xdhG (T ) c1 κβpn/ log n,

x∈Ex(BX ) G

where N = card Ex(BXn ). In particular, it can be shown that an irreducible group G acting on Rn belongs to M(n/20, 1/4, 1/5). Thus, if X ∈ Xn,[δn] is one of the spaces given by Theorem 27 (with an appropriate choice of parameters), then we have sup

x=1 G

√ T x dhG (T ) c1 n/400 log n,

for every irreducible group G ⊂ L(Rn ). A similar inequality (but without the logarithmic factor) concerning a class of groups connected with the unconditional basis structure was proved by Ball in [1]. Results parallel to Theorems 28 and 27 for random families of quotients Yn,m , with m = nσ , can be found in [27]. Recall a question of Pełczy´nski which relaxes the complementation requirement in the context of Theorem 21. It was asked in [42] whether every n-dimensional Banach space E is isometric to a subspace of Y , with bc(Y ) C and m = dim Y C n, for some universal constants C, C > 1. The next theorem shows that C cannot be close to 1.

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T HEOREM 29 ([31]). Let 0 < ε < 1/2 and δ = 2−11 ε. For every integer n 1 there exists −6 an n-dimensional subspace E of N ∞ , where N = (1 + 2 )n, with the property that whenever E is a subspace of Y with m = dim Y (1 + δ)n, then bc(Y ) cn1/2−ε . Moreover, if Z is a subspace of E with k = dim Z (1 − δ)n, then bc(Z) cn1/2−ε . Here c > 0 is a universal constant. In fact, the theorem holds with E = X∗ , a for a majority of spaces X ∈ Xn,[2−6 n] . Note that the moreover part establishes additional properties of subspaces of E. The question of existence of a universal C < ∞ remains open. In the context of n-dimensional subspaces of 2n nski also asked whether ∞ , Pełczy´ such subspaces can be well-embedded into m ∞ , for every m proportional to n. This was answered in the negative in [16], where it was shown that for a majority of nm dimensional subspaces E ⊂ 2n ∞ , the minimal distance to a subspace of ∞ is larger than c((2n − m)/ log m)1/2 , for m < 2n − log n, where c > 0 is a universal constant. (The majority is understood with respect to the Haar measure on the Grassman manifold.) To conclude this section let us mention that very recently a similar random method was used in [11] to construct an n-dimensional (not √ centrally symmetric) convex body B for which all projections P B, with rank P c n log n are “far from being centrally symmetric”, where c > 0 is a universal constant. The proof is similar to the argument of Theorem 25 given in [10]. For the precise quantitative statement and more details we refer the interested reader to [11]. 7. Random quotients of convex bodies in special position We would like to apply the fundamental line of arguments used in the previous sections for random quotients Xn,m of n+m to investigate quotients of an arbitrary ﬁnite-dimensional 1 Banach space. Properties essential for the arguments in Sections 3 and 4 are: the unit ball BXn of each Xn ∈ Xn,m contains an orthonormal system, and is contained in 2B2n ; the and Xn have uniformly bounded volume ratio (see the deﬁnition below). spaces n+m 1 When all these requirements together are satisﬁed, we talk about a special position of a body. These conditions may seem to be pretty restrictive, but in fact, luckily, a special position can be achieved for an arbitrary ﬁnite-dimensional Banach space by passing to a judiciously selected quotient of proportional dimension. This is a consequence of several deep fundamental results from the asymptotic theory of normed spaces that are presented in other articles of this Handbook (cf. [8]). One of the geometric concepts which plays a fundamental role in studies of random subspaces (and quotients) of arbitrary ﬁnite-dimensional Banach spaces is the volume ratio ([46,54], see also [44], Chapter 6). Let X = (RN , · ) and let B = BX . Let E ⊂ B be the John ellipsoid of maximal volume contained in B, and let · E be the Euclidean norm determined by E. Then the volume ratio of X is deﬁned by vr(X) = vr(B) = (vol B/ vol E)1/N .

(18)

The next proposition establishes the existence and some properties of special positions of quotients of arbitrary ﬁnite-dimensional convex bodies ([30,32]). Its proof depends upon

Quotients of ﬁnite-dimensional Banach spaces

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the existence of an M-ellipsoid ([36,37], see also [44], Chapter 7 and [8]) and the proportional Dvoretzky–Rogers factorization theorem (see the remark after Theorem 23). Some more properties of special positions are discussed in [33]. P ROPOSITION 30. For every 0 < λ < 1 there exist ρ = ρ(λ) 1 and τ = τ (λ) 1 such that the following holds. Let X be an N -dimensional Banach space. There exists a λN dimensional quotient Y of X and a Euclidean norm | · |2 on Y , with the unit ball B2 , such that (i) vr(Y ) ρ, −1 (ii) d(X, N 2 ) B2 ⊂ BY ⊂ 2B2 , (iii) there is an orthogonal basis {xi } in (Y, | · |2 ) such that maxi xi Y τ . An analysis of probabilistic estimates in the previous sections shows that these estimates for Xn ∈ Xn,m depend on the ratio of volumes of unit balls of further quotients of Xn (by orthogonal projections) and the volume of the Euclidean ball of the corresponding dimension. On the other hand, ratio of volumes of this type is well known to play a crucial role in the existence of Euclidean sections of proportional dimension of arbitrary convex bodies. We ﬁrst brieﬂy discuss this latter connection. Let E ⊂ RN be an arbitrary ellipsoid such that E ⊂ B (then we also have x xE for all x ∈ X). Set ρ = (vol B/ vol E)1/N . The volume ratio argument proved in ([46]) (see also [44], Chapter 6) shows that for every 1 k < N there exists a k-dimensional subspace E ⊂ RN such that xE (4πρ)N/(N−k) x, for x ∈ E. In particular, d(E, k2 ) (4πρ)N/(N−k) . The set of all such k-dimensional subspaces has Haar measure larger than 1 − 2−N . So the dependence of the Euclidean distance of k-dimensional subspaces of X on the ratio of volumes ρ is poor; it is of polynomial type, with the degree of polynomial tending to ∞ when k = γ N and γ → 1. As easy examples show, this dependence cannot be improved in general. We shall therefore need more sophisticated volumetric invariants, closely connected with so-called volume ratio numbers (cf. [44], Chapter 9, also [40]). For 1 k N set

1/

vk, (X) = vk, (B) = sup inf vol F ∩ PE (B) / vol B2 , E

F

(19)

where the supremum runs over all k-dimensional subspaces E ⊂ RN and the inﬁmum runs over all -dimensional subspaces F ⊂ E. The reader may observe that the deﬁnition above differs slightly from the one in [30], and yields a smaller value of the invariant. Thus the results below imply corresponding results from [30]. Let us illustrate this invariant on a simple example. Let E be an ellipsoid on RN given by N E = {x = (ti ) ∈ R | |ti /λi |2 1}, with the lengths of semiaxes satisfying λ1 λ2 · · · λN > 0. Then it is not difﬁcult to see that for any 1 k N we have vk, (E) =

k $ i=k−+1

1/ λi

.

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We shall write vk (B) for vk,k (B). Clearly, v1 (B) is the norm of the formal identity operator I : B → B2N . Also, vN (B) = (vol B/ vol B2N )1/N . Using classical geometric inequality of Alexandrov, or Steiner–Minkowski formula, it can be shown that v1 (B) · · · vN (B) ([40], see also [44], Chapter 9). Using Santaló’s and inverse Santaló’s inequalities (cf., e.g., [44], Chapter 8) we get that vk, (B) is equivalent, up to universal constants, to the expression, supE (1/v (B 0 ∩ E)), where B 0 denotes the polar of B and the supremum runs over all k-dimensional subspaces E ⊂ RN . For a ﬁxed k this expression is increasing in . In particular we have vk, (B) Cvk, (B) for 1 k N , where C 1 is a universal constant. If B ⊂ B2N , then Santaló’s and inverse Santaló’s inequalities imply that vN (B) (vol B2N / vol B 0 )1/N CvN (B), where C > 1 is a universal constant. Therefore our previous discussion about Euclidean sections of the polar B 0 suggests by duality, that our volumetric invariant can be used for ﬁnding Euclidean quotients of B of proportional dimension. The following lemma shows that it is indeed so. In part (i) the dependence on a (simpler version of) the volumetric invariant is polynomial, while in part (ii), the dependence on the (more complicated) invariant is linear. L EMMA 31. Let X = (RN , · ) be an N -dimensional space and assume that B = BX ⊂ B2N . (i) Let 1 k N . For every σ > 0 there exists a quotient space Y of X with m = −(1+σ ) , dim Y = σ k/(1 + σ ) for which the Euclidean distance d(Y, m 2 ) Cvk (B) where C is an absolute constant. (ii) Let 0 < ζ 1/2. Let 1 k N and = ζ k. There exists a quotient space Y of X with m = dim Y = k − 2 so that the Euclidean distance d(Y, m 2) Cζ −2 vk, (B)−1 , where C is an absolute constant. The proof of (i) easily follows by the volume ratio method. The proof of (ii) is more involved and uses the existence and properties of an M-ellipsoid. (This proof can be found, e.g., in [30], Proposition 3.4, where it is written in terms of s-numbers.) It is noteworthy that case (i) admits a random version that follows from the volume ratio argument, while the proof in case (ii) provides no information on randomness of quotients constructed there. Let us mention that it has been proved very recently in [34] that it is possible to obtain simultaneously a linear dependence on an invariant of a volumetric type and a randomness of quotients in the geometric form. In addition of using an M-ellipsoid, the argument heavily depends on geometric considerations on the Grassman manifold. For more details we refer the interested reader to [34]. Finally, we pass to probabilistic estimates for arbitrary convex bodies. The heart of technical arguments analogous to Sections 3 and 4 will be done when the image body is a ran. For a dom quotient of an arbitrary body, and the domain body is a random quotient of n+m 1 Banach space W = (Rn+m , · ) consider a random quotient Z = qn,m (ω)(W ) ∈ Zn,m,BW as deﬁned in Section 2. By XZ we denote qn,m (ω)(n+m ), that is, XZ is a quotient of n+m 1 1 given by the same quotient map as Z. T HEOREM 32. Let ρ 1, and 0 < δ < 1/4. Let n 1 be an integer and let m = [δn]. Let W = (Rn+m , · ) be a Banach space such that the volume ratio satisﬁes vr(BW ) ρ.

Quotients of ﬁnite-dimensional Banach spaces

1227

(i) Let α > 0 be sufﬁciently small (depending on ρ and δ). A majority of pairs of random quotients (X, Z) ∈ Xn,m × Zn,m,BW satisfy T : X → Z αvk (BW )−1 , for all operators T ∈ L(Rn ) with at least n/2 s-numbers larger than or equal to 1, where k = [n/2 − δn]. (ii) Let 0 < κ < 1/2 and assume additionally that δ < κ/4. Let α > 0 be sufﬁciently small (depending on ρ, δ and κ/δ). A majority of random quotients Z ∈ Zn,m,BW satisfy T : XZ → Z αvk (BW )−1 , for all mixing operators T ∈ Mixn (κn, 1) where k = [κn/4 − δn]. O UTLINE OF THE PROOF. (i) The proof consists of two steps. For arbitrary ﬁxed Z ∈ Zn,m,BW and T ∈ L(Rn ) deﬁne the set A(α, Z, T ) = X ∈ Xn,m | T : X → Z αvk (BW )−1 .

(20)

We shall start by estimating the probability of A(α, Z, T ) from above, for relevant operators T and a ﬁxed Z. Next, we estimate the minimal cardinality of an ε-net in a certain set of operators. Combining these two estimates for every ﬁxed Z ∈ Zn,m,BW we get that the set of X ∈ Xn,m such that the inequality in (i) is violated for some relevant T , has measure less than or equal to exp(−an), where a = a(δ, α, ξ ) > 0. The ﬁnal statement will follow by Fubini’s theorem. Fix Z = qn,m (ω)(W ). The estimate for the probability of A(α, Z, T ) is obtained by the method of Lemma 4. However, applying this method directly to Z would result in an estimate in terms of the invariant vk evaluated on BZ rather than on BW . To overcome this difﬁculty, we ﬁrst apply an additional orthogonal projection Q in Rn such that Q(BZ ) is equal to a certain orthogonal projection of BW . Speciﬁcally, set Q to be the orthogonal projection in Rn with kernel ker Q = span[gn,i (ω) | 1 i m] and observe that R = Qqn,m (ω) is an orthogonal projection in Rn+m . The identity Q(BZ ) = R(BW ) will allow us to use the invariant of BW . Fix an operator T ∈ L(Rn ) with at least n/2 s-numbers larger than or equal to 1. Then QT ∈ L(Rn ) has at least n/2 − δn s-numbers greater than or equal to 1. Using essentially the same line of an argument as in the proof of Lemma 4, we set E = span[ui (QT ) | 1 i k], where the vectors ui (QT ) belong to the range of QT and are obtained from the polar decomposition of QT (see Fact 2). Then E ⊂ Q(Rn ) and we get, by Fact 1,

P A(α, Z, T ) P X ∈ Xn,m | QT : X → Q(Z) αvk (BW )−1 m $

P QT gn,i ∈ αvk (BW )−1 Q(BZ ) i=1

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P. Mankiewicz and N. Tomczak-Jaegermann

#

k vol PE Q(BZ ) m α en/kvk (BW )−1 vol B2k #

km α en/k .

(21)

The latter inequality follows from the fact that PE Q(BZ ) = PE Qqn,m (ω)(BW ) = PE R(BW ) is a rank k orthogonal projection of BW , and the deﬁnition of vk (BW ). An ε-net argument is described in a separate lemma. L EMMA 33. Let K ⊂ Rn be a convex symmetric body, with volume ratio vr(K). Then, for every 0 < ε 1, the set of operators T = T ∈ L Rn T : n1 → K 1 admits an ε-net N in the operator norm from n2 to K with cardinality card(N ) 2 (C/ε)n vr(K), where C is an absolute constant. Let E ⊂ K be the ellipsoid of maximal volume contained in K, and let u ∈ L(Rn ) be an isomorphism such that u(E) = B2n . Then it sufﬁces to prove the statement with K replaced by u(K). This case follows directly from Corollary 8 and the fact that B2n ⊂ u(K). Combining (21) with the lemma above, as in Lemma 9, we get the probability estimate with a ﬁxed space Z, and (i) follows by Fubini theorem. (ii) The proof of this case is a suitable adaptation of the proof of Theorem 12, where one deals with the norms T : X → Z. In the future we may want to use the estimates analogous to Theorem 32 to ﬁnd proportional dimensional Euclidean quotients of the body BW , with good dependence of the distance on the given proportion close to 1. As was explained earlier (see Proposition 31 and the remarks that precede it), the best results in this direction are achieved by using the invariant vk, instead of vk . The theorem below shows that this is possible and one can replace vk by vk, in Theorem 32. T HEOREM 34. Let 0 < ξ 1. Theorem 32 holds for vk, replacing vk , for = [ξ k], with α depending additionally on ξ . O UTLINE OF THE PROOF. Since vk, (BW ) Cvk (BW ) for all k, where C is a universal constant, we need to strengthen the estimates of probability in (21). To this end let E be the k-dimensional subspace from the previous proof, and let F = span[u¯ i (QT ) | 1 i k], where the vectors u¯ i (QT ) belong to the domain of QT and are obtained from the polar decomposition of QT (see Fact 2). Note that PE QT = QT PF and that S = QT |F is an isomorphism from F to E and S −1 has norm less than or equal to 1. Fix 1 i m

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1229

and observe that

P QT gn,i ∈ αvk, (BW )−1 Q(BZ )

P PE QT gn,i ∈ αvk, (BW )−1 PE Q(BZ )

P QT PF gn,i ∈ αvk, (BW )−1 PE Q(BZ ) . Let E0 ⊂ E be an -dimensional subspace such that

vol E0 ∩ PE Q(BZ ) = inf vol H ∩ PE Q(BZ ) | H ⊂ E, dim H = . Let F0 = S −1 E0 . Then PF = PF0 + PF1 , where F1 is an orthogonal complement of F0 in F . Therefore the probability above is equal to

P QT PF0 gn,i ∈ −QT PF1 gn,i + αvk, (BW )−1 PE Q(BZ ) . Since QT PF0 takes values in E0 , and PF0 gn,i and PF1 gn,i are independent Gaussian variables, by rescaling PF0 gn,i and using Brunn’s inequality we get, as usual, that the latter probability is less than or equal to

P QT PF0 gn,i ∈ E0 ∩ −QT PF1 gn,i + αvk, (BW )−1 PE Q(BZ )

# #

en/ en/

vol(E0 ∩ (−QT PF1 gn,i + (αvk, (BW )−1 )PE Q(BZ ))) vol B2

vol(E0 ∩ (αvk, (BW )−1 )PE Q(BZ ))

vol B2 #

#

en/ αvk, (BW )−1 vk, (BW ) = α en/ . Hence

m # P X ∈ Xn,m | T : X → Z αvk, (BW )−1 α en/ , which is the required analogue of (21). The rest of the proof is the same as in Theorem 32. , set τ = To consider operators from quotients of W rather than from quotients of n+m 1 → W (i.e., τ = max e ). Then a direct application of the previous Id : n+m i 1in+m 1 results shows that T HEOREM 35. Under the hypothesis of Theorems 32 and 34 we have that (i) a majority of pairs of random quotients (Z1 , Z2 ) ∈ Zn,m,BW × Zn,m,BW satisfy

−1 T : Z1 → Z2 α τ vk, (BW ) ,

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P. Mankiewicz and N. Tomczak-Jaegermann

for all operators T ∈ L(Rn ) with at least n/2 s-numbers larger than or equal to 1, where k = [n/2 − δn]; (ii) a majority of random quotients Z ∈ Zn,m,BW satisfy

−1 T : Z → Z α τ vk, (BW ) , for all mixing operators T ∈ Mixn (κn, 1) where k = [κn/4 − δn]. As a free bonus we get lower estimates for the Banach–Mazur distances and basis constants: a majority of random quotients from Zn,m,BW satisfy

−2 d(Z1 , Z2 ) α 2 τ vk, (BW ) ,

−1 bc(Z) α 2 τ vk, (BW ) .

(22) (23)

Inequalities (22) and (23) recover (and, due to the presence of a smaller invariant vk, , are stronger than) main technical estimates in [30] (Theorem 3.6) and [32] (Theorem 2.2). We conclude with the important observation that the lower estimates presented in this section are in general optimal (up to multiplicative constants). This is the case if W = N 1 . Indeed, this follows from Theorems 3 and 10 by letting k = βN and = ξ k, for some ﬁxed proportions 0 < β, ξ 1, and using Lemma 5 to see that vk, (B1N ) ∼ vk (B1N ) ∼ √ C(β, ξ )/ N .

8. Spaces with large Euclidean subspaces and quotients This section is devoted to the interplay between purely geometric properties of convex bodies and the structure of the underlying Banach spaces, and its impact on the local theory of Banach spaces. It was Bourgain who ﬁrst realized that ideas behind the proof on the diameter of Minkowski compactum can be adopted to a general context and used for a solution of the ﬁnite-dimensional version of the homogeneous space problem. Further development of the theory led to a deeper understanding of the relationship between ﬁner volumetric invariants and structure of Banach spaces, in particular producing several interesting results in the local theory of Banach spaces. The local concept which is relevant is called weak cotype 2 ([38]); for more details we refer the reader to [44], Chapter 10, and references therein. We will use the following geometric deﬁnition of weak cotype 2 spaces: a Banach space X is of weak cotype 2 whenever there exist 0 < δ0 < 1 and D0 1 such that every ﬁnite-dimensional subspace E ⊂ E with dim E δ0 dim E and with the Banach–Mazur of X contains a subspace E satisfying dE D0 . distance dE to the Euclidean space of dimension equal to dim E The main strategy of proofs common for several results of this section can be described in the following way. Let n 1 and m = [δn], for some δ > 0. Assume that we are given some regularity conditions on quotients of an (n + m)-dimensional Banach space W , in the form of an upper bound for certain invariants evaluated on all n-dimensional quotients

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1231

of W . We would like then to deduce that W has a well-Euclidean quotient of an a priori given proportional dimension. For reasons which become clear in the next paragraph, the actual construction begins from an arbitrary (n + [δ1 n])-dimensional quotient W1 of W , for a certain 0 < δ1 δ. Next, one passes to a quotient of W1 to obtain a (n + [δ2 n])-dimensional quotient W2 which is in a “special position” (see Proposition 30) for a certain δ2 > 0 sufﬁciently close to 0. Applying to the latter quotient estimates (22) and (23) (with k = , if less precision is required), one arrives at a random n-dimensional quotient Z ∈ Zn,[δ2 n],BW2 for which the invariants in question admit a lower estimate in terms of (vk, (BW ))−1 . Our assumptions on the upper bounds for these invariants yield lower estimates for vk, (BW ), which in turn allow the use of volume-type arguments in the dual to get the existence of 0 < γ < 1 and γ n-dimensional well Euclidean subspaces in (W1 )∗ (see Lemma 31). By dualizing back we conclude that every (n + [δ1 n])-dimensional quotient W1 admits a γ n-dimensional well Euclidean quotient. If the dimension γ n is larger than required, one could have skipped the ﬁrst step, and right away pass to an (n + [δ2 n])-dimensional quotient in a special position. If γ is too small, one can ﬁnd Euclidean quotients of higher dimensions using a method from [38] (cf. also [3,30]) of constructing well Euclidean subspaces of large proportional dimensions in spaces saturated with smaller well Euclidean ones. In such a case, the resulting dimension and the Euclidean distance also depend on a smart choice of δ1 , δ2 and . The following theorem on ﬁnite-dimensional homogeneous Banach spaces was proved in [3] for α sufﬁciently small, and in [30] for arbitrary α ∈ (0, 1). To follow the pattern of this exposition, we state the theorem in a clearly equivalent form, for quotients spaces rather then subspaces. T HEOREM 36. Let 0 < α < 1, and let W be a ﬁnite-dimensional Banach space. Set K = sup d(Z1 , Z2 ), where Z1 , Z2 are quotients of W with dim Z1 = dim Z2 = [α dim W ]. Then

cK 3/2 if 0 < α < 2/3, dim W d W, 2 if 2/3 α < 1, cK 2 where c = c(α). O UTLINE OF THE PROOF. We sketch the argument for α = 1/2. Setting n = [α dim W ] = [dim W/2], we proceed as outlined at the beginning of this section with δ = 1, δ1 = 3/32 and δ2 = 1/16, arriving at W2 . Inequality (22) applied for n and m = [δ2 n] implies the existence of two n-dimensional quotients of W2 whose distance is larger than or equal to cvk, (BW2 )−2 , where k = [n/8 − n/64] = [7n/64], = k/4 and c depends on parameters related to the good position, and hence in our case is an absolute constant. By our assumptions we get vk,k/4 (BX2 ) cK −1/2 . Thus, by Lemma 31, there exists a√ quotient of W2 , and hence of W1 as well, with dimension k/2 = [7n/128], which is C K-Euclidean. Since k/2 is clearly smaller than [α dim W ], we still need to ﬁnd a well Euclidean quotient of dimension [α dim W ], which can be done by the method described before, √ noting that the choice of the quotient W1 was arbitrary. Therefore W itself admits a C K-Euclidean quotient of dimension n. Thus, by the triangle inequality, every ndimensional quotient of W is C K 3/2 -Euclidean, and it is well known (see, e.g., [56],

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P. Mankiewicz and N. Tomczak-Jaegermann

Proposition 26.1) that this implies that W itself is C

K 3/2 -Euclidean as well (where C

is an absolute constant). In the context of Theorem 36 let us mention the following recent result, [34] T HEOREM 37. For every 0 < λ < 1, and 0 < ε < 1/2 there exists c = c(ε, λ) > 0 with the property: let K1 and K2 be two symmetric convex bodies in RN1 and RN2 respectively such that for i = 1, 2 the Euclidean unit ball B2Ni is the ellipsoid of minimal volume containing Ki . Let 1 n λ min(N1 , N2 ) and set m = (1/2 − ε)n. Then

GN2 ,n

GN1 ,n

c GN1 ,m

d PH1 (K1 ), PH2 (K2 ) dμN1 ,n (H1 ) dμN2 ,n (H2 )

d PL1 (K1 ), PL1 B2N1 dμN1 ,m (L1 )

× GN2 ,m

d PL2 (K2 ), PL2 B2N2 dμN2 ,m (L2 ),

where for 1 k, Gk, denotes the Grassman manifold of all -dimensional linear subspaces in Rk and μk, denotes the normalized Haar measure on Gk, . Let us note that in contrast with results of the previous section, Theorem 37 is valid for quotients of arbitrary dimension 1 n λ min(N1 , N2 ), not just for quotients of proportional dimension. Moreover, one does not require passing to quotients in a “special position”. This allows to consider the geometric representation of quotients as orthogonal projections of Ki ’s, and obtain “randomness” with respect to the natural measures μNi ,n . The proof of Theorem 37 is based upon classical properties of contact points between the ellipsoid of minimal volume containing a given body K and the body itself, delicate estimates of the distribution of s-numbers of operators on RN given by random matrices with independent Gaussian entries, and relations of such operators to some geometric properties of Grassman manifolds. The following result is a counterpart of Theorem 36 for basis constants. Let us just mention that the notion of weak type 2 is a bit stronger than weak cotype 2 for the dual space, and a space is called weak Hilbert if it is simultaneously of weak type 2 and weak cotype 2. For the details, and in particular for the deﬁnitions of the corresponding invariants, we refer the reader to [44], Chapters 10, 11 and 12. T HEOREM 38 ([33]). Let 0 < α < (1 + 2−8 )−1 , and let W be a ﬁnite-dimensional Banach space. Set K = max(sup bc(E), sup bc(Z)), where the suprema are taken over all [α dim W ]-dimensional subspaces E ⊂ W and all [α dim W ]-dimensional quotients Z of W . Then W is a weak Hilbert space and the weak type 2 and weak cotype 2 constants satisfy wT2 (W ) CK 63

and wC2 (W ) CK 62 ,

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1233

where C = C(α). Moreover, if X is a ﬁnite- or inﬁnite-dimensional Banach space such that all of its ﬁnite-dimensional subspaces have basis constant uniformly bounded above by some K < ∞, then X is of a weak cotype 2, with weak-cotype 2 constant depending on K only. The additional interest of the ﬁnite-dimensional part of the theorem lies in the fact that from the assumptions on a ﬁxed proportional level one gets information about all subspaces of W of arbitrary dimensions. In view of the deﬁnition of weak cotype 2 it is a legitimate question to ask whether one can replace the existence of proportional-dimensional Euclidean subspaces inside every ﬁnite-dimensional subspace by a similar saturation with subspaces satisfying weaker properties. This can be done, for example, for the basis constant; or, indeed, for any invariant considered in Section 5 ([33]). Namely, there exists 0 < γ0 < 1 such that a ﬁnite- or inﬁnitedimensional Banach space X is of weak cotype 2 if and only if there exists K < ∞ such that every ﬁnite-dimensional subspace E ⊂ X contains E0 ⊂ E, with dim E0 γ0 dim E, such that bc(E0 ) K. We now pass to the description of some other properties of random subspaces of ﬁnitedimensional spaces and related inﬁnite-dimensional facts. They will be connected with some special factorizations of operators, mostly with the Gordon–Lewis property discussed in Section 6. The search for proportional-dimensional subspaces with large GL-constant inside an arbitrary ﬁnite-dimensional Banach space is based on the following fact ([6]). There exist a ﬁxed proportion 0 < γ0 < 1 and a constant C 1 such that for every n-dimensional Banach space W and every invertible operator T : W → n2 , there exists a subspace E ⊂ W with dim E γ0 n such that π1 (T |E) Cπ2 (T ). Here πp (·) stands for the p-summing norm (see [14], Section 10). Moreover, if W = (Rn , · W ) and the unit ball B2n is the ellipsoid of maximal volume contained in BW and T is a formal identity operator, then E can be chosen to be a random subspace in the corresponding Grassman manifold. As the next result shows, there is a relation between a lower estimate for the GLconstant of random subspaces of W and the existence of further subspaces which are wellEuclidean. For a ﬁnite-dimensional space W , let k(W ) denote the maximal dimension k such that d(F, k2 ) 2 for some k-dimensional subspace F ⊂ W . T HEOREM 39 ([6]). There exist 0 < c, γ0 , γ1 < 1 such that whenever W = (Rn , · ) is an n-dimensional Banach space for which B2n is the ellipsoid of maximal volume, then a random subspace (in the sense of Grassman manifold) E ⊂ W with dim E γ0 n satisﬁes: (i) gl(E) c(n/k(W ))1/2 , (ii) there exists a further subspace F ⊂ E with m = dim F γ1 dim E such that gl(E) cd(F, m 2 ). These results are proved in [6]: (i) is Corollary 3.7, (ii) is a combination of Propositions 3.4 and 3.3. It is worthwhile to mention that the above condition (i) applied to a random subspace E∞ of n∞ does not yield the sharp estimate from Theorem 26. On the other hand, the same condition gives the sharp estimate for proportional-dimensional subspaces of np ,

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with 2 < p < ∞. For this it is enough to recall that k(np ) Cp n2/p , where Cp depends 1/2−1/p for all m-dimensional subspaces on p (see, e.g., [39], 5.4), and that d(E, m 2)m n E ⊂ p . (see, e.g., [56]). The arguments involved here also yield an interesting result concerning projection constants. This is a classical invariant of ﬁnite-dimensional Banach spaces connected with the factorization of the identity operator through ∞ , i.e., λ(X) = γ∞ (IdX ). A very precise asymptotic information on the projection constant λ(np ) is known for 1 p 2 ([19]); here √ let us only recall that, up to a multiplicative constant, this order is n (see, e.g., [56]). On √ the other hand, it is well known that λ(W ) n for any n-dimensional Banach space W (see, e.g., [56]). It turns out that for an arbitrary ﬁnite-dimensional Banach space the projection constant of a random subspace of an a priori given ﬁxed proportion is of the maximal order ([6]). (Randomness is taken in the sense of the ellipsoid of maximal volume.) Passing to inﬁnite-dimensional spaces, note that the GL-property is local, which means that if the GL-constants are uniformly bounded on all ﬁnite-dimensional subspaces of a given space X, then X has the GL-property. T HEOREM 40. (i) If all subspaces of a Banach space X have the GL-property, then X is of weak cotype 2. (ii) There exists 0 < γ0 < 1 such that a Banach space X is of weak cotype 2 if and only if all ﬁnite-dimensional subspaces E ⊂ X contain further subspaces E0 ⊂ E, with dim E0 γ0 dim E, with all gl(E0 ) uniformly bounded. Condition (i) easily follows from Theorem 39(i). Condition (ii) follows from the results of [6], and it appeared in [44] (Theorem 10.13).

9. An inﬁnite-dimensional construction The idea of using ﬁnite-dimensional tools to constructions of inﬁnite-dimensional Banach spaces is quite natural in the local theory and was widely used for a long time. We describe in very general terms the strategy of one such technique. We want to construct an inﬁnite-dimensional Banach space X which fails a property expressed in terms of the existence of bounded operators on X of a speciﬁc type. To localize a problem of this nature we look for X in the form of a direct sum of ﬁnite-dimensional spaces Xn , with dim Xn → ∞, satisfying two conditions. Firstly, there are no operators of the required type acting on an individual space XN , nor on partial sums YN = N n=1 Xn , with an upper bound for the norms uniform in N . Secondly, no non-trivial operator on YN can be factored through the tail ∞ n=N+1 Xn , with a uniform bound on the norms of factorizations. Such spaces can be found by considering appropriate ﬁnite-dimensional parameters. Typically, arguments depend on deep distinctions in geometry and structure of ﬁnite-dimensional p -spaces for varying p (1 p ∞). We shall apply the described approach using as building blocks random spaces studied in Sections 3 and 4. Resulting inﬁnite-dimensional Banach spaces will fail some seemingly elementary and natural structural properties (see Theorems 46, 44, 45).

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The ﬁrst argument based on glueing together ﬁnite-dimensional random spaces is due to Bourgain who in [2] disproved the complex version of Mazur–Ulam theorem (see Theorem 47). His idea was picked up by Szarek [49,50]. In this section we present a more general argument from [25] based on the construction of linear-multiplicative functionals on the algebra of all bounded operators. The following proposition presents a collection of tools needed in our construction. P ROPOSITION 41. For every n 64 there exists a scale of Banach spaces Xn,p = (Rn , · n,p ), for p ∈ [1, 2), such that for every operator T ∈ L(Rn ) there exist subspaces ET , FT with dim ET 7n/8, dim FT 3n/4 and λT ∈ R such that setting RT = T − λT IdRn and m(Xn,p ) = m(Xn,p , 1/32), for every p ∈ [1, 2) we have (i) 2−1 (2n)1/p−1 x2 xXn,p xp n1/p−1/2 x2 for every x ∈ Rn , (ii) RT |ET 2→2 (8/ m(Xn,p ))T : Xn,p → Xn,p , (iii) T |FT 2→2 (32/ m(Xn,p ))γ2 (T : Xn,p → Xn,p ). Moreover, (iv) αn1/p−1/2 m(Xn,p ) 2n1/p−1/2 , (v) |λT | CT : Xn,p → Xn,p , where C > 1 and α > 0 are numerical constants. P ROOF. First consider the case p = 1. For every integer n 32 let Xn,1 = qn,n (ω)(2n 1 )∈ Xn,[δn] be any space satisfying the assertion of Theorem 12 with δ = 1, κ = 1/32 and some ﬁxed α ∈ (0, 1), depending on κ. Clearly, the condition √(i) is satisﬁed. (ii) and (iii) directly follow from Theorem 16. The inequality m(Xn,1 ) α n is a consequence of Theorem 12. On the other hand,√since by (i) the norm in X√ n,1 of any rank[n/2] orthogonal projection is not greater than 2 n, we have m(Xn,1 ) 2 n and it remains only to prove (v). To this end, pick x ∈ ET with xXn,1 = 1. Then, by (i), (ii) and (iv), |λT | = λT IdxXn,1 T xXn,1 + RT xXn,1 T : Xn,1 → Xn,1 + n1/2 RT x2 (1 + 8/α)T : Xn,1 → Xn,1 . This completes the argument for p = 1. For p ∈ (1, 2), set Xn,p = qn,n (ω)(2n p ). Since 1−1/p B , the same inclusion holds for the unit balls of X ⊂ (2n) and Xn,p . B2n ⊂ B2n 2n n,1 1 p 1 n n Also, since B1 ⊂ BXn,1 we infer that Bp ⊂ BXn,p , which together with (i) for p = 1 completes the proof of (i) for p ∈ [1, 2). (ii) and (iii) follow from Theorem 16. The inclusion of balls yields that for any operator T ∈ L(Rn ), we have T : Xn,p → Xn,p (2n)1/p−1T : Xn,1 → Xn,1 . Hence m(Xn,p ) (2n)1/p−1 m(Xn,1 ) 21/p−1αn1/p−1/2 . The remaining inequality in (iv) follows by the same argument as in the case p = 1. To prove (v) observe that, by Lemma 5, there is a numerical constant c1 > 0 such that for every p ∈ [1, 2) vol BXn,p (2n)n(1−1/p) vol BXn,1 c1 nn(1/2−1/p) vol B2n .

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On the other hand, n1/2−1/p B2n ⊂ BXn,p . Thus, by the standard volume ratio argument, there exists a numerical constant c2 > 0 such that for every p ∈ [1, 2) there exists a subspace Hp ⊂ Rn with dim Hp n/2 on which the norms · 2 and · Xn,p are c2 equivalent. More precisely, for every x ∈ Hp we have c2 n1/p−1/2x2 xXn,p n1/p−1/2x2 . (In fact, a slightly more careful argument shows that there is one subspace H which is good for all p’s.) Now, (v) with C = 1 + 8(c2 α)−1 follows by the same argument as in the case p = 1, applied to x ∈ ET ∩ Hp . Construction of the space X ([25,49,50]). Deﬁne by induction increasing sequences ∞ (nk )∞ k=1 ⊂ N and (pk )k=1 ⊂ (1, 2) in such a way that for every k 1 1/p −1/2

nk−1k

−1/2 1/pk −1/2

2 and nk−1 nk

>k

(24)

and set X=

∞

Xnk ,pk

k=1

,

(25)

2

where for k 1, the space Xnk ,pk satisﬁes Proposition 41. Let Pk : X → Xnk ,pk for k 1 be the canonical projection onto the k-th term, and let Yk = ker Pk . In the sequel, we shall write Xk instead of Xnk ,pk . Notice that the space X is superreﬂexive and admits a ﬁnite-dimensional decomposition (FDD). An important feature of X is a good control of the Euclidean distance of lowdimensional subspaces and quotients of the tails. Indeed, this is exactly the reason which prevents a factorization of non-trivial operators on an initial component of X through the corresponding tail. For an arbitrary Banach space Y and n 1 deﬁne

(26) ∂n (Y ) = sup d E, n2 | E ⊂ Y, dim E = n . It is clear that for every n-dimensional Banach space X and every Banach space Y and every T ∈ L(X) one has γ2 (T : X → X) γY (T : X → X)∂n (Y ),

(27)

where γY denotes the factorization constant through Y . Observe that for a ﬁxed k, the distance of the space k−1 j =1 Xj to a Euclidean space 1/2

is less than or equal to nk−1 . On the other hand, recall that by the well known result of Lewis, an m-dimensional subspace of a quotient space F of Lp , for p ∈ [1, 2], satisﬁes 1/p−1/2 (cf., e.g., [56], Sections 27 and 28 and page 232). This and (24) d(F, m 2)m yield that nk -dimensional quotients and subspaces of ∞ j =k+1 Xj are 2-Euclidean. Thus we have

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L EMMA 42. For every integer k 1 and a subspace E1 and a quotient space E2 of 1/2 Yk , with dim Ei nk (i = 1, 2), one has d(Ei , 2dim E ) nk−1 (i = 1, 2). In particular, 1/2

∂nk (Yk ) nk−1 . Now we pass to the main result about the structure of the algebra of all bounded operators L(X). T HEOREM 43 ([25]). Let X be the space deﬁned in (25). Then the Banach algebra L(X) admits a Banach algebra homomorphism h onto the Banach algebra C(βN), where βN is ˇ the Stone–Cech compactiﬁcation of N. First note that the algebra C(βN) can be identiﬁed with l∞ with the coordinatewise multiplication. Therefore, if ψk for k ∈ N denotes the k-coordinate functional on l∞ then the maps ψk ◦ h : L(X) → R are non-trivial linear multiplicative functionals on L(X) for k = 1, 2, . . . . In fact it is proved in [25] that L(X) admits continuum different linear multiplicative functionals. However, in most applications of Theorem 43 it sufﬁces to have just one such functional (cf. Theorems 44, 45 and 47). Also note that for every m 2, the algebra of m × m matrices, identiﬁed with L(m 2 ), does not admit a non-trivial linear multiplicative functional. Therefore there is no homon morphism from L(m 2 ) into L(X), mapping Id2 onto IdX . Before we embark on the proof of the existence of non-trivial linear multiplicative functionals let us recall basic facts concerning the notion of ultraﬁlters within the (very limited) setting needed here. Let I be an arbitrary inﬁnite set. A family U of subsets of I is said to be a ﬁlter if and only if it satisﬁes the conditions: (i) U does not contain an empty set, (ii) if U1 , U2 ∈ U then U1 ∩ U2 ∈ U and (iii) if U0 ∈ U and U0 ⊂ U then also U ∈ U . A ﬁlter can be interpreted as a family of subsets of measure 1 for some ﬁnitely additive normalized measure on I . For a ﬁxed ﬁlter U on I and a function f : I → R we write limU f = α if and only if for every ε > 0 the set f −1 ((α − ε, α + ε)) belongs to U . The ﬁlter U0 (I ) consisting of all complements of ﬁnite subsets of I is called Fréchet ﬁlter. A maximal ﬁlter with respect to the order induced by the inclusion is called an ultraﬁlter. It is said to be free if it contains the Fréchet ﬁlter U0 (I ). It follows from the maximality of an ultraﬁlter U that for every partition of I into a ﬁnite number of disjoint subsets U1 , U2 , . . . , Uk exactly one of the Ui ’s belongs to U . It is an easy exercise to check that this property implies that for every bounded function f : I → R the limit limU f exists. P ROOF OF T HEOREM 43. For every integer k 1 and T ∈ L(X), ﬁx an arbitrary λPk T |Xk satisfying Proposition 41 for the operator Pk T |Xk . Moreover, assume, as we clearly may, that λS = λ if Pk S|Xk = λIdXk , for λ ∈ R. This deﬁnes, for every k 1, the function φk : L(X) → R by the formula φk (T ) = λPk T |Xk . Let (Ni )∞ i=1 be a partition of N into disjoint inﬁnite subsets and let Ui for i ∈ N be an arbitrary free ultraﬁlter on N with Ni ∈ Ui . By Proposition 41, for every k 1 and T ∈ L(X) we have |φk (T )| CT : X → X,

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and therefore for every ultraﬁlter U on N the limit ΦU (T ) = limU φk (T ) exists for every T ∈ L(X). Finally, deﬁne h : L(X) → C(βN) by setting

∞ h(T ) = ΦUi (T ) i=1 . It is not difﬁcult to see that h is onto. The proof will be completed once we show that for every free ultraﬁlter U on N the function ΦU is a non-trivial linear multiplicative functional. First observe that ΦU (IdX ) = 1. Thus ΦU is non-trivial. We shall prove the multiplicativity only; the proof of linearity goes along the same lines. Fix a free ultraﬁlter U and assume to the contrary that ΦU is not multiplicative. This means that there exists ε > 0, U ∈ U and operators T1 , T2 , T3 ∈ L(X) with T3 = T1 T2 such that φk (T1 )φk (T2 ) − φk (T3 ) > ε for every k ∈ U. (28) Now, ﬁx k ∈ U with k > 500C(αε)−1 T1 : X → XT2 : X → X. Consider X as an l 2 product of Xk and Yk , where Xk and Yk have been deﬁned above, and for i = 1, 2, 3 write Ti in a matrix form with operator entries; & ' Ai Ci Xk . Di Bi Yk By the deﬁnition of φk , for i = 1, 2, 3 we can write Ai = φk (Ti )IdRnk + RAi . By multiplying matrices we get A3 = A1 A2 + C1 D2 . Hence

φk (T1 )φk (T2 ) − φk (T3 ) IdRnk = −φk (T1 )RA2 − φk (T2 )RA1 − RA1 RA2 + RA3 − C1 D2 . By (28), for every x ∈ Rnk

φk (T1 )φk (T2 ) − φk (T3 ) IdRnk x > εx2 . 2

(29)

(30)

On the other hand, by Lemma 42 and (27) γ2 (C1 D2 ) C1 : Yk → Xk D2 : Xk → Yk ∂nk (Yk ) 1/2

T1 : X → XT2 : X → Xnk−1 . Thus, by Proposition 41(iii), there is a subspace F ⊂ Rnk with dim F 3nk /4 such that

1/2 C1 D2 |F 2 32/ m(Xk ) nk−1 T1 : X → XT2 : X → X.

(31)

Similarly, by Proposition 41(ii), for i = 1, 2, 3 there exist subspaces Ei ⊂ Rnk of dimension at least 7nk /8 such that

Ai |Ei 2→2 8/ m(Xk ) Ti : X → X. (32)

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−1 Let E = F ∩ E1 ∩ E2 ∩ E3 ∩ RA (E1 ) and note that dim E nk /4. By the choice of k, 2 and by (31) and (32), an easy calculation show that for every x ∈ E we get

−φk (T1 )RA − φk (T2 )RA − RA RA + RA − C1 D2 x εx2 /2. 2 1 1 2 3 2→2 This estimate combined with (29) and (30) contradicts (28), which completes the proof. Now we present several applications of Theorem 43 and of its proof. Linear topological properties of inﬁnite-dimensional Banach spaces which we are about to consider are deﬁned in terms of the existence of bounded linear operators, or sequences of operators, with natural algebraic properties (such as a sequence Pn of commuting rank n projections or T 2 = −Id). As we shall see, even one linear multiplicative functional on L(X) prohibits their existence. We start with an inﬁnite-dimensional version of Theorem 17. T HEOREM 44 ([49]). The space X from Theorem 43 is not real isomorphic to any complex Banach space. Indeed, otherwise it would admit a bounded operator A ∈ L(X) such that A2 = −Id. Thus for every non-trivial linear multiplicative functional Φ on L(X) we would have −1 = Φ(A2 ) = (Φ(A))2 , which is impossible. T HEOREM 45 ([25]). The space X from Theorem 43 is not isomorphic to any Cartesian power of a Banach space. This follows from the easy fact that the space of operators on an m-th power of a Banach space admits a non-trivial homomorphism from the algebra of m × m matrices, combined with the remark after Theorem 43. Using the line of argument from the proof of Theorem 43 one can prove the inﬁnitedimensional analogue of Theorem 4. Namely T HEOREM 46 ([50]). The space X from Theorem 43 does not admit a Schauder basis. P ROOF. By Theorem 16(ii), in the notation of the proof of Theorem 43, for every k 1 we have

φk (T1 ) − φk (T2 ) 8/ m(Xk ) T1 : X → X + T2 : X → X ,

(33)

for every pair of operators T1 , T2 ∈ L(X) such that ker(Pk (T1 − T2 )|Xk ) > nk /4. ∞ Assume that {xi }∞ i=1 is a Schauder basis in X with bc{xi }i=1 = b. Let Pm for m 1 −1 2 denote the m-th basis projection. Pick k > 2500Cα b . By (33) and (24), since φk (P1 ) 8b/ m(Xk ) and since, by Proposition 41(ii), lim supi→∞ φk (Pi ) .9, one can easily deduce that for some m0 we have 7/16 < φk (Pm0 ) < 9/16. Now, by the argument from the proof of Theorem 43, we get |(φk (Pm0 ))2 − φk (Pm2 0 )| 1/10. This is a contradiction, since (φk (Pm0 ))2 < (9/16)2 while φk (Pm2 0 ) = φk (Pm0 ) > 7/16.

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In fact, even a stronger result is valid. With a more sophisticated choice of sequences {nk } and {pk } one can prove that for every subspace Z ⊂ X the space X ⊕ Z has no Schauder basis ([28]). In contrast, note that since X has a ﬁnite-dimensional decomposition, by a general theorem ([41]) there exists an inﬁnite-dimensional Banach space Z1 such that X ⊕ Z1 has a basis. Thus to ﬁnd such a space Z1 one has to leave the space X itself. Using as the building blocks the complex spaces from Theorem 18 instead of the Xk ’s one can obtain a result of Bourgain that initiated the development presented here. T HEOREM 47 ([2]). There exists a complex superreﬂexive Banach space Y with a ﬁnitedimensional decomposition such that Y and its complex conjugate Y are not isomorphic. To conclude this section let us mention a result that circulated several years ago in a preprint form ([45]). It presents a construction of a Banach space Y with commuting bounded approximation property (BAP) and without a Schauder basis. It is based on a clever interlocking of spaces considered in Proposition 41.

10. Constructions in a general setting This section presents an argument which leads to a characterization of a Hilbert space in terms of subspaces of quotients of 2 -sums, due to the present authors ([32]). This is done by gluing together ﬁnite-dimensional quantitative phenomena, described in Section 7. On the ﬁnite-dimensional level, one may compare the result of Gluskin on the diameter of Minkowski compacta versus the theorem on homogeneous spaces; likewise, there is an obvious analogy between the basis constant result (Theorem 4) and Theorem 40. Namely, for a given ﬁnite-dimensional Banach space W , either one can mimic random constructions known for n1 ’s, or, alternatively, the space W satisﬁes strong geometric regularity conditions. We shall show here that for inﬁnite-dimensional Banach spaces a similar dichotomy occurs. Compared with inﬁnite-dimensional constructions involving np ’s (p = 2), here we shall concentrate on ﬁnding building blocks with analogous volumetric properties within the l2 -sum of an given arbitrary Banach space (non-isomorphic to a Hilbert space). We are going to consider linear topological properties of inﬁnite-dimensional Banach spaces similar to that considered in the previous section. There are natural numerical invariants assigned to them; with the convention that failing the property make the invariant equal to ∞. The basic constant of a Banach space is a role model. These invariants make sense in the ﬁnite-dimensional case as well, however they suffer a discontinuity syndrome. Even in the best case when an inﬁnite-dimensional space is a direct sum of ﬁnite-dimensional ones, such an invariant can be uniformly bounded on all ﬁnite-dimensional summands, and still to be inﬁnite on the whole space. On the other hand, the ﬁnite-dimensional invariants may be unbounded on the summands, while the invariant on the whole space remains ﬁnite. Thus to pass from the ﬁnite to the inﬁnite-dimensional case stronger structural properties are needed. For the case of the Schauder basis this is illustrated by the theorem below.

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T HEOREM 48. Let F = (Rn , · ) be an n-dimensional Banach space with n 64. Then the basis constant of F ⊕ 2 satisﬁes

1/2 bc(F ⊕ 2 ) c m(F ) , where c > 0 is a universal constant and m(F ) = m(F, 1/32). Its proof is in fact an abstract counterpart of the argument in Theorem 46. Moreover, it yields that if in the above theorem one replaces 2 by an arbitrary Banach space W then

1/2 bc(F ⊕ W ) c m(F ) /∂n (W ) (recall that ∂n (W ) = sup{d(E, n2 ) | E ⊂ W, dim E = n}). This leads to a general method of construction of Banach spaces without Schauder bases. T HEOREM 49. For k 1, let Fk = (Rnk , · Fk ) such that m(Fk ) → ∞ as k → ∞, and ∂nk−1 (Fk ) 2 for all k. Then there exists a subsequence {km } such that the space ( Fkm )2 does not have a Schauder basis. ( k−1 Indeed, let {F k } be a subsequence of {Fk } such that m(F k ) i=1 dim F i → ∞, as

1/2 k−1 k → ∞. For each k 1 one has ∂m (( i=k F i )2 ) 2 i=1 dim F i . Hence by the remark preceding the theorem we have ) 1/2 k−1 bc F i = bc F k ⊕2 Fi dim F i → ∞. c m(F k ) i

2

i=k

2

i=1

Thus our construction depends on the possibility of ﬁnding sequences {Fk } of spaces satisfying the hypothesis of the theorem. As we have seen in the previous section one can ﬁnd such spaces as subspaces of np for 2 < p ∞. Our aim here is to ﬁnd such a sequence {Fk } starting from an arbitrary space W and performing only three structural operations natural in the Banach space theory: passing to subspaces and quotients and taking 2 -sums. We begin this construction with an inﬁnite-dimensional Banach space W such that W ∗ fails the weak cotype 2 property ([44], Chapter 10). By the volume ratio characterizations of spaces of weak cotype 2 and duality arguments, including Santaló’s inequality, one can get a sequence {Ek } of ﬁnite-dimensional quotients of W , each of them in a special position, and satisfying vmk (Ek ) → 0 as k → ∞, where mk = dim Ek /2. Thus by Theorem 35, for a majority of quotients Fk of Ek of proportional dimensions we have m(Fk ) → ∞; and so the ﬁrst part of the assumptions of the theorem above is satisﬁed. However at this point we have no information on the behaviour of ∂dim Ek−1 (Ek ). This obstacle can be circumnavigated by the use of interpolation. The fact that ∂· (·) interpolates follows from general theory of L2 -factorable operators. However, the interpolation causes additional problems since neither vmk (Ek ) nor m(Fk ) interpolate well. We therefore use a well-known stronger invariant related to Levy mean, usually denoted by

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M.∗ (·). It controls vmk (Ek ) (hence also m(Fk )) and, what is crucial here, can be controlled during interpolation. Therefore we are forced to modify our argument and to use the failure of the weak cotype 2 property for W ∗ to ﬁnd quotients of subspaces E k of W in special positions ∗ (E ) → 0. Choosing for each k a suitable interpolation space between E such that Mm k k k and the Euclidean space we obtain a sequence {Fk } of ﬁnite-dimensional Banach spaces which satisﬁes both conditions in Theorem 49 simultaneously. Finally, spaces resulting from various interpolation schemes can be in general thought of as subspaces of quotients of direct sums of suitably chosen sequence of spaces. In our case, the spaces Fk above, can be isometrically identiﬁed with suitable subspaces of quotients of 2 (W ). Therefore, their 2 -sum is isometric to a subspace of a quotient of 2 (W ) as well. Thus we got P ROPOSITION 50. For a Banach space W , if every subspace of every quotient of 2 (W ) admits a Schauder basis then W ∗ is of weak cotype 2. Now we are ready to present the ﬂagship of this section, namely an unexpected structural characterization of a Hilbert space. T HEOREM 51. If X is a Banach space for which every subspace of every quotient of 2 (X) admits a Schauder basis, then X is isomorphic to a Hilbert space. Let us point out that the fact that every subspace of every quotient of a space itself has a Schauder basis does not imply that such a space is isomorphic to 2 . As an example, as shown by Johnson, one can take the so-called 2-convexiﬁed Tsirelson space ([13]). To deduce Theorem 51 from Proposition 50, we use well-known facts about so-called weak Hilbert spaces, which can be found, e.g., in [44], Chapter 12. In particular, for a Banach space Z, 2 (Z) is a weak Hilbert space if and only if Z is isomorphic to a Hilbert space. Let W = 2 (X), so that 2 (W ) = W . Proposition 50 implies that W ∗ is of weak cotype 2. Assuming that W ∗∗ is not of weak cotype 2, we would be able to construct a sequence {Fk } of ﬁnite-dimensional subspaces of quotients of 2 (W ∗ ) whose 2 -sum, denoted by Y , has no Schauder basis. It is easy to see that the Fk∗ ’s are subspaces of quotients of (2 (W ∗ ))∗ = (2 (W ))∗∗ , hence, by the principle of local reﬂexivity, we may assume that the Fk∗ ’s are subspaces of quotients of 2 (W ). Therefore its 2 sum, equal to Y ∗ , has a basis. By a general theorem ([17]), this implies that Y has a basis, a contradiction. This shows that W ∗∗ , hence also W , is of weak cotype 2. Finally, W cannot contain n1 ’s uniformly; otherwise, by the Johnson–Schechtman theorem on embedding kp into 1m (cf. [15]), W would contain np ’s uniformly for 1 < p < 2, allowing for construction of Theorem 46 in 2 (W ). In conclusion, W is a weak Hilbert space, and thus X is Hilbert. In a similar way, one can prove several other characterizations of a Hilbert space in terms of structural properties of quotients of subspaces of 2 (X) (cf. [32]). By adopting the proof of Theorem 51 one can obtain that for a space X being isomorphic to a Hilbert space is characterized, for example, by the property that every inﬁnite-dimensional subspace Z of every quotient of 2 (X) is isomorphic to a Cartesian square, or any other Cartesian power given in advance. The same characterizations hold for complex spaces as well. Moreover,

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the uniqueness of a complex structure on every such Z yields another isomorphic description of a Hilbert space. Finally, as an example of an interaction between real and complex structures in Banach spaces, let us note that for real Banach spaces, the possibility of introducing complex structure on every Z as above also provides a characterization of a Hilbert space.

11. Updates and recent results In this ﬁnal section we would like to bring to the reader’s attention several very recent developments, as of September 2002, in the area discussed in this survey. We begin with very recent results by Szarek and Tomczak-Jaegermann ([55]). They are concerned with the existence of normed √ spaces X which are well saturated by an arbitrary ﬁxed normed space. Given 1 k c n (where c > 0 is an absolute constant) and any kdimensional normed space W there exists an n-dimensional normed space X such that for of X contains a 1-complemented isometric copy m ∼ n/2, every m-dimensional quotient X of W . In fact the same construction works for saturation of√all m-dimensional quotients, for all m0 m n, where m0 is any ﬁxed integer satisfying n log n ) m0 n, and then the conditions on (non-trivial) k depend on m0 . Clearly, the dual space Y = X∗ has a similar saturation property for all m-dimensional subspaces of Y ; and in this case we can also force Y to have some cotype (if the space V = W ∗ being saturated with has cotype); in this case the complementation constant of copies of V in Y may get slightly larger than 1. This way, taking, for example, W = k∞ , we obtain an n-dimensional space X whose any mdimensional quotient contains a copy of k∞ , in particular, the cotype q constant of X, for any 2 < q < ∞, cannot be improved by passing to m-dimensional quotients. In the dual setting, the space Y may have good cotype, while none of its m-dimensional subspaces is K-convex (i.e., the K-convexity constant is not bounded independently on m). These examples answer in the negative a series of questions raised by Milman ([37]) in the mideighties. The basic construction of X is random, similar to the construction of Yn,m in the notation of Sections 2 and 6, but with several new twists. In most general terms, one considers the space Z = N 1 (W ), the 1 -sum of N copies of W , and then X is a random n-dimensional quotient of Z given by a rectangular n × Nk Gaussian matrix. This makes the kernels of the quotient maps uniformly distributed on the Grassman manifold of all (Nk − n)dimensional subspaces of RNk . Furthermore, we may let, for example, N ∼ n1+ε , for some ε > 0 (or even for any ε > 0). Another progress in the area has been made in a recent paper by Mankiewicz and Szarek ([29]) who proved: (∗) For every κ ∈ (0, 1/2] and every δ > 0 there exist c = c(κ, δ) > 0 and N = N(κ, δ) > 1 such that for n > N a majority of Yn ∈ Yn,[(1+δ)n] has the property: for every T ∈ Mixn (κn, √ 1) the cardinality of the set of all i ∈ {1, 2, . . . , [(1 + δ)n]} such that T gi > c n is greater than or equal to δn/2. √ In particular, a majority of Yn ∈ Yn,[(1+δ)n] satisﬁes m(Yn , κ) > c n. So this can be viewed as a counterpart of Theorem 12 for the random space Yn,[(1+δ)n] . (Recall that the difference in the behaviour between the random spaces X·,· and Y·,· was pointed out at

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the beginning of Section 6.) As a consequence, all the results of Section 5 for random spaces X·,· , which depend on the lower estimates for the norms of mixing operators, remain valid if the spaces X·,· are replaced by suitable random spaces Y·,· . Moreover, since random spaces Y·,· are “typically” well isomorphic to random spaces generated by corresponding orthogonal projections (cf. [34,29]), one can show that an estimate similar to that in (∗) holds for the random spaces of the form {PH (B1[(1+δ)n] ) | H ∈ G[(1+δ)n],n }. Thus, the corresponding versions of results in Section 5 are valid in this context as well. We conclude with results by the authors ([35]) which are counterparts of Theorem 37 for the mixing invariant m(·, κ). In the setting of Theorem 37, for every κ ∈ (0, 1/2], we have

m PH (K), κ dμN,n (H ) c d PL (K), PL B2N dμN,m (L), GN,n

GN,m

where m = (κ − ε)n. Compared with [34] the main progress was achieved by a better understanding of volumetric invariants of a body K, like those deﬁned in (19), and their geometric interpretation through the lengths of semi-axes of so-called M-ellipsoid for K. We refer the reader to [35] for details.

Acknowledgements The authors thank Alexandr Litvak for his careful reading of parts of the manuscript and Krzysztof Oleszkiewicz for his useful comments on Gaussian variables.

References [1] K. Ball, Normed Spaces with a Weak-Gordon–Lewis Property, Lecture Notes in Math. 1470, Springer (1991). [2] J. Bourgain, Real isomorphic complex spaces need not to be complex isomorphic, Proc. Amer. Math. Soc. 96 (1986), 221–226. [3] J. Bourgain, On ﬁnite-dimensional homogeneous Banach spaces, GAFA Israel Seminar 1986–87, Lecture Notes in Math. 1317, Springer, 232–239. [4] J. Bourgain and S.J. Szarek, The Banach–Mazur distance to the cube and the Dvoretzky–Rogers factorization, Israel J. Math. 62 (1988), 169–180. [5] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press (1995). [6] T. Figiel and W.B. Johnson, Large subspaces of n∞ and estimates of the Gordon–Lewis constant, Israel J. Math. 37 (1980), 92–112. [7] T. Figiel, S. Kwapie´n and A. Pełczy´nski, Sharp estimates for the constants of local unconditional structure of Minkowski spaces, Bull. Acad. Polon. Sci. 25 (1977), 1221–1226. [8] A.A. Giannopoulos and V.D. Milman, Euclidean structure in ﬁnite dimensional normed spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 707–779. [9] E.D. Gluskin, The diameter of Minkowski compactum roughly equals to n, Functional Anal. Appl. 15 (1981), 57–58 (English translation). [10] E.D. Gluskin, Finite-dimensional analogues of spaces without basis, Dokl. Akad. Nauk SSSR 216 (1981), 1046–1050.

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[11] E.D. Gluskin, A.E. Litvak and N. Tomczak-Jaegermann, An example of a convex body without symmetric projections, Israel J. Math. 124 (2001), 267–277. [12] Y. Gordon and D.R. Lewis, Absolutely summing operators and local unconditional structure, Acta Math. 133 (1974), 27–48. [13] W.B. Johnson, Banach spaces all of whose subspaces have the approximation property, Special Topics in Applied Mathematics (Proceedings GMD, Bonn 1979), Seminaire Analyse Fonctionelle, Ecole Polytechnique, Palaiseau, 1978/79, North-Holland (1980), 15–26. [14] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. n [15] W.B. Johnson and G. Schechtman, Embedding m p into 1 , Acta Math. 149 (1982), 71–85. [16] W.B. Johnson and G. Pisier, The proportional U.A.P. characterizes weak Hilbert spaces, J. London Math. Soc. 44 (1991), 525–536. [17] W.B. Johnson, H.P. Rosenthal and M. Zippin, On bases, ﬁnite-dimensional decompositions, and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488–506. [18] B.S. Kashin, Sections of some ﬁnite-dimensional bodies and classes of smooth functions, Izv. Akad. Nauk SSSR 41 (1997), 334–351. [19] H. König, C. Schütt and N. Tomczak-Jaegermann, Projection constants of symmetric spaces and variants of Khintchine’s inequality, J. Reine Angew. Math. 511 (1999), 1–42. [20] D.R. Lewis, Finite dimensional subspace of Lp , Studia Math. 63 (1978), 207–112. [21] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, New York (1977). √ [22] P. Mankiewicz, Finite-dimensional Banach spaces with symmetry constant of order n, Studia Math. 79 (1984), 193–200. [23] P. Mankiewicz, Subspace mixing properties of operators in Rn , with applications to Gluskin spaces, Studia Math. 88 (1988), 51–67. [24] P. Mankiewicz, Factoring the identity operator on a subspace of N ∞ , Studia Math. 95 (1989), 133–139. [25] P. Mankiewicz, A superreﬂexive Banach space X with L(X) admitting a homomorphism onto the Banach algebra C(βN), Israel J. Math. 65 (1989), 1–16. [26] P. Mankiewicz, Compact groups of operators on proportional quotients of n1 , Israel J. Math. 109 (1999), 75–91. [27] P. Mankiewicz, Compact groups of operators on subproportional quotients of l1m , Canad. J. Math. 52 (2000), 999–1017. [28] P. Mankiewicz and N.J. Nielsen, A superreﬂexive Banach space with a ﬁnite dimensional decomposition so that no large subspace has a basis, Israel J. Math. 70 (1990), 188–204. [29] P. Mankiewicz and S.J. Szarek, On the geometry of proportional quotients of l1m , Studia Math. 155 (2003), 51–66. [30] P. Mankiewicz and N. Tomczak-Jaegermann, A solution of the ﬁnite-dimensional homogeneous Banach space problem, Israel J. Math. 75 (1991), 129–159. [31] P. Mankiewicz and N. Tomczak-Jaegermann, Embedding subspaces of n∞ into spaces with Schauder basis, Proc. Amer. Math. Soc. 117 (1993), 459–465. [32] P. Mankiewicz and N. Tomczak-Jaegermann, Schauder bases in subspaces of quotients of l2 (X), Amer. J. Math. 116 (1994), 1341–1363. [33] P. Mankiewicz and N. Tomczak-Jaegermann, Structural properties of weak cotype 2 spaces, Canad. J. Math. 48 (1996), 607–624. [34] P. Mankiewicz and N. Tomczak-Jaegermann, Families of random projections of symmetric convex bodies, Geom. Funct. Anal. 11 (2001), 1282–1326. [35] P. Mankiewicz and N. Tomczak-Jaegermann, Volumetric invariants and operators on random families of Banach spaces, to appear in Studia Math. [36] V.D. Milman, Inégalité de Brunn–Minkowski inverse at applications à le théorie locale des espaces normés, C.R. Acad. Sci. Paris Sér. I 302 (1986), 25–28. [37] V.D. Milman, The concentration phenomenon and linear structure of ﬁnite-dimensional normed spaces, Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, CA, 1986), Amer. Math. Soc., Providence, RI (1987), 961–975.

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[38] V.D. Milman and G. Pisier, Banach spaces with a weak cotype 2 property, Israel J. Math. 54 (1986), 139– 158. [39] V.D. Milman and G. Schechtman, Finite-Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, New York (1986). [40] A. Pajor and N. Tomczak-Jaegermann, Volume ratio and other s-numbers of operators related to local properties of Banach spaces, J. Func. Anal. 87 (1989), 273–293. [41] A. Pełczy´nski, Any separable Banach space with the bounded approximation property is a complemented subspace of a space with a basis, Studia Math. 40 (1971), 133–139. [42] A. Pełczy´nski, Geometry of ﬁnite-dimensional Banach spaces and operator ideals, Notes in Banach Spaces, E. Lacey, ed., Austin Univ. Press (1980), 81–182. [43] A. Pietsch, Operator Ideals, VEB, Berlin (1979) and North-Holland (1980). [44] G. Pisier, Volumes of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press (1989). [45] C. Read, Different forms of the approximation property, Preprint (1988). [46] S.J. Szarek, On Kašin’s almost Euclidean orthogonal decomposition of n1 , Bull. Acad. Polon. Sci. 26 (1978), 691–694. [47] S.J. Szarek, The ﬁnite-dimensional basis problem with an appendix on nets of Grassman manifolds, Acta Math. 151 (1983), 153–179. [48] S.J. Szarek, On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc. 293 (1986), 339–353. [49] S.J. Szarek, A superreﬂexive space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (1986), 437–444. [50] S.J. Szarek, A Banach space without a basis which has the bounded approximation property, Acta Math. 159 (1987), 81–98. n and random matrices, Amer. J. Math. 112 (1990), 899–942. [51] S.J. Szarek, Spaces with large distance to l∞ [52] S.J. Szarek, Condition numbers of random matrices, J. Complexity 7 (1991), 131–149. [53] S.J. Szarek and M. Talagrand, An “isomorphic” version of the Sauer–Shelah lemma and the Banach–Mazur distance to the cube, GAFA Israel Seminar 1987–88, Lecture Notes in Math. 1376, Springer, 105–112. [54] S.J. Szarek and N. Tomczak-Jaegermann, On nearly Euclidean decompositions of some classes of Banach spaces, Compositio Math. 40 (1980), 367–385. [55] S.J. Szarek and N. Tomczak-Jaegermann, Saturating constructions for normed spaces, in preparation. [56] N. Tomczak-Jaegermann, Banach–Mazur Distances and Finite Dimensional Operator Ideals, Pitman Monographs, Longman, Harlow (1989). [57] E. Wigner, Characteristic vectors of bordered matrices with inﬁnite dimensions, Ann. Math. 62 (1955), 548–564; On the distribution of the roots of certain symmetric matrices, Ann. Math. 67 (1958), 325–327.

CHAPTER 29

Banach Spaces with Few Operators Bernard Maurey Laboratoire d’Analyse et Mathématiques Appliquées, UMR 8050, Université de Marne la Vallée, Boulevard Descartes, Cité Descartes, Champs sur Marne, 77454 Marne la Vallée Cedex 2, France E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . 1.1. Other spaces with few operators . . . . . . 2. Ancestors . . . . . . . . . . . . . . . . . . . . . 3. Inevitable behaviours . . . . . . . . . . . . . . . 4. Coding with inevitable sets . . . . . . . . . . . . 5. HI spaces, spectral properties and consequences 6. Sequence spaces . . . . . . . . . . . . . . . . . . 7. A class of examples . . . . . . . . . . . . . . . . 7.1. We have a HI space! . . . . . . . . . . . . . 8. Factorization through a HI space . . . . . . . . . 9. Additional results . . . . . . . . . . . . . . . . . 9.1. The shift space . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction Several natural questions about the linear structure of inﬁnite-dimensional normed spaces, that were asked since the early days of the theory, remained without answer for many years. Here are two examples (in this article, Banach space means inﬁnite-dimensional Banach space, real or complex): A. Is it true that every Banach space is isomorphic to its (closed) hyperplanes? B. If a Banach space X is isomorphic to every inﬁnite-dimensional subspace of itself, does it follow that X is isomorphic to 2 ? These two questions come from Banach’s book [4]. Question B was called the homogeneous Banach space problem in modern times. Question A is attributed to Banach; actually, Banach’s book contains a weaker question, formulated in terms of linear dimension, which amounts to asking whether every Banach space embeds isomorphically in its hyperplanes; this simply asks whether X is isomorphic to some proper subspace of itself. Let us formulate two other general questions, less ancient than the ﬁrst two. Recall that a sequence (en )n0 of non-zero vectors is unconditional if there exists a constant C such that m m ±ai ei C a i ei i=1

i=1

for every m 0, all scalars (ai )m i=1 and every choice of signs ±1; the best constant C is called unconditionality constant of (en )n0 . An unconditional sequence is a basis for its closed linear span; unconditional bases were earlier called absolute bases, see [7]. C. Does every Banach space contain an inﬁnite unconditional basic sequence? D. Is it possible to decompose every Banach space as a topological sum of two inﬁnitedimensional subspaces? Question D was formulated around 1970 by Lindenstrauss [37]. Question C appears in Bessaga and Pełczy´nski [7] in 1958, but was considered several years before, since it asks for a natural improvement of the classical result from Banach’s book, according to which every Banach space contains a subspace with basis. All these questions have been answered during the last decade, most of them in the negative direction that seems to indicate that there is no hope for a structure theory of general Banach spaces. There is one notable example though of a positive answer, the homogeneous space problem; interestingly enough, one of the “negative” objects discovered during the period 1990–95 plays a little rôle in the positive solution of Question B: at some point in the proof, one has to exclude the possibility that the homogeneous space could be hereditarily indecomposable. Despite my rather pessimistic comments above, the results and examples obtained since 1990 represent a signiﬁcant progress of our understanding of inﬁnite-dimensional Banach spaces. The solutions of the different problems have various points of contact, and introduce new notions that underlie several of the constructions and proofs. Let us agree that for the rest of this paper, the word subspace will always indicate an inﬁnite-dimensional vector subspace of a Banach space (but not necessarily a closed subspace). Beside the preceding problems, one of the main questions that remained a mystery

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for years is the following: does every Banach space contain a subspace isomorphic to some p (for 1 p < +∞) or to c0 ? This question was settled negatively in 1974 by the famous example of Tsirelson [54], who constructed a reﬂexive Banach space that does not contain any p . Tsirelson’s solution has had an enormous inﬂuence on most of what is discussed here. After being disappointed by Tsirelson’s example, the structure-seekers had to look for more modest questions, for example, the following. E. Does every Banach space contain a reﬂexive subspace, or else a subspace isomorphic to either c0 or 1 ? This question conforms to the common experience that non-reﬂexivity is often related to the presence of c0 or 1 ; indeed, a theorem of James [29] says that a Banach space X with unconditional basis is reﬂexive if and only if it does not contain isomorphs of c0 or 1 . There is therefore a loose connection between Questions E and C; any counterexample to E has to be a counterexample to C as well. As we have said before, it is obvious when we look back that the ﬁrst giant step in the direction of all the results mentioned in this article was done around 1974 by Tsirelson (in his only paper about Banach spaces, as he likes to point out!). As far as I know, Tsirelson’s space was the ﬁrst example of a space where the norm is deﬁned by an inductive procedure that “forces” some speciﬁc property to hold, but somehow, nothing more than the desired property. The same year, Krivine proved the ﬁnite-dimensional counterpart, that goes in the opposite (positive) direction and says roughly that every Banach space contains np ’s of arbitrary large ﬁnite dimension n. Almost 20 years passed before Tsirelson’s breakthrough was extended to a solution of the above mentioned problems; during these years, it was still hoped by many that techniques using Krivine’s ideas could lead, for example, to a positive solution of Question C. A difﬁculty common to these questions is that one has to analyze whether or not some particular phenomenon will occur in every subspace of a given Banach space X; this rather vague question can be put to precise terms as follows: what do we know about subsets A of the unit sphere of X that meet every inﬁnite-dimensional vector subspace? Here, I want to call such a set A a (linearly) inevitable set (these sets were called in [26] by the inexpressive term asymptotic; I feel that one should reform this poor terminology). Can we say that two inevitable sets A1 , A2 are in a sense so big that they must intersect, or at least almost intersect, meaning that dist(A1 , A2 ) = 0? An equivalent question asks whether every enlargement Aε of an inevitable set A must contain an inﬁnite-dimensional vector subspace. This type of problem reminds Ramsey theory, but nobody could exploit this analogy before Gowers, in his dichotomy theorem (see below). It was realized by Milman (see [42,43]), a few years before Tsirelson’s example, that if we can prove that any two inevitable sets A1 and A2 in the unit sphere of a given space X almost intersect, then X must contain some p , p ∈ [1, ∞), or c0 . In [42], Milman deﬁnes a notion of spectrum for a uniformly continuous function on the unit sphere, and shows that when the spectrum of every such function is non-empty, then X contains some p or c0 ; if inevitable sets almost intersect, then this spectrum is non-empty. This question about intersecting inevitable sets is not totally absurd, since the answer is positive in one case: the result of Gowers’ paper [20] implies that any two inevitable subsets of the unit sphere of c0 almost intersect. It follows indirectly that Tsirelson gave the ﬁrst example of two inevitable sets A1 and A2 that are separated by some δ > 0, that is a1 − a2 δ for all a1 ∈ A1 , a2 ∈ A2 .

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Tsirelson’s space thus gives us a ﬁrst clue, but the real start was the modiﬁcation of Tsirelson’s space constructed by Schlumprecht [49], which provides us with an inﬁnite sequence (An )n0 of such separated inevitable sets, giving a lot of possibilities for coding things in the sphere. This will be explained in Section 4. For Gowers and the author of this article, this was the decisive information for constructing a space with no inﬁnite unconditional sequence; during the summer of 1991, after hearing Schlumprecht at the Banach space conference in Jerusalem, we both constructed an example of a space with no unconditional sequence; the two examples X(G) and X(M) were very similar. Gowers’ example was the ﬁrst to be presented to a few specialists. The construction will be indicated in Section 7; it is rather intricate, but the fundamental principle, that seems to me very clear, is presented in a separated section (Section 4) containing the following partial result: if we have a sequence of well separated inevitable sets in a Banach space X, then for any given C 1 we can renorm X in such a way that any inﬁnite basic sequence in X has unconditionality constant larger than this C in the new norm. Obviously we are on the way to a negative answer to Question C. The reader who wants to get an idea about what is going on with the failure of unconditionality must read Section 4. The technical problem for solving Question C itself is to mix the simple idea from Section 4 with an inductive deﬁnition of a norm a la Tsirelson–Schlumprecht. This will be done in Section 7. Schlumprecht’s example is an ad hoc example of a space containing a sequence of separated inevitable subsets (An ), but this strange situation did not seem likely to happen in the most regular of all spaces, namely 2 . It was therefore a big surprise when Odell and Schlumprecht showed that one can move the sets (An ) from Schlumprecht’s space S to 2 by a non-linear procedure, and get a sequence (Bn ) of inevitable subsets in the unit sphere of 2 that are somewhat orthogonal [46]. On the other hand, it is still unknown whether Tsirelson’s space T contains a sequence of well separated inevitable subsets, or whether T satisﬁes the opposite property of having bounded distortion. This subject of distortion will not be discussed here. The reader may consult [5, Chapter 13] for a complete description, that includes the results of [46]. The space X(G) (or X(M)) was basically intended to be a counterexample to the unconditional sequence problem C, but it quickly appeared to have a very radical property: when seeing Gowers’ preprint about X(G), Johnson observed that the space had the additional property that for every pair (Y, Z) of subspaces, we have that inf y − z is zero, when y and z run in the unit spheres of Y and Z. This means that Y and Z can never form a topological sum; in other words, every subspace of X(G) is indecomposable. The paper [26] therefore introduced the ﬁrst example Xgm of a hereditarily indecomposable space (in short: HI), thus solving negatively Question D. As it happens sometimes, Question D was solved by proving much more than asked. Obviously, a HI space cannot contain an inﬁnite unconditional sequence, since the span of such a sequence is clearly decomposable (in odd and even, for example), and Xgm of course solves negatively Question C. Finding examples of indecomposable spaces that are not HI took some more time (the shift space Xs mentioned later in this introduction is such an example; other examples are given by non-HI duals of HI spaces, as given by Ferenczi or Argyros and Felouzis, see below); at this point of the story, it seems easier to get a HI space than a genuine indecomposable space (one that is not HI)!

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The HI spaces are very rigid, in the following sense: every bounded linear operator T on a complex HI space X has the form λId + S, where λ ∈ C and S is a strictly singular operator. Let us recall that a famous open question is the existence of a Banach space X such that every operator on X would have the form λId + K, with K compact. The existence of HI spaces is far from solving this “λId + K problem”, but it does give comparable spectral consequences, because it is known that strictly singular operators allow to extend the classical Riesz spectral theory of compact operators (see [39, 2.c], for example). It follows that operators on a complex HI space X have a discrete spectrum, consisting of a converging sequences of eigenvalues, together with the limit (that ought to be λ, if T = λId + S); if T is Fredholm on a HI space X, it has always index 0 (this result also holds in the real case). This implies easily that a HI space is not isomorphic to any proper subspace, thus solving negatively Question A (it solves exactly the question from Banach’s book, but the question had become more popular under the weaker form that asks whether hyperplanes are isomorphic to the whole space; as for Question D, this weaker form was solved by proving much more). These relatively easy spectral consequences of the HI property are presented in Section 5; the interested reader can jump directly to that section. Gowers has constructed several further examples; the most striking is a space Xg not containing any reﬂexive subspace, and containing no c0 or 1 [22]; this is a counterexample to Question E. The construction is partly similar to that of [26], but in a much more difﬁcult context which requires new ideas; I like to think that this example Xg is a sort of generalized James-tree space (see [31,38]), where every vector in the unit ball of X∗ is a potential node for the tree. To some extent, constructing this space Xg is like building Xgm on this abstract tree. It was very tempting to relate the HI property of a Banach space X to the fact that X does not contain any inﬁnite unconditional sequence. This was done by Gowers in his beautiful “dichotomy theorem” [24], see also [25]. Let X be an arbitrary inﬁnite-dimensional Banach space. Either X contains an inﬁnite unconditional sequence, or X contains a HI subspace. For proving this theorem, Gowers has found a very satisfactory way to extend Ramsey theory to a linear setting. This result also explains why it is not so surprising to get a HI space when one simply looks for a space with no inﬁnite unconditional sequence. It also gives a profound reason for introducing HI spaces. Some questions about general Banach spaces can then be divided into two cases: the “usual” case, where unconditional bases exist, and the “exotic” case, where we may ﬁnd a HI space in the middle of our road. This dichotomy was the missing piece for the solution of Question B, for which Komorowski and Tomczak had proved the following result: a Banach space X with unconditional basis, non-trivial type and not containing 2 , contains a subspace not isomorphic to X (see [34, 35]). They deduce the following partial solution to Question B: if a homogeneous Banach space X contains an inﬁnite unconditional sequence, then X is isomorphic to 2 . Now, by the dichotomy theorem, a homogeneous space X must contain an unconditional sequence: otherwise, it contains a HI subspace, hence it is HI itself by homogeneity, but clearly a HI space is not homogeneous! Combining [34] and [24] solves Question B: every homogeneous space X is isomorphic to 2 . As we have said, every operator on a complex HI space X has the form λId + S, where S is strictly singular. Ferenczi [12] proved a more general result (which was previously

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checked by hand in [26] for the speciﬁc example Xgm ): every operator from a subspace Y ⊂ X to X has the form λi + S, where i is the inclusion map and S is strictly singular. Actually, this property of a complex Banach space X is clearly equivalent to the fact that X is HI. One could be fully satisﬁed to have examples of spaces that have as few operators as possible. However, there are other questions which assume some structure for the space and then ask whether further structure follows. Roughly speaking, the results of [27] state that given an algebra of maps satisfying certain conditions, one can replace the multiple of the inclusion map in the statement above by the restriction to Y of some element of the algebra. These examples illustrate the following principle: you will ﬁnd inside the space constructed in this way, nothing more than what you decide to put at the start; we obtain in this way a space Xs with an isometric right shift S but no inﬁnite unconditional sequence; every operator T ∈ L(Xs ) is a perturbation of an absolutely summable series of iterates of the shift S and its adjoint. In this space all complemented subspaces are trivial (ﬁnite-dimensional or ﬁnitecodimensional), which makes this space a bizarre example of a prime space (a Banach space isomorphic to every inﬁnite-dimensional complemented subspace; here the shift and its iterates provide isomorphisms between Xs and the ﬁnite-codimensional subspaces). Recall that the “normal” known prime spaces are c0 , p , for p ∈ [1, +∞) (see [47]) and also ∞ (see [36]). These results from [27] are described in Section 9. This paper [27] also gives a space isomorphic to its subspaces of codimension two but not to its hyperplanes and a space isomorphic to its cube but not to its square (Gowers had previously given in [23] the ﬁrst example of this cube-not-square phenomenon). The shift space Xs provides a good illustration for Gowers’ dichotomy theorem. Indeed, this space Xs does not contain any inﬁnite unconditional sequence; but Xs is not HI, because it has a non-trivial operator, the shift S; one can also see directly that for every λ ∈ C with modulus one, we may ﬁnd a subspace Yλ of X on which the shift S is almost equal to λId (we generate Yλ from a sequence of almost eigenvectors of S, corresponding to the spectral value λ); when μ = λ, the two subspaces Yλ and Yμ can be chosen to form a topological direct sum, and this explains why X is not HI. By Gowers’ dichotomy theorem, every subspace of Xs must contain a further subspace which is HI. One can check that the subspaces Yλ are examples of such HI subspaces. Let us mention further results in the HI direction. Kalton [32] has constructed an example of a quasi-Banach space X with the very strange property that there is a vector x = 0 such that every closed inﬁnite-dimensional subspace of X contains x. It follows that this quasi-Banach space does not contain any inﬁnite basic sequence. This example is related to an example of Gowers [21] of a space with unconditional basis not isomorphic to its hyperplanes; Kalton’s construction uses the technique of twisted sums together with the properties of the space in [21]. Argyros and Deliyanni [1] constructed HI spaces without using Schlumprecht’s space, by a technique called mixed Tsirelson’s norms; their example is also an asymptotically-1 space. Ferenczi [11] constructed a uniformly convex HI space, by adding to the tools from [26] the tool of complex interpolation for families of Banach spaces developed by Coifman, Cwikel, Rochberg, Sagher and Weiss. Habala [28] constructed a space such that no inﬁnite-dimensional subspace has the Gordon–Lewis property (this property is a weak form of unconditional structure for a Banach space), and Ferenczi–Habala unite in [15].

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Ferenczi showed that the dual of the example Xgm in [26] is also HI and that every quotient of this space is still HI [14]. This type of question is not yet clariﬁed in general. What is clear is that the dual of a reﬂexive indecomposable space is indecomposable; therefore, if every quotient of a reﬂexive space is HI, then every subspace of a quotient is indecomposable and this property clearly passes to the dual. However Ferenczi [14] gave an example of a HI space such that the dual is not HI. This phenomenon was widely extended by Argyros and Felouzis [2], who showed that for every p > 1, the space p (or c0 when p = +∞) is isomorphic to a quotient of some HI space Xp (the corresponding result is obviously false for 1 ). The dual space Xp∗ , which contains q , is clearly not a HI space. This sheds more light on the non-stability of the HI property under duality. Argyros and Felouzis obtain this quotient result as a consequence of a factorization theorem through HI spaces: every operator which is thin in some sense factors through a HI space. This is, in a way, a very perverse result, as one is normally trying to factor operators through nice spaces, like what Davis, Figiel, Johnson and Pełczy´nski did in [10] for weakly compact operators! A sketch for the results of [2] is presented in Section 8. A variant of the interpolation method of [10] will be used for the general factorization result. Obtaining p as quotient of a HI space requires a clever construction which is also sketched in Section 8.

1.1. Other spaces with few operators As we have said, complex HI spaces have few operators. But HI spaces are not the ﬁrst examples of spaces that have, in some sense, few operators, and there is a series of works on this theme. The space constructed in [26] could be regarded as the inﬁnite-dimensional analogue of the random ﬁnite-dimensional spaces introduced by Gluskin [18,19]. It was shown by Szarek, that operators on Gluskin’s spaces all approximate, in a certain precise sense, multiples of the identity. Although the proofs in [26] and [18] are very different, there are some points of contact, such as the idea of constructing a unit ball with just a few “spikes”. Since these spikes must, under a well-bounded operator, map to other spikes, if they can be chosen in a very non-symmetrical way, a well-bounded operator is forced to approximate a multiple of the identity. Gluskin achieved the lack of symmetry by choosing the spikes randomly, whereas in [26] they were constructed directly (in the dual space) using some inﬁnite (and not too difﬁcult) combinatorics. Gluskin’s spaces were “glued” together to produce several inﬁnite-dimensional examples of interest by, amongst others, Bourgain [8], Szarek [52,53], Mankiewicz [40] and Read [48]. Some of these gluing methods were not at all straightforward. Several of the properties of these spaces are shared by the spaces constructed in [26] and [27]. For instance, Bourgain’s example is a complex Banach space X such that X and its opposite space X are not isomorphic complex spaces. Szarek’s example is a real reﬂexive Banach space X that does not admit a complex structure (because X does not have any operator T such that T 2 = −Id). The complex HI space from [26] has Bourgain’s property, while its real version satisﬁes Szarek’s conclusion (but these two facts are not stated in [26]). A related direction is the search for examples in Banach algebras. One particular question is the existence of non-zero homomorphisms from the algebra of bounded linear operators L(X) to C, or equivalently of closed ideals I such that L(X)/I is C, or more

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generally a commutative Banach algebra. The result on the “shift space” in [27] can be compared to those of Mankiewicz [40]: we have in [27] another example of a complex Banach space such that there exists a (unital) algebra homomorphism from L(X) into a commutative Banach algebra. It follows (as is recalled in [40]) that X is not isomorphic to any power Y n of a Banach space Y , for any n 2 (see also Figiel [16] for an early related example of a reﬂexive space not isomorphic to its square). Indeed, if ϕ is a non-zero multiplicative functional from L(X) to C, and if X = Y n , there is a natural homomorphism i from Mn to L(X). But then ϕ ◦ i would be a non-zero multiplicative functional on Mn , which is not possible, as soon as n 2. Let us mention two results that have little in common with the present paper, apart from their statements. Kalton and Roberts [33] have given results for quasi-Banach spaces. They constructed subspaces of Lp , 0 < p < 1, where the only continuous linear operators are the multiples of the identity. Shelah [50] has constructed examples of non-separable Banach spaces for which every bounded linear operator has the form λId+S, where S has separable range. The proof used some heavy machinery from logic (diamond axiom, or V = L) which was avoided later in [51]. In the fall of 1999, Argyros and Tolias announced that they can obtain the same conclusion as Shelah, from a space Xa that is a relative of the HI family (see [3]). This new example lies somewhere between the spaces from [1] and [22]. As in the case of the James tree space, the space Xa is a space of sequences indexed by a tree. The dual Xa∗ is non-separable, but every operator on Xa∗ has the form λId + S, where S has separable range. At the end of this introduction, I must confess that part of this paper has been realized by the well-known cut-and-paste technique, applied to the two papers [26] and [27]; as a result, some portions of the present paper may sound curiously too english to the reader: they were stolen from Gowers’ writing of our papers. 2. Ancestors In order that the reader sees where our methods come from, we have to say a few words about the ancestor of this story, namely the space T constructed by Tsirelson [54] (see also [17,9]). Let us ﬁrst ﬁx some notation about sequence spaces. Let c00 denote the space of ﬁnitely supported scalar sequences. Given two subsets E, F ⊂ N, we say that < F if E +∞ be the standard basis of c . Given a vector x = max E < min F . Let (en )∞ 00 n=0 xn en n=0 its support, denoted supp(x), is the set of n such that xn = 0. If x, y ∈ c00 , we write x < y when supp(x) < supp(y). We also write n < x when n ∈ N and n < min supp(x). If x1 < · · · < xn , then we say that the vectors x1 , . . . , xn are successive. Let C > 1 be ﬁxed and let BT∗ be the smallest convex subset of B(c0 ) ∩ c00 that contains ±ei for each i 0 and such that x1∗ + · · · + xn∗ ∈ CBT∗ whenever x1∗ , . . . , xn∗ are successive in BT∗ and n < x1∗ . The Tsirelson norm is then deﬁned on c00 by xT = sup x ∗ (x): x ∗ ∈ BT∗

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and T is the completion of c00 under this norm. This point of view agrees with Tsirelson’s original presentation. A dual formulation, given by Figiel and Johnson [17], introduces the Tsirelson norm as the smallest norm on c00 satisfying ei = 1 for every i 0 and n 1 1 x1 + · · · + xn xi n C i=1

for every n 2 and all n < x1 < · · · < xn . The original choice of C was C = 2; very interesting effects can be achieved by varying the constant C and mixing the norms obtained in this way (see [1] as one example). Recall brieﬂy why T is a couterexample to the p -containment problem: since x1 + · · · + xn 12 n whenever n norm one vectors satisfy n < x1 < · · · < xn , it is quite clear that among p spaces, only 1 can embed into T ; Tsirelson excluded this possibility by showing that T is reﬂexive; Figiel and Johnson gave a quantitative proof, which is closer to the spirit of this paper; they showed that T does not contain a (9/8)-isomorph of 1 (see also [39, 2.e.1]). But James [30] proved (by the well-known James’ blocking technique) that if X contains an isomorph of 1 , then X contains a (1 + ε)-isomorph of 1 for every ε > 0. Therefore T does not contain 1 , and as a consequence, T does not contain any subspace isomorphic to an p space or to c0 . A second example, extremely important for us, is the space S constructed by Schlumprecht [49], which is a very useful variation of the construction of T . The constant factor C in Tsirelson’s construction x1∗ + · · · + xn∗ ∈ CBT∗ is replaced by a variable value f (n) depending upon the number n of vectors in the sum; this function f should tend to inﬁnity, but slowlier than any power nα , α > 0. Schlumprecht chooses f (n) = log2 (n+1). Let BS∗ be the smallest convex subset of B(c0 )∩c00 containing ±ei for each i 0 and such that x1∗ + · · · + xn∗ ∈ f (n)BS∗ whenever x1∗ , . . . , xn∗ are successive in BS∗ and n 2. The norm is then deﬁned on c00 by xS = sup x ∗ (x): x ∗ ∈ BS∗ and S is the completion of c00 for this norm. It is useful to observe that BS∗ is obtained as the union of an increasing family of convex sets (Bn )n0 , where B0 is the intersection of c00 with the unit ball of 1 , and Bn+1 is obtained from Bn by adding all vectors x of the form x = f (m)−1 (x1 + · · · + xm ) with m 2 arbitrary and x1 , . . . , xm successive elements of Bn , and letting Bn+1 be the convex hull of this extended set. This remark helps to show several properties of the space, for example, the fact that the unit vector basis is 1-unconditional in S, by checking this inductively for · n = sup{|x ∗ (x)|: x ∗ ∈ Bn }.

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An alternative description of S deﬁnes the norm as the solution of some implicit equation. We could say that the Schlumprecht norm on c00 is the smallest norm · on c00 such that the unit vector basis is normalized and n xi /f (n) x1 + · · · + xn i=1

for every integer n 2 and every sequence of n successive vectors in S. In other words, the Schlumprecht norm is the solution to the implicit equation n xS = max xc0 , sup f (n)−1 xi S : i=1

n 2, x =

n

xi , x1 < · · · < xn

.

i=1

3. Inevitable behaviours In order to support some intuition for the notion of inevitable set (formally deﬁned in the next section), we recall some easy facts, together with a few words about Schlumprecht’s space S. We begin by a well known blocking procedure for constructing n1 , originating in James [30]. L EMMA 1. Let n 2 be an integer and 0 < ε < 1; suppose that N is an integer that can be written as N = nk for some k 1, and let (xi )N i=1 be norm one vectors in a normed space X, such that N ±xi (1 − ε/n)k N = (n − ε)k i=1

for all signs ±1. There exists a sequence of n blocks y1 , . . . , yn from (xi )N i=1 that is n −1 (1 − ε) -equivalent to the unit vector basis of 1 . Let us brieﬂy sketch the proof: we consider successive generations of blockings, the ﬁrst one being the obvious blocks xi of length one, i = 1, . . . , nk , and for = 0, . . . , k − 1 the next generation (numbered + 1) consists of nk−−1 new vectors z = nj=1 εj zj , that are blocks of n consecutive elements z1 , . . . , zn from the preceding generation, with some signs εj = ±1 chosen such that nj=1 εj zj is minimal. The last generation is just one single block using all nk vectors. If on our way from = 0 to = k we never encounter an inequality of the form n εj zj (n − ε) max zj , 1j n j =1

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k k we get at the end a choice of signs (εi )ni=1 such that N i=1 εi xi < (n − ε) , which contradicts the hypothesis. We must therefore get at some stage a family of n blocks (zj ) that give, after we rescale them into (yj ) yj 1,

j = 1, . . . , n;

n ±yj n − ε j =1

for all signs. It is easy to conclude, using simple convexity arguments, that this sequence (yj ) is well equivalent to the unit vector basis of n1 . Let us give an application of this lemma. Suppose that X is a Banach space with basis, and that f is a non-decreasing function on [1, +∞) that tends to +∞ at inﬁnity, but slowlier than any power function t α (α > 0, for example, f (t) = log t). Suppose that for every N 2, all sequences of successive norm one vectors x1 , . . . , xN in X satisfy N i=1 ±xi N/f (N). This is of course true for 1 , or for an Orlicz sequence space M with M(t) ∼ t/ log(1/t) as t → 0; it is also true, and more interesting in our context, for the Tsirelson space T or Schlumprecht’s space S. If ε > 0 and n 2 are given, we can choose k large enough so that f (nk ) (1 − ε/n)−k . If N = nk , we are in a position to apply Lemma 1. It follows easily that every subspace Y of X contains almost isometric copies of n1 , spanned by small perturbations of successive vectors. We can describe a scheme L1 for getting subsets of the unit sphere of X which intersect every subspace, namely we can describe a decreasing sequence of sets Ln1 that intersect every subspace: let Ln1 consist of all norm one vectors x ∈ X such that there exist successive vectors y1 < · · · < yn that are (1 + 2−n )-equivalent to the unit vector basis of n1 , and n 1 yi < 2−n . x − n i=1

Then Ln1 intersects every subspace of X. Of course the intersection of the classes (Ln1 )n1 is in general empty; what we call L1 is not a class of vectors, but a symbolic notation that is meant to represent the “idea” of any Ln1 with large n. In the present case, we say that the scheme L1 is inevitable in X. The next natural thing to do, if we want to know whether X contains, not only n1 s, but n also 1 , is to look for sums x + y, where x is in Lm 1 and y in some L1 , with n much larger than m and much larger than the “size” of x; if (x + y)/x + y is again in some Lk1 , with k large, then we are in the right direction for building 1 . Let us write symbolically this set n m n of vectors x + y, with x ∈ Lm 1 , y ∈ L1 , as L1 ∗ L1 , and let the notation L1 ∗ L1 represent m n the scheme limm limn L1 ∗ L1 . This scheme L1 ∗ L1 is related to the limit behaviour of the norm of sums x + y, where a ﬁrst large m is given, with a vector x ∈ Lm 1 , and where n is then chosen much larger than m and than the “size” of x, with a vector y ∈ Ln1 ; the notion of size of a vector x as to be precised: it could be, for example, the 1 -norm of x, or the product of the ∞ -norm of x by the smallest N such that supp(x) N . When the scheme L1 is inevitable, then obviously L1 ∗ L1 , L1 ∗ L1 ∗ L1 , and so on, are also inevitable (extending here the use of the word inevitable from subsets of the unit sphere to bounded

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sets A ⊂ X on which the norm function is bounded below by some κ > 0). When we look for 1 , our ﬁrst step is to check whether the schemes 2L1 and L1 ∗ L1 have something in n common. Indeed, when x and y are blocks of a given 1 basis, with x ∈ Lm 1 , y ∈ L1 , x < y k(m) and m < n, then 12 (x + y) belongs to L1 , with k(m) → +∞. This is precisely this ﬁrst step that goes wrong with T . When we look for the scheme S2 = L1 ∗ L1 , we get a new inevitable scheme, distinct from 2L1 . Indeed, the Figiel– Johnson argument for proving that T does not contain 1 shows that the inevitable set n k Lm 1 ∗ L1 is well separated from the inevitable sets 2L1 , when n is very large and k > 20, say. We may go further, and look for S3 = L1 ∗ L1 ∗ L1 , and so on; in T , we do not seem to get much more by having this inﬁnity of possibilities; what we do get is that, in some n sense, we cannot get more from T than what was input at the start: if we have xj ∈ L1j , j = 1, . . . , k, with nj +1 very large with respect to the “size” of x1 , . . . , xj , then (taking the deﬁning constant C equal to 1/2) k k k xj (1 + ε) . 2 2 j =1

Let us turn now to Schlumprecht’s space S. Since f (n) = log2 (n + 1) grows slowlier than any power of n, we get n1 s everywhere, hence the scheme L1 is inevitable in S, and so are the next schemes Sk , k 2. But now we get something new and very interesting. L EMMA 2 (Schlumprecht’s lemma). If x1 , . . . , xk is a successive sequence in S, such that n xj ∈ L1j for j = 1, . . . , k, with nj +1 very large compared to ε−1 , to the size of x1 , . . . , xj and to n1 , . . . , nj , then k k . x1 + · · · + xk (1 + ε) f (k) f (k) A sequence such as x1 , . . . , xk will be called a rapidly increasing sequence (of n1 s), in short RIS. Schlumprecht’s lemma states that for a RIS, we almost get an equality between the norm of the sum ki=1 xi and the lower bound k/f (k) that was imposed by the deﬁnition of the space. Recall the observation made before the statement of the lemma, that implies that for every k 1 and every subspace Y ⊂ S, we may ﬁnd a RIS of length k consisting of small perturbations of vectors in Y . In other words, the normalized sums of RIS of length k form a nearly inevitable set Ak in S. Since f (k) tends to inﬁnity with k, it is clear that we get now inﬁnitely many different inevitable classes in S. This was the ﬁrst known example of this situation, and it was used (implicitely) by Schlumprecht in order to show that S is arbitrarily distortable. We obtain in this way what is for us the most important feature of Schlumprecht’s example: on one hand, we can ﬁnd n1 in every subspace; on the other hand, we can always combine very n different 1i in a RIS and get a behaviour arbitrarily far from the 1 behaviour. There is an endless interplay between these classes of vectors: we see that Sk deviates more and more from the k1 behaviour; if x1 , . . . , xk is as above then clearly it is not a good k1 basis, since √

f (k) can get as big as we want, but by Lemma 1 it can be blocked to give a good 1 k , say.

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Applying this blocking procedure with successive RIS of increasing lengths kj will give a √ k

new RIS built from pieces in L1 j ; the sum of this RIS can again be blocked to n1 , and so on . . . . This remark is crucial for the construction of the example Xgm in [26], and it is used intensively in Section 7. We can precise what we mean by “well separated” inevitable sets, in the framework of S. Schlumprecht’s lemma does not only say that RIS deviate more and more from the n1 behaviour, but it gives a precise estimate for the norm of the sum. With this information, it is possible to associate a class of functionals that almost norm the sums of RIS. For every n > 1, let An denote the set of normalized vectors which are multiple of the sum of a RIS sequence of length n; we deﬁne a class A∗n of functionals on S, consisting of all functionals of the form f (n)−1 (x1∗ + · · · + xn∗ ), with xi∗ 1, i = 1, . . . , n. If x ∈ An , ∗ ∗ ∗ then we may neasily ﬁnd x ∈ An such that x (x) > 1/2. Indeed, this vector x has the form −1 cn f (n) i=1 xi , where (xi ) is a RIS and 1/2 < c 1 (apply Lemma 2 with ε < 1); we select for each i a norming functional xi∗ for xi , with supp(xi∗ ) = supp(xi ), and we set x ∗ = f (n)−1 (x1∗ + · · · + xn∗ ). We have x ∗ 1 by the deﬁnition of S, and x ∗ (x) c > 1/2. ∗ (x )| is small Furthermore, it can be shown (see [26,46], or [5, Theorem 13.30]) that |xm n ∗ ∗ – depending upon min(m, n) – when xm ∈ Am , xn ∈ An and m, n are very different. This gives a weak form of orthogonality, which is what we name almost biorthogonal inevitable system in the next section.

4. Coding with inevitable sets Let X be a normed space and let S(X) be its unit sphere. We shall say that a subset A ⊂ S(X) is inevitable if A ∩ S(Y ) = ∅ for every inﬁnite-dimensional (not necessarily closed) subspace Y ⊂ X. It is sometimes more convenient to work with the notion of a nearly inevitable set A ⊂ S(X), that has the property that inf{d(y, A): y ∈ Y } = 0 for every inﬁnite-dimensional subspace Y ⊂ X. When X has a basis, it is easy to check that every subspace contains a further subspace which is spanned by a perturbation of a block sequence (see [39, 1.a.11]). It follows that A is nearly inevitable in X when the above condition is true for Y an arbitrary block subspace. Observe that if we replace a nearly inevitable set A by the enlargement Aε consisting of all x ∈ S(X) such that dist(x, A) < ε, we get an inevitable set Aε . The most obvious example of an inevitable set is the unit sphere S(Y ) of a ﬁnitecodimensional subspace Y . On the sphere of 2 , I would not be able to show any interesting example that can be described and proved inevitable using only pre-’90s ideas. The discussion of the preceding section shows that in a space with basis, close to 1 in the sense above, all the classes Ln1 are inevitable; we also said that in Schlumprecht’s space S, we can even ﬁnd a sequence of distinct inevitable sets. A key observation made in [26] is that if a space X contains inﬁnitely many inevitable sets that are all well disjoint from one another, then these can be used to construct an equivalent norm on X such that no sequence is C-unconditional in this norm. The idea is to use a certain coding, that has some similarity with what was done in [41] for getting a statement about every subsequence of a given sequence; this is extended now to general vectors

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(general, as opposed to vectors from a given weakly null sequence); here, the coding action will be related to the numbering of a sequence of inevitable sets, while the coding action in [41] was simply related to the numbering of the given sequence; we shall get in this way a statement about every subspace of a given space. Let us explain this obvious coding principle in a simpliﬁed setting: let Δ be a countable set, and let (Bn )n0 be a sequence of disjoint non-empty subsets of Δ; a coding function is an injective map σ , from the countable set of ﬁnite sequences of elements of Δ, to the natural numbers; a coding number N = σ (d1 , . . . , dm ) is thus associated in a 1-1 way to every ﬁnite sequence (d1 , . . . , dm ) of elements of Δ, where m varies in N; the coding action builds a tree of ﬁnite sequences, by saying that (d1 , . . . , dm , d) is a successor of (d1 , . . . , dm ) if and only if d belongs to BN with N = σ (d1 , . . . , dm ), and saying that (d1 , . . . , dm ) is a node of the tree if (d1 , . . . , dj ) is a successor of (d1 , . . . , dj −1 ) for j = 2, . . . , m. If (d1 , . . . , dm ) and (e1 , . . . , en ) are two nodes of this tree and if dj , ek belong to the same set B , then it follows that j = k and that d1 = e1 , . . . , dj −1 = ej −1 . If C is a class of subsets of Δ such that every C ∈ C intersects every set Bn , then clearly, for every C ∈ C we can construct arbitrarily long nodes (d1 , . . . , dn ) such that dj ∈ C for j = 1, . . . , n. In this way, we have a tool that can affect every C ∈ C. Below, Δ will be a rich enough countable subset of the unit ball of the dual of a separable Banach space X and each C ∈ C will be a family of functionals which are norming for some subspace YC of X. Let A1 , A2 , . . . be a sequence of subsets of the unit sphere of a normed space X and let A∗1 , A∗2 , . . . be a sequence of subsets of the unit ball of X∗ . We shall say that A1 , A2 , . . . and A∗1 , A∗2 , . . . are an almost biorthogonal inevitable system with constant δ > 0 if the following conditions hold for every integer n 1: (i) the set An is inevitable; (ii) for every x ∈ An there exists x ∗ ∈ A∗n such that x ∗ (x) > 1/2; (iii) for every m 1 with n = m, every x ∈ An and every x ∗ ∈ A∗m , we have |x ∗ (x)| < δ. The deﬁnition is interesting only when δ > 0 is small. It is not at all obvious that any Banach space contains an almost biorthogonal inevitable system with constant δ < 0.01 say. As far as I know, no example was known before the Schlumprecht space S appeared. The main result of this section is the following theorem, whose proof is taken almost verbatim from [26]. T HEOREM 3. Let r be an integer > 9 and let X be a separable normed space containing an almost biorthogonal inevitable system with constant δ = r −2 . Then there is an equivalent norm on X such that no sequence is r/9-unconditional. P ROOF. We shall write the proof in the real case. Let · be the original norm on X and let A1 , A2 , . . . and A∗1 , A∗2 , . . . be the almost biorthogonal inevitable system with constant δ = r −2 . For each n 0 let Zn∗ be a countable subset of 0<λ<1 λA∗n such that for every x ∈ An there exists x ∗ ∈ Zn∗ with 0 < x ∗ (x) − 1/2 < δ. Obviously the almost orthogonality ∞ ∗ and A when m = n. Let Δ = ∗ relations (iii) still hold between the sets Zm n n=1 Zn . Next, let σ be an injection to the natural numbers from the countable collection of ﬁnite sequences of elements of Δ. We shall now deﬁne a collection of functionals which we call special functionals. A special sequence of functionals is a sequence of the form z1∗ , z2∗ , . . . , zr∗ , where z1∗ ∈ Z1∗ and,

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∗ ∈ Z∗ ∗ ∗ ∗ for 1 i < r, zi+1 σ (z1∗ ,...,zi∗ ) . A special functional is simply the sum z = z1 + · · · + zr of a special sequence of functionals. We shall let Γ stand for the collection of special functionals. Let us deﬁne an equivalent norm ||| · ||| on X by

|||x||| = max x, r sup z∗ (x): z∗ ∈ Γ .

Let x1 , x2 , . . . be any sequence of linearly independent vectors in X. We shall show that it is not r/9-unconditional in the norm ||| · |||. We shall do this by constructing a sequence of vectors z1 , . . . , zr , generated by x1 , x2 , . . . and disjointly supported with respect to these vectors, with the property that r r zi . r (−1)i zi < 9 i=1

i=1

Let X1 be the algebraic subspace generated by (xi )∞ i=1 . Since A1 is an inevitable set, we can ﬁnd z1 ∈ A1 ∩ X1 . This implies that z1 has norm 1 and is generated by ﬁnitely many of the xi . Next we can ﬁnd z1∗ ∈ Z1∗ such that 0 < z1∗ (z1 ) − 1/2 < δ. Now let X2 be the algebraic subspace generated by all the xi not used to generate z1 . Since Aσ (z1∗ ) is inevitable, we can ﬁnd z2 ∈ Aσ (z1∗) ∩ X2 of norm 1. We can then ﬁnd z2∗ ∈ Zσ∗ (z∗) such that 1 0 < z2∗ (z2 ) − 1/2 < δ. Continuing this process, we obtain sequences z1 , . . . , zr and z1∗ , . . . , zr∗ with the following properties. Let n1 = 1, and ni+1 = σ (z1∗ , . . . , zi∗ ) for 1 i < r. First, zi ∈ Ani (thus zi = 1) for each i. Second, zi∗ ∈ Zn∗i for each i (i.e., z1∗ , . . . , zr∗ is a special sequence of functionals). Third, zi∗ (zi ) ∼ 1/2 for each i. Fourth, since σ is an injection, the zi∗ belong to different Zn∗ s, so |zi∗ (zj )| < δ when i = j since zi∗ ∈ Zn∗i and zj ∈ Anj . Let us now estimate ||| ri=1 zi |||. Since z1∗ , . . . , zr∗ is a special sequence, the triple norm is at least r

r i=1

zi∗

r

zi > r r/2 − δr 2 = r(r/2 − 1) > r 2 /3

i=1

(since r > 6). On the other hand, if (wi∗ )ri=1 is any special sequence, let t be the maximal index i such that wi∗ = zi∗ (or let t = 0 if w1∗ = z1∗ ). Then r t r ∗ w (zi ). (−1)i wi∗ (zi ) (−1)i wi∗ (zi ) + wt∗+1 (zt +1 ) + i i=1

i=1

i=t +2

Since σ is an injection, wi∗ and zj∗ are chosen from different sets Zn∗ whenever i = j or i = j > t + 1. By property (iii) this tells us that |wi∗ (zj )| < δ. In particular, r |w∗ (z )| < δr = 1/r. When i t we know that wi∗ = zi∗ , hence |1/2 − wi∗ (zi )| i=t +2 t i i i ∗ δ, so | i=1 (−1) wi (zi )| 1/2 + t/r 2 . It follows that r i ∗ (−1) wi (zi ) 1/2 + 1 + 2/r < 2. i=1

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We also know that i=j |wi∗ (zj )| δr 2 = 1. Putting all these estimates together, we ﬁnd that r ∗ (−1)i zi 3r. r sup w ∗ w ∈Γ i=1

Finally, by the triangle inequality, ri=1 (−1)i zi r and ||| ri=1 (−1)i zi ||| 3r, from which it follows that the sequence x1 , x2 , . . . was not r/9-unconditional in the equivalent norm. 5. HI spaces, spectral properties and consequences D EFINITION 4. Let X be an inﬁnite-dimensional Banach space, real or complex. We say that X is indecomposable if X cannot be written as the topological direct sum of two inﬁnite-dimensional closed subspaces Y1 and Y2 . We say that X is hereditarily indecomposable (in short, HI) if every closed inﬁnite-dimensional subspace Y of X is indecomposable, that is if no subspace Y of X can be written as the topological direct sum of two inﬁnite-dimensional closed subspaces Y1 and Y2 of X. Obviously, if X is HI then every subspace Y ⊂ X is HI. It is easy to see that a Banach space X is HI if and only if for all subspaces Y and Z of X, we have inf y − z: y ∈ Y, z ∈ Z, y = z = 1 = 0. We see that X is HI when the “angle” between any two subspaces Y and Z of X is equal to 0. Since every Banach space contains a subspace with basis, it is formal from the existence of any HI space that there exist HI spaces with monotone basis; the example Xgm in [26] has a basis, and it is also reﬂexive; much more difﬁcult is another example due to Gowers [22] of a space Xg without any reﬂexive subspace, and not containing c0 or 1 ; it follows by James [29] that Xg does not contain any subspace with unconditional basis, hence by the dichotomy theorem, Xg must contain a HI subspace Yg ; of course Yg has no reﬂexive subspace; saying that there exists a HI space Yg with no reﬂexive subspace seems to be a clean way to present things, but [22] left open the point of deciding whether the space Xg is already HI. D EFINITION 5. A bounded linear operator T ∈ L(X, Y ) is strictly singular when there is no inﬁnite-dimensional subspace X0 ⊂ X such that T restricted to X0 is an into isomorphism; in other words, for every X0 and ε > 0 there exists x ∈ X0 such that T x < εx. It is standard to deduce that when T is strictly singular, we can construct in any subspace X0 of X a further subspace X1 ⊂ X0 , spanned by a normalized basic sequence, such that T|X1 ε (see [39, 2.c.4]). Observe that X is HI if and only if for every subspace Y ⊂ X, the quotient map πY : X → X/Y is strictly singular.

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Let X be a complex Banach space and let T be a bounded linear operator on X. Let ΩT be the set of all μ ∈ C such that there exist c > 0 and a ﬁnite-codimensional subspace Z ⊂ X such that T x − μx cx for every x ∈ Z. Clearly ΩT is open in C and contains the complement of the spectrum Sp(T ) of T . Saying that μ ∈ ΩT implies that ker(T −μId) is ﬁnite-dimensional and that the range of T − μId is closed. This indicates that T1 = T − μId is semi-Fredholm when μ ∈ ΩT , with generalized index ind(T1 ) = dim(ker T1 ) − codim(T1 (X)) ﬁnite or equal to −∞. Let FT = C \ ΩT ; it is not difﬁcult to show that this closed subset of Sp(T ) is not empty. Let us give a rather simple argument for this. It is well known that U ∈ L(X) is Fredholm of U in the quotient algebra L(X)/K(X) is invertible (where if and only if the class U K(X) denotes the ideal of compact operators); this quotient Banach algebra (the Calkin in algebra) is not trivial when X is inﬁnite-dimensional. It follows that the spectrum of T the Calkin algebra is not empty; this spectrum is the essential spectrum σess (T ) of T . Let λ be a point of σess (T ) with maximal modulus. Then T − μId is Fredholm for every μ ∈ C such that |μ| > |λ|; by continuity of the index, all operators T − μId for |μ| > |λ| have index 0, because this is true when μ is large enough to make T − μId invertible. If T − λId was semi-Fredholm, then all T − μId for μ close to λ would be also semi-Fredholm, with the same (generalized) index; but this index must be 0 by the preceding argument, and T − λId would then be Fredholm, which was excluded from the beginning by the fact that λ belongs to the essential spectrum. This show that λ ∈ FT . If λ ∈ FT , that is if λ ∈ / ΩT , we may construct by induction a normalized basic sequence (xn ) such that T xn 2−n ε, and get a subspace Yε ⊂ X such that T y − λy εy for every y ∈ Yε (see again [39, 2.c.4]). Suppose now that X is a complex HI space. It is easy to see that FT contains exactly one element λ in this case: indeed, if λ, μ ∈ FT , we may ﬁnd two subspaces Y and Z in X such that T ∼ λId on Y and T ∼ μId on Z; since the unit spheres of Y and Z almost meet, it follows that λ = μ. Let λ0 be this unique value, and let U = T − λ0 Id. Let Z be any subspace of X. Since λ0 ∈ FT , there exists for every ε > 0 a subspace Y such that U|Y < ε. Since X is HI, the unit spheres of Z and Y almost meet, thus we may ﬁnd z in the unit sphere of Z such that U z < ε. This shows that U is strictly singular. We already obtained most of the next theorem. T HEOREM 6. Let X be a complex HI space. Then every T ∈ L(X) can be written as T = λId + S, where λ ∈ C and S is strictly singular. Thus every T ∈ L(X) is either strictly singular or Fredholm with index 0. Furthermore, the spectrum of T is either ﬁnite, or consists of a sequence (λn ) converging to λ. In this second case, each λn = λ is an eigenvalue of T with ﬁnite multiplicity. P ROOF. We proved that there exists a number λ ∈ C such that T − λId = S is strictly singular. If λ = 0, it is classical that T = λId + S is Fredholm with index 0, because it is a strictly singular perturbation of λId (see, for example, [39, 2.c.10]). The property of eigenvalues is also well known. By the discussion above, if μ = λ belongs to the spectrum of T , then μ is not in the essential spectrum of T , hence T − μId is Fredholm and not invertible, therefore μ is an eigenvalue of T with ﬁnite multiplicity.

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In the real case, we have: C OROLLARY 7. Let X be a real HI space. Every T ∈ L(X) is either strictly singular or Fredholm with index 0. P ROOF. Let X be a real HI space and consider its complexiﬁcation XC = X ⊕ X. This space XC need not be a complex HI space, but it has the property that given three subspaces, the sphere of one of them almost meets the sum of the two other subspaces. This implies that the essential spectrum of any U ∈ L(XC ) contains at most two points, say λ and μ (using the three subspaces property from above). Let T ∈ L(X), and consider its complexiﬁcation U = TC on XC . The spectrum of the ¯ Either complexiﬁed operator TC is invariant under complex conjugation, therefore μ = λ. λ is real, and T − λId is strictly singular; then T = λId + (T − λId) is Fredholm with index 0 if λ = 0, or T is strictly singular when λ = 0. Otherwise, we have λ ∈ C \ R; the part of the spectrum of TC contained in the upper half plane is ﬁnite or consists of a ¯ convergent sequence of eigenvalues with ﬁnite multiplicity, together with its limit λ or λ. ¯ / {λ, λ}; in particular, TC is Fredholm Then TC − νId is Fredholm with index 0 for every ν ∈ with index 0, and the same holds for T . C OROLLARY 8. Let X be a HI space, real or complex. Then X is not isomorphic to any proper subspace. In particular, X is not isomorphic to its hyperplanes. P ROOF. Let T be an isomorphism from X into itself; then T is not strictly singular, hence it must be Fredholm with index 0 by Theorem 6 or Corollary 7 and thus T X = X. These properties of operators on HI spaces were not immediately seen by the authors of [26]; Gowers actually constructed, before [26] was written, a modiﬁcation Xu of Xgm and he showed that Xu is not isomorphic to its hyperplanes [21]; this space Xu was thus the ﬁrst declared example of a space not isomorphic to its hyperplanes. This example Xu is a very intriguing example of space with unconditional basis. If X is a reﬂexive HI space, then the facts that the spectrum is countable and that T − μId is Fredholm with index 0 for all but one value μ = λ ∈ C hold true for operators on X∗ , although X∗ need not be HI. As a consequence, the dual of a reﬂexive HI space is not isomorphic to any proper subspace, and a reﬂexive HI space is not isomorphic to any proper quotient. However, these results do not seem to answer the following question: does there exist a HI space X isomorphic to its dual X∗ ? We can say that such an X must be reﬂexive, because X∗∗ X is HI and X ⊂ X∗∗ . An obvious remark is that the usual way to get simple examples of spaces isomorphic to their dual, namely X = Y ⊕ Y ∗ , Y reﬂexive, has no chance to yield a HI space X! Ferenczi [12] has shown that, given a complex HI space X and a bounded linear operator T from a subspace Y of X to X, one can write T = λiY,X + S, where λ ∈ C, S is strictly singular and iY,X denotes the inclusion map from Y to X. Clearly, Ferenczi’s result is a characterization of complex HI spaces. Let us sketch Ferenczi’s proof. In some sense all subspaces of a HI space X intersect; we may deﬁne a net of subspaces of X that captures a good part of the structure of this HI space; the order of this net is not the

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inclusion, as it is not strictly true that any two subspaces have an inﬁnite-dimensional intersection, but almost . . . . We say that Y Z if there exists a compact operator K : Y → X such that (iY,X + K)(Y ) ⊂ Z. Given Z1 , Z2 ⊂ X there exists Y such that Y Z1 and Y Z2 . We could call “germ” of HI space an equivalence class of such nets, in a way to be made precise. An interesting class of examples is the family of spaces containing no inﬁnite unconditional sequence but having only a ﬁnite set of germs of HI spaces [13]. The example Xg of Gowers [22] of a HI space with no reﬂexive subspace gives an example of “non-reﬂexive germ”. Ferenczi’s proof consists essentially in showing that the space of “germs” of operators on X is a Banach ﬁeld, hence isomorphic to C. A germ of operator is an equivalence class for the relation where T1 ∈ L(Z1 , X) and T2 ∈ L(Z2 , X) are equivalent if we can ﬁnd a subspace Y Z1 , Z2 such that the operator T1 ◦ (iY,X + K1 ) − T2 ◦ (iY,X + K2 ) is compact on Y , where K1 and K2 are the compact operators that appear in the deﬁnition of the order.

6. Sequence spaces The scalar ﬁeld is K = R or C. Since we want to include some of the results of Argyros and Felouzis, we shall work in an extended setting where the space of sequences is a space of sequences of vectors taken from some separable space V , or even more generally, from a sequence V = (Vn ) of separable spaces. We let K denote the sequence (Vn ) where Vn = K for every n 0. The reader may decide that V = K until he wants to study the section about [2]. The norm on Vn is denoted · n . The exposition is slightly simpler if we assume that each Vn is reﬂexive, but one can modify the construction in order to take care of the case Vn separable but not reﬂexive. Let us denote by c00 (V) the space of vector sequences x = (vn )n0 , where vn ∈ Vn for every n 0, and such that vn = 0 for only ﬁnitely many values of n. If x ∈ c00 (V), we call support of x the (ﬁnite) set supp x of integers n such that vn = 0. If x, y ∈ c00 (V), we say that x < y if supp(x) < supp(y). We also write n < x when n ∈ N and n < min supp(x). If x1 < · · · < xn , then we say that the vectors x1 , . . . , xn are successive. An inﬁnite sequence of successive non-zero vectors is also called a block basis and a subspace generated by a block basis is a block subspace. An interval of integers is a set of the form {n, n + 1, . . . , m} and the range of a vector x, written ran(x), is the smallest +∞ interval containing supp(x). It is convenient to write x = (vn ) ∈ c00 (V) as x = n=0 vn ⊗ en . Given a subset E ⊂ N and a vector x as above, we write Ex for the vector n∈E vn ⊗ en . We let Pn denote the projection corresponding to the set En = {0, . . . , n}; thus Pn is the natural projection from c00 (V) onto nk=0 Vk ⊗ ek . If V∗ is the sequence of duals (Vn∗ ), we have a natural duality between c00 (V) and c00 (V∗ ); the second space will be considered as space of functionals; we extend our terminology to functionals, for example, we shall talk about successive functionals. Let X (V) stand for the set of Banach spaces X obtained as the completion of c00 (V) for a norm · such that Ex x for every interval E, and vn ⊗ en = vn n for every n 0 and vn ∈ Vn ; when V = K, this means that (en ) is a normalized bimonotone basis for X.

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The reader must pay attention to the following fact: when V = K, every subspace Y of X contains a normalized sequence (yn ) which is a perturbation of a successive sequence (xn ) ⊂ X. This is clearly not the case in the vector case, as Y could be equal to V0 ⊗ e0 , for example. However, if we assume that Y is a subspaceof X such that for every n 0, the projection Pn is not an into isomorphism from Y to nk=0 Vk ⊗ ek , then the standard glidind hump procedure can be extended to the vector setting in an obvious way. We shall say in this case that Y is a GH-subspace. We shall obtain interesting information only about GH-subspaces. This is a vacuous limitation in the scalar case, but it is an important one in the vector case. When the restriction to Y of every Pn , n 0, is strictly singular, we get that every subspace of Y is a GH-subspace. Given a “slowly increasing” function f from [1, +∞) to [1, +∞) and a space X ∈ X (V), we shall say that X satisﬁes a lower f -estimate if, given any vector x ∈ X and any sequence of intervals E1 < · · · < En , we have that x f (n)−1 ni=1 Ei x. In the dual formulation, this property means that whenever x1∗ , . . . , xn∗ are successive functionals with norm 1, then ∗

x + · · · + x ∗ /f (n) ∗ 1. n 1 X Let X ∈ X (V) and x ∈ X. For every n 1, let x(n) = sup

n

Ei x,

i=1

where the supremum is extended to all families E1 < · · · < En of successive intervals. This quantity is clearly increasing with n, and x = x(1) by the monotonicity property of (ei ), since X ∈ X (V). Observe that the basis (ei ) satisﬁes vi ⊗ ei (n) = vi i for every n 1. Clearly, · (n) is an equivalent norm on X, with x x(n) nx; also, Ex(n) x(n) for every interval E. The value of this norm at x is related to the fact that x can be broken into blocks that look like the unit vector basis of n1 , or more accurately, into blocks that look like a n1+ basis; this norm will be used in the deﬁnition of RIS as a substitute for the notion of n1+ 2

average used in [26]. It is easy to check that when (xi )ni=1 are successive and have norm 2 2 1, then x = n−2 ni=1 xi satisﬁes x(n) 1 + 1/n. When (xi ) is a n1+ -sequence with constant C, we get the additional fact that x 1/C. This type of vectors x for which the original norm is well equivalent to x(n) for some large n will play an essential rôle later. Not surprisingly, the proof of the next lemma, which states the existence of such vectors x when X ∈ X (V), is essentially identical to that of Lemma 1. L EMMA 9. Let X ∈ X (V) satisfy a lower f -estimate. Given a positive integer n and ε > 0, . . , xN of successive norm there exists an integer N(n, ε) such that for every sequence x1 , . one vectors with N N(n, ε), we can ﬁnd x of the form x = λ i∈A xi , where A is some subinterval of {1, . . . , N}, such that x = 1 and x(n) 1 + ε.

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C OROLLARY 10. Let X ∈ X (V) satisfy a lower f -estimate. Then for every n 1 and ε > 0, every GH-subspace Y of X contains a vector y such that y = 1, y(n) 1 + ε and Pn (y) < ε. P ROOF. By the gliding hump procedure, we may select for every integer N a normalized sequence y1 , . . . , yN of vectors in Y and successive vectors n < x1 < · · · < xN in X such that yi − xi < ε/nN . The result follows from Lemma 9 and an obvious approximation argument.

7. A class of examples The contents of this section come from [26] and [27]. The general strategy is as follows: we want to use the coding principle from Section 4 and build special functionals that will somehow distinguish between sums like x + y + z and x − y + z; the difference in this section is that the inevitable sets are not given in advance, but must be constructed together with the norm, by an inductive procedure similar to the construction of Schlumprecht’s space; in short, our example is a Schlumprecht space with special functionals. Another difference with Section 4 is that in order to kill unconditionality, we want to push the unconditionality constant beyond C, not only for a given big C, but for every C. In Section 4, we used special functionals of a ﬁxed length, depending upon C. Here, we shall need special sequences with different lengths k, tending to inﬁnity. Each length k will be used to prove that every basic sequence has unconditionality constant Ck , with limk Ck = +∞. In order to make the behaviour of the space easier to understand, we try to ensure that the different types of special functionals, of lengths k1 and k2 = k1 , have as little interaction as possible; for this we shall prove various lemmas that give almost orthogonality of several classes of vectors and functionals. These lemmas are easy. Also, special functionals should not ruin the possibility of having some form of Schlumprecht’s lemma in our space X; this will require some harder work. The deﬁnition of special sequences is essentially taken from Section 4, except that we must use sets (A∗ ) which are not given in advance, but are enriched step by step as the induction proceeds; we have to guess from the beginning that the chosen sets (A∗ ) will satisfy the needed properties at the end of the construction. However our choice is simple and inspired by what we saw about Schlumprecht’s space S at the end of Section 3: the set A∗ will consist of functionals of the form f ()−1 (x1∗ + · · · + x∗ ), where f is some logarithmic function, ﬁxed in the whole chapter. But in order to make sure that some set A (consisting of normalized sums of RIS of length ) will be inevitable and normed by A∗ , we have to have some sort of Schlumprecht’s lemma here (as explained at the end of Section 3, it is important to have a precise estimate for the norm of the sum of a RIS, in order to be able to predict a class of almost norming functionals for the vectors in A ). An obvious start is to force a lower f -estimate, which ensures that the scheme L1 is inevitable in X, as well as all sets A obtained from RIS of length . But the presence of the special functionals will ruin the possibility that X has a behaviour as regular as that of the Schlumprecht space, and this regular behaviour seemed important for getting Schlumprecht’s lemma.

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This difﬁculty will be solved in the following way: the usual Schlumprecht lemma will hold for RIS that have length in some thin subset L of N, and the corresponding norming functionals will be of the form f ()−1 (x1∗ + · · · + x∗ ) as before. On the other hand, the special sequences y1∗ , . . . , yk∗ will have lengths k in another thin subset K, chosen “very far” from L. Then, the “special normalization” puts the special functional f (k)−1/2 (y1∗ + · · · + yk∗ ) in the dual unit ball, √ so that a RIS sequence of length k ∈ K may have a sum whose norm is as big as k/ f (k) (to be compared to the “usual” smaller value k/f (k)). The failure of unconditionality will appear exactly as in Section 4, by constructing together two sequences (xj )kj =1 , (xj∗ )kj =1 , in such a way that xj ∈ Anj , where nj is the coding number assigned to the beginning x1∗ , . . . , xj∗−1 of the special sequence x1∗ , . . . , xk∗ , and (xj ) is also a RIS vector (a notion to be deﬁned, essentially the normalized sum of a RIS); the special functional f (k)−1/2 (x1∗ + · · · + xk∗ ) will then give to the sum kj =1 xj an abnormally large norm, that would not be achieved by the alternate sum kj =1 (−1)j xj , for which the special functionals will fail to push the norm far beyond the usual √ k/f (k) estimate. In this way we show that the unconditionality constant is larger than f (k), and this can be done with arbitrarily large k ∈ K. In order to generalize Schlumprecht’s lemma, we analyze what properties of a slowly increasing function g are needed. We shall show that the class of possible functions is ﬂexible √ enough to allow the construction of a function √ g which agrees with f on L and with f on K (or on part of K), and stays between f and f everywhere. To this end we introduce the family F of functions g : [1, ∞) → [1, ∞) satisfying the following conditions: (i) g(1) = 1 and g(t) < t for every t > 1; (ii) g is strictly increasing and tends to inﬁnity; (iii) limt →∞ t −ε g(t) = 0 for every ε > 0; (iv) the function t/g(t) is concave and non-decreasing; (v) g(st) g(s)g(t) for every s, t 1. We shall give a convenient representation formula for a subclass F0 of F . Let us denote by L the class of real functions on [0, +∞) that are non-decreasing and 1-Lipschitz on [0, +∞[, and tend to +∞ at +∞. Suppose that M belongs to L; the reader will easily check that the formula gM (t) = exp 0

ln t

du 1 + eM(u)

(F )

deﬁnes gM ∈ F provided gM tends to inﬁnity at inﬁnity, which means that +∞ a function M(u) )−1 du = +∞. The only unpleasant point is to check that t/g(t) is concave, e (1 + 0 see the appendix. The idea of using this subclass F0 is taken from Habala’s paper [28]. One nice point about this subclass F0 is that it is extremely easy to glue together different functions from the class L: if we divide [0, +∞) into successive intervals (In ), it is obvious that M belongs to L if and only if M is continuous, tends to +∞ at inﬁnity and coincides on each interval In with a function Mn ∈ L. Therefore, a function g is in F0 when it is C 1 , tends to +∞ at inﬁnity and coincides on each interval eIn with a function gn ∈ F0 . If we let M(u) = ln(a + bu), with 0 < b a, we get a function in L, for which gM (t) = b (1 + 1+a ln t)1/b . We shall use the two special cases M0 (u) = ln(1 + u), with f0 (t) =

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1 + 12 ln t (corresponding to a = b = 1) and M1 (u) = ln(3 + 2u), with f1 (t) = √ (1 + 12 ln t)1/2 = f0 (t) (corresponding to b = 2, a = 3). For the rest of the paper we √ set f (t) = f0 (t) = 1 + 12 ln t; then f = f1 also belongs to the class F0 . Notice that t −1/4 ln(t) decreases when t > e4 . Checking the value at t = e16 , we obtain that 4t −1/4 f (t) 1 when t e16 .

(F1 )

We need a technical lemma. 2

4u0 u0 L EMMA 11. there √ For every t0 = e with u0 5 and t1 = e √ exists a function g ∈ F0 such that f g f on [1, +∞), g =√ f on [1, t0 ] and g = f on [t1√ , +∞). Similarly, there exists a function g ∈ F0 such that f g f on [1, +∞), g = f on [1, t0 ] and g = f on [t1 , +∞).

For the proof, see the appendix. We begin by gathering some lemmas that do not make use of any “special” construction. Amongst them is a version of Schlumprecht’s lemma (Lemma 14); Lemma 16 shows how to rebuild an n1+ from a RIS, and Lemma 12 deals with almost orthogonality. The ﬁrst ingredient is the notion of a rapidly increasing sequence, in short RIS; it was already mentioned in Sections 2 and 3, but not in precise terms. Let X ∈ X (V). We say that a sequence x1 , . . . , xr of successive non-zero vectors in X satisﬁes the RIS condition 3 if there is a sequence n1 < · · · < nr of integers such that e2r < n1 , xi (ni ) 1 for each i = 1, . . . , r and i−1 # f (ni ) > ran xj , i = 2, . . . , r. j =1

If E is an interval, then the non-zero vectors of the sequence Ex1 , . . . , Exr clearly form a new RIS of length r1 r. Also, λx1 , . . . , λxr is a RIS when 0 < |λ| 1 and every subsequence of a RIS is again a RIS. Given g ∈ F , q 1 and X ∈ X (V), a (q, g)-form q on X is deﬁned to be a functional x ∗ of norm at most one which can be written as j =1 xj∗ for a sequence x1∗ < · · · < xq∗ of successive functionals all of which have norm at most g(q)−1 . Observe that if x ∗ is a (q, g)-form then x ∗ ∞ g(q)−1 and |x ∗ (x)| g(q)−1 x(q) for any x ∈ X. Observe the obvious fact that a (q, g1 )-form is a (q, g2 )-form when g1 g2 . We are looking for two kinds of orthogonalities: the ﬁrst says that a long q-form x ∗ has a small action on a much longer n1+ -average x, where q ) n; this is simply given by the preceding relation |x ∗ (x)| g(q)−1 x(q) . The second kind says that a very long form has a moderate action on the sum of a RIS (thus, by Lemma 14 below, a small action on the normalized sum of this RIS). L EMMA 12. Suppose that (x1 , . . . , xr ) satisﬁes the RI S condition in a space X ∈ X (V). √ 2 Let g ∈ F satisfy g f . If x ∗ is a (q, g)-form on X and q e2r , then we have ∗ x (x1 + · · · + xr ) 3.

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P ROOF. Let n1 < n2 < · · · < nr be the sequence of integers associated to the RIS property of the sequence x1 , . . . , xr . Let i ∈ {0, . . . , r} be such that ni < q ni+1 (consider that n0 = 0 and that nr+1 is larger than q and nr ). Observe that the RIS condition implies xj ∞ xj 1 for each j = 1, . . . , r. The result follows from three easy inequalities, i−1 i−1 ∗ ∗ xj x ∞ ran xj g(q)−1 f (ni )1/2 1, x j =1

j =1

|x ∗ (xi )| xi 1, and for j i + 1, ∗ x (xj ) g(q)−1 xj (q) g(q)−1 xj (n ) g(q)−1 , j so that |x ∗ (x1 + · · · + xr )| 2 + rg(q)−1 . When we have q e2r , we get that g(q) √ f (q) = (1 + 12 ln q)1/2 r. 2

R EMARK 13. It is possible to prove a (1 + ε)-version of Lemma 12 (see [27]), as well as (1 + ε)-versions of all lemmas that follow; we make the deliberate choice of giving simpler proofs, to the cost of introducing ridiculous constants 5, 15, 45, 75, . . . in what follows. The next lemma is a variation on Schlumprecht’s lemma. We need a more general statement than the one from Section 3, that allows to play with different functions from√the family F . In our example, the next lemma will√be applied either to g = f or to g = f , or actually to a variety of functions g between f and f . L EMMA 14 (Variant of Schlumprecht’s lemma). Let X ∈ X (V), g ∈ F , and let p 2 be r an integer. r Suppose that x1 < · · · < xr in X satisfy xi (p ) 1 for every i = 1, . . . , r. Let x = i=1 xi and suppose that Ex 1 ∨ sup x ∗ (Ex): x ∗ is a (q, g)-form, 2 q p for every interval E. Then x rg(r)−1 . The painful proof is deferred to the appendix. C OROLLARY 15. Suppose that V = K. Let X ∈ X (K) and g ∈ F , g X satisﬁes a lower f -estimate and that

√

f ; suppose that

x xc0 ∨ sup x ∗ (x): x ∗ is a (q, g)-form, 2 q for every x ∈ X. Then X is reﬂexive. When V = K, this is also true if each space Vn is reﬂexive.

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P ROOF. It follows immediately from the fact that X satisﬁes a lower f -estimate that the standard basis e0 , e1 , . . . is boundedly complete. Now suppose that it is not a shrinking basis. Then we can ﬁnd ε > 0, a norm-1 functional x ∗ ∈ X∗ and a sequence of successive ∗ normalized blocks that y1 , y2 , . . . such that x (yn ) 2ε for every n 1. It is easy to see 1 −1 2 , and if x = 2 |A| y , then x 1/2 + 1/n for every A ⊂ N such that |A| n (n) i∈A i x ∗ (x) ε. It is clear that for every r, we may construct a RIS x1 , . . . , xr with vectors xi of the preceding form. By Lemma 12, we have x ∗ ( ri=1 xi ) 3 for every long form, therer 1 fore Lemma 14 can be applied to the sequence i=1 xi 3r/g(r). 3 (xi ), proving that r For r sufﬁciently large, this contradicts x ∗ ( i=1 xi ) > rε. For the vector setting V, the reader may consult [2]. The next simple lemma is useful in conjunction with Lemma 14. It explains how to “rebuild” a good n1+ basis from a RIS. L EMMA 16. Let X ∈ X (V), let m, r be integers such that e16 < m < r m4 ; let x1 < · · · < xr in X be such that xi 1,

i = 1, . . . , r,

|A| and xi f (|A|) i∈A

for every interval A ⊂ {1, . . . , r} such that |A| m. Then for every integer n such that mn 2r 3/4 we have r xi i=1

5 (n)

r . f (r)

P ROOF. Let x = ri=1 xi and let (Eh )nh=1 be a sequence of n successive intervals. For every h, let Eh be the largest interval contained in Eh such that Eh x = i∈Ah xi for some interval Ah ⊂ {1, . . . , r}; then, Eh is the union of Eh and two small intervals at both ends of Eh , and Eh x 2 + Eh x. Let H = {h: |Ah | m}. We get Eh x 2 + (m − 1) when h ∈ / H , and n

Eh x n(m + 1) +

h=1

4r 3/4 +

h∈H

Eh x 2nm +

h∈H

|Ah | f (|Ah |)

r r h |Ah | 4r 3/4 + 4r 3/4 + 4 f (m) f (m) f (r)

since f (r) f (m4 ) 4f (m) (using the conditions for the family F ). The result follows, because the condition r e16 implies that 4r 3/4 r/f (r), by formula (F1 ). The construction of our examples uses a lacunary subset J of N. Let us write J in increasing order as {j1 , j2 , . . .}. We assume that each jn is the fourth power of some integer.

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This set J should be such that between two successive elements jn and jn+1 of J , there is enough room to apply Lemma 11; to be on the safe side, let us assume that j1 e256

2

and ∀n 1,

e256jn < jn+1 .

(L1)

This implies by induction that 3

j n en

(L2) 3

3

for every n 1, because j1 > e and e(n+1) e8n e8jn < jn+1 for n 1. Now we check 1/4 that Lemma 11 can be applied between jn4 and jn+1 . If we let u = 4jn , then u 5 and 1/4

2

2

jn4 e4jn = eu ; by (L1), jn+1 e64jn = e4u . With Lemma 11 and the conditions on J it is rather clear that √ L EMMA 17. Let K0 ⊂ K. There exists a function g ∈ F0 such that f g f , g(k) = √ f (k) whenever k ∈ K0 and g(t) = f (t) whenever j ∈ J \ K0 and t is in the interval [j 1/4, j 4 ]. Since we want to include some of the results of Argyros and Felouzis, we shall work in the extended setting V. Also, we want to present a part of [27], so we are trying to build a space with a given algebra of operators generated by some set S of basic operators on our space. Here again, the reader may decide to consider that S = {Id} (the trivial case), which is what is needed for constructing a HI space. Given two inﬁnite sets A, B ⊂ N, deﬁne the spread from A to B to be the map on c00 deﬁned as follows. Let the elements of A and B be written in increasing order respectively / A, and eai maps to ebi for as {a0 , a1 , . . .} and {b0 , b1 , . . .}. Then en maps to zero if n ∈ every i 0. Denote this map by SA,B . Note that SB,A is (formally) the adjoint of SA,B . Observe that for every interval projection E and any U ∈ S, there exist two intervals F1 and F2 such that EU = U F1 , U E = F2 U . Given any set S of spreads containing the identity map, we shall say that it is a semigroup if it is closed under composition (note that this applies to all compositions and not just those of the form SB,C SA,B ). If we want to deﬁne extensions of the SA,B ’s in the V setting, we need to assume more about the relations between the different spaces Vn . We say that V and S are compatible if whenever U ∈ S and U em = en for some m and n, then Vm ⊂ Vn and vn vm for every vector v ∈ Vm . The trivial semi-group S = {Id} is of course compatible with any family V. Given any set S of spreads, compatible with V, all maps SA,B in S are extended to c00 (V) in the usual way, by tensoring with the identity of Vn , giving the set S(V). For example, SA,B (vn ⊗ en ) = vn ⊗ SA,B (en ). The compatibility assumption means that SA,B 1 for the c00 (V) norm (or 1 (V)), for every SA,B ∈ S. The main tool for our construction, already seen in Section 4, is the notion of special sequence. We split the lacunary set J into two disjoint parts K and L: let K, L ⊂ J be the sets {j1 , j3 , j5 , . . .} and {j2 , j4 , j6 , . . .}. The set K is used for the lengths of special sequences, while m ∈ L or m “close” to Lis used for lengths of “regular” RIS x1 , . . . , xm that will satisfy the ordinary inequality m i=1 xi 3m/f (m); the sum of such RIS will

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be therefore normed by (m, f )-forms; this explains our choice for the members of special sequences below. For every n 0, let us choose a countable subset Δn ⊂ Vn∗ dense in Vn∗ (this is where assuming Vn reﬂexive makes our life easier; if not, we would have to work with countable norming sets for Vn ). Let Q ⊂ c00 (V∗ ) be the (countable) set of sequences with the n-th coordinate in Δn for every n and maximum at most 1 in Vn∗ -norm. Let σ be an injection from the collection of ﬁnite sequences of successive elements of Q to the set L introduced above. Given X ∈ X (V) and given an integer m 1, let A∗m (X) be the set of (m, f )-forms X, i.e., the set of all functionals x ∗ of norm at most 1 of the form m on ∗ −1 ∗ ∗ ∗ ∗ x = f (m) i=1 xi , where x1 < · · · < xm and xi 1 for each i = 1, . . . , m. If k ∈ K, let ΓkX be the set of sequences y1∗ < · · · < yk∗ such that yi∗ ∈ Q for each i, y1∗ ∈ A∗j2k (X) ∗ ∈ A∗ and yi+1 σ (y1∗ ,...,yi∗ ) (X) for each 1 i k − 1. We call these special sequences. Let ∗ Bk (X) be the set of all functionals y ∗ of the form k 1 ∗ y∗ = √ y f (k) j =1 j

such that (y1∗ , . . . , yk∗ ) ∈ ΓkX is a special sequence. These, when k ∈ K, are the special functionals (on X of size k). The idea behind this notion of special functionals is that their normalization is different from the usual normalization of functionals obtained by the “Schlumprecht operation” (x1∗ + · · · + xn∗ )/f (n), so that they produce “spikes” in the unit ball of X∗ ; but special functionals are extremely rare, and they are easy to trace, as it was explained in Section 4. Now, given a semi-group S of spreads, compatible with V, we consider the smallest norm on c00 (V) satisfying the following equation, x = xc0 ∨ sup f (n)

−1

n

Ei x: n 2, E1 < · · · < En intervals

i=1

∨ sup x ∗ (Ex): k ∈ K, x ∗ ∈ Bk∗ (X), E ⊂ N an interval ∨ sup U x: U ∈ S . We deﬁne the space X(S, V) as the completion of c00 (V) under this norm. In the case S = {Id} the fourth term drops out and the deﬁnition reduces – when V = K – to that of the space constructed in [26]. The fourth term is there to force every spread U = SA,B ∈ S to deﬁne a bounded operator on X (actually, we get SA,B 1. This restrictive choice could perhaps be relaxed to produce more examples). The second term ensures that X satisﬁes a lower f -estimate. It is useful to understand the construction of X in a way similar to what we have said about the spaces T and S in Section 2: we construct a norming subset in X∗ in a sequence of steps, producing an increasing sequence (Bn ) of convex subsets of Bc0 (V∗ ) . We start with B0 = B1 (V∗ ) ∩ c00 (V∗ ). After Bn is deﬁned, we enlarge it as follows: ∗ – for any integer m 2, we add all functionals x ∗ = f (m)−1 m j =1 xj built from ele∗ ments xj ∈ Bn ,

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– for every k ∈ K, we add all functionals λEU ∗ x ∗ where|λ| = 1, E is an interval, U ∈ S and x ∗ is any special functional x ∗ = f (k)−1/2 kj =1 yj∗ with yj∗ ∈ Bn for j = 1, . . . , k. n ⊃ Bn , and we let ﬁnally Bn+1 be the convex We obtain in this wayan expanded set B hull of Bn . We let B = n Bn and we can check that the above deﬁned norm is equal to x = sup x ∗ (x): x ∗ ∈ B . Observe that the images of successive functionals by a spread are still successive; observe also that the adjoint operation of a spread is again a spread, and that for every interval F and spread U , the operator U F can also be expressed as EU for some interval E; it follows by induction that Bn is stable under the adjoints of the spreads in S and under projections on intervals, and that Bn is contained in the unit ball of c0 (V∗ ), for every n 0. All this implies that X belongs to the family X (V). We summarize the preceding discussion in the following statement. P ROPOSITION 18. Let S be a semi-group of spreads, compatible with the family V. The space X(S, V) belongs to X (V) and satisﬁes a lower f -estimate. Every spread U ∈ S veriﬁes U 1. If we want to compute the norm of x ∈ X = X(S, V), either x = xc0 or, given ε > 0 such that xc0 < x − ε, there exists a ﬁrst n 0 such that |x ∗ (x)| > x − ε n that was adjoined to Bn in the construction of Bn+1 , namely either an for some x ∗ ∈ B (m, f )-form or some EU ∗ y ∗ , with y ∗ a special functional of some length k ∈ K, E an interval and U ∈ S. Let us call surface functional any functional x ∗ on X which is either √ a (m, f )-form for some m 2 or a (k, f )-form EU ∗ y ∗ , with k ∈ K and y ∗ a special functional. We may summarize the lines above by saying that for every vector x in X, either x has the c0 (V)-norm, or x is the supremum of |x ∗ (x)|, when x ∗ runs over the set of surface functionals. Note that if g ∈ F and g(k) = f (k)1/2 , then a special functional y ∗ of size k ∈ K and norm 1 is also a (k, g)-form, and the same is true for each EU ∗ y ∗ , for every interval E and every U ∈ S. A trick which will be repeated several times is √ √ that, when g ∈ F satisﬁes f g f and g = f on K, then all surface functionals √ are g-forms of a certain length 2, either because g f or because g = f on K. This remark explains why the generalized Schlumprecht lemma (Lemma 14), applied to suitable functions g ∈ F , will be our main tool for estimating norms in X(S, V). The preceding paragraph illustrates an aspect of our class of examples that is simpler than what happens with the example of [1], produced by mixed Tsirelson norms. When working with Tsirelson norms, it is often necessary to analyze how a vector was constructed, in a tree of operations corresponding to the inductive deﬁnition of the space. This is not the case We will not need to look “below the surface”. here.1/4 Let L = ∈L [ , 4 ]. The next lemma talks about “regular” RIS. For them, the norm of the sum behaves essentially as in Schlumprecht’s space.

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L EMMA 19. Let x1 , . . . , xr be a RIS in X(S, V) with r ∈ L. Then r r . xi 3 f (r) i=1

√ L, to f on K, and satisfying P ROOF. Use Lemma 17 to get g ∈ F0 equal to f on √ f g f . Then all surface functionals are g-forms of a certain length. Let n1 , . . . , nr 3 be the integers associated to x1 , . . . , xr by the deﬁnition of a RIS. We have e2r < n1 2 3 and xi (ni ) 1. Let p be the integral part of e2r . Then pr e2r < n1 ni , thus r xi (pr ) xi (ni ) 1 for every i = 1, . . . , r. Let x = i=1 xi ; by Lemma 12, we know that |x ∗ (Ex)| 3 for every interval E, when x ∗ is a (q, g)-form with q > p, and by our remarks, we know that Ex is less than the supremum of |x ∗ (Ex)|, when x ∗ runs in the set of (q, g)-forms (unless Ex has the c0 norm, in which case Ex 1). We see that Lemma 14 applies to the RIS ( 13 xi )ri=1 , hence x 3rg(r)−1 = 3rf (r)−1 . L EMMA 20. Let ∈ L and let x1 , . . . , xr be a RIS in X(S, V) with r 4 . Then r r . xi 15 √ f (r) i=1

( )

1/4 1/4 P ROOF. Let m denote the smallest integer larger than r . Then m and we by construction, hence i∈A xi 3|A|/f (|A|) when |A| m, know that [m, r] ⊂ L by Lemma 19. The result follows from Lemma 16, applied to the sequence 13 (xi ) with √ √ 1/4 n = , because e16 < j2 m r m4 , m 2r 1/4 , hence mn = m 2r 3/4 .

A RIS vector is a vector x of the form x =r

−1

f (r)

r

xi ,

i=1

where x1 , . . . , xr is a RIS. We say that r is the length of the RIS vector x. By deﬁnition 3 of a RIS, xi xi (ni ) 1 for some ni > e2r . The most interesting case is when the vectors (xi ) have norms bounded below, by 1/2 say. In this case the lower f -estimate gives x 1/2. If furthermore r ∈ L, then we know that x 3 by Lemma 19. The next lemma is absolutely crucial for understanding what happens in the space X(S, V). This lemma says roughly the following. Suppose that we construct together a special sequence (xj∗ )kj =1 of length k ∈ K and a sequence (xj )kj =1 of RIS vectors, in such a way that the number j ∈ L such that xj∗ is a (j , f )-form, coincides with the length of the RIS vector xj . Then the special functional x ∗ = f (k)−1/2 ki=1 xi∗ and its images U ∗ x ∗ by the adjoints of the spreads U ∈ S will be essentially the only functionals that can force the norm of the vector x = ki=1 xi to exceed signiﬁcantly the usual bound k/f (k). The corresponding lemma was not correctly stated in [26] (the condition ran(xi ) ⊂ ran(xi∗ ) was missing there).

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L EMMA 21. Let k ∈ K and let x1∗ , . . . , xk∗ be a special sequence of length k; let 1 = j2k , √ ∗ ). Assume that f ( ) > | ran( i−1 x ∗ )|2 for and for 2 i k let i = σ (x1∗ , . . . , xi−1 i j =1 j i = 2, . . . , k. Let x1 , . . . , xk be a sequence of successive vectors in X(S, V) such that every xi is a RIS vector of length i and ran(xi ) ⊂ ran(xi∗ ), i = 1, . . . , k. Suppose that k k ∗ ∗ U xi Exi 16 i=1

i=1

for every interval E and every U ∈ S. Then k k . xi 45 f (k) i=1

P ROOF. We know by Lemma 20, applied to the decomposition of xi into a RIS, that √ 3 xi (mi ) 15, where mi = i . Also 1 = j2k e8k by the lacunarity condition (L2 ), 3 1/2 thus 1 = m1 e2k . Since ran xi ⊂ ran xi∗ for each i, this implies that #

i−1 i−1 f (mi ) ran xj∗ ran x j , j =1

i = 2, . . . , k.

j =1

1 This and the lower bound for m1 ensure that 15 (x1 , . . . , xk ) is a RIS of length k, prepar ing thus to apply Lemma 12 to the sequence (xi ) = (xi /15). Let x = kj =1 xj and x = kj =1 xj = x/15. √ By Lemma √ 17, we may select a function g ∈ F equal to f on L ∪ {k}, to f on K \ {k} and such that f g f . Then all f -forms and all ∗-spreads of special forms of length = k are g-forms (by ∗-spread, we mean the adjoint of a spread in S). In order to prove Lemma 21 we shall apply Lemma 14 to the sequence (xi ). We observe that all vectors Ex are either normed by (q, g)-forms or by ∗-spreads of special functionals of length k, or they have norm at most 1. The ﬁrst observation is that “long” forms have a small action on x : indeed by Lemma 12 we know that

∗ z Ex 3

(∗)

whenever z∗ is a (q, g)-form with q > p, where p denotes the integral part of e2k . On the 3 other hand, each vector xi satisﬁes xi (mi ) 15 with mi e2k pk , which puts us in a position to apply Lemma 14 to x . Before this, we need to show that Ex is not given by |z∗ (Ex )|, where z∗ is a ∗-spread of a special functional of length k (the only kind of surface functionals which are not g-forms, in the present situation). k and We shall show that if z1∗ , . . . , zk∗ is any special sequence of functionals of length ∗ z∗ (Ex )| 1 for every U ∈ S, where z∗ is the (k, √f )-form E is any interval, then |U f (k)−1/2 ki=1 zi∗ . Indeed, let t be maximal such that zt∗ = xt∗ or zero if no such t exists. 2

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Suppose i = j or one of i, j is greater than t + 1. We show that |(U ∗ zi∗ )(Exj )| < k −2 . ∗ Since σ is an injection, we can ﬁnd λ1 = λ2 ∈ L such that √ zi is a (λ1 , f )-form, xj is a

RIS vector of length λ2 and xj (m 2 ) 15, where m2 = λ2 . If λ1 < λ2 , it follows from the lacunarity properties of the set J ⊃ L that λ1 < m 2 . This yields that |(U ∗ zi∗ )(Exj )| xj (λ1 ) /f (λ1 ) f (λ1 )−1 . We know that λ1 j2k since λ1 appears in a special sequence of length k. The conclusion in this case now follows from the fact that f () k 2 when j2k (implied by the lacunarity condition (L2 )). If λ2 < λ1 , we apply Lemma 12 to the vector xj

= λ2 f (λ2 )−1 xj equal to the sum of the 2

vectors in the RIS deﬁning xj . The deﬁnition of L gives us that e256λ2 < λ1 , so Lemma 12 gives |(U ∗ zi∗ )(Exj

)| 3. It follows that |(U ∗ zi∗ )(Exj )| f (λ2 )/λ2 . The conclusion follows because j2k implies that f ()/ −3/4 e−6k k −2 (by condition (F1 )). Now choose an interval F (depending upon U ) such that 3

t k U ∗ zi∗ (Ex) = U ∗ xi∗ (F x) 16. i=1

i=1

It follows (since xt +1 3 by Lemma 19) that k

∗ ∗ U zi (Ex) 16 + U ∗ zt∗+1 (Ext +1 ) + 15k 2.k −2 45. i=1

We ﬁnally obtain that |U ∗ z∗ (Ex )| (45/15)f (k)−1/2 < 3f (j1 )−1/2 < 1 as claimed. It follows from (∗) and from what we have just shown about special sequences of length k that

Ex 3 ∨ sup x ∗ Ex : 2 q p, x ∗ is a (q, g)-form whenever E is an interval. Since xj (pk ) 1 for each j = 1, . . . , k, Lemma 14 applied to x /3 implies that x 3k/g(k) = 3k/f (k).

7.1. We have a HI space! In this paragraph, we assume that S = {Id}, so that any family V is compatible with S. We are primarily interested in the case V = K, and we shall prove that the resulting space X = X({Id}, K) is HI. However, we shall keep the V framework and prove at the same time that any two GH-subspaces of X(V) = X({Id}, V) almost intersect. When V = K, every subspace Y of X is a GH-subspace, and we get that X is HI. Let Y, Z be two GH-subspaces of X. Let us choose δ > 0 and let k ∈ K be an integer such that f (k)−1/2 < δ/360. We want to show that the distance between the unit spheres of Y and Z is less than δ. By the gliding hump property of Y and Z, we may assume that both Y and Z are spanned by block bases. Since X satisﬁes a lower f -estimate, Corollary 10 tells us that every block

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subspace of X contains, for every n 1, a vector x such that x(n) 1 and x > 1/2. We already observed that every vector Ex either has the supremum norm or satisﬁes the inequality Ex sup x ∗ (Ex): q 2, x ∗ is a (q, g)-form , where g is the function obtained from Lemma 17 in the case K0 = K. This allows us to make the following construction. Using Corollary 10, we may ﬁnd in Y (or in Z) for any given integer ∈ L, a sequence y1 , . . . , y which satisﬁes the RIS condition, and also such that yj > 1/2 for j = 1, . . . , . We are going to apply this fact k times, with increasing values of , and alternating our choice between Y and Z at each step. Let ﬁrst 1 = j2k ∈ L, and let x1 = 1 −1 j =1 yj ∈ Y be a RIS vector of length 1 , with yj > 1/2 for every j . For each 1 f (1 ) j between 1 and 1 let yj∗ be a functional such that yj∗ 1 and 0 < yj∗ (yj ) − 1/2 < k −1 ; let x ∗1 be the (1 , f )-form f (1 )−1 j 1=1 yj∗ . Then 0 < x ∗1 (x1 ) − 1/2 < k −1 . By continuity and the density of Δn in the unit ball of Vn∗ , we may assume that there exists an (1 , f )-form x1∗ ∈ Q such that 0 < x1∗ (x1 ) − 1/2 < k −1 and ran(x1∗ ) = ran(x1 ) (in the real case; in the complex case, we ask for 1/2 < !x1∗ (x1 ) and |x1∗ (x1 ) − 1/2| < k −1 ); since we have an inﬁnite sequence of possible choices for x1∗ ∈ Q and since σ is injective, we may √ choose x1∗ such that 2 = σ (x1∗ ) satisﬁes f ( 2 ) > | ran(x1∗ )|2 . Also, note that there is no difference between an (1 , g)-form and an (1 , f )-form, because g = f on L. Now let 2 = σ (x1∗ ) and pick a RIS vector x2 ∈ Z of length 2 such that x1 < x2 , similarly to the ﬁrst step. As above, we can ﬁnd an (2 , g)-form x2∗ ∈ Q such that √ 0 < x2∗ (x2 ) − 1/2 < k −1 , ran(x2∗ ) = ran(x2 ) and f ( 3 ) > | ran(x1∗ + x2∗ )|2 , where 3 = σ (x1∗ , x2∗ ). Continuing in this manner, we obtain a pair of sequences x1 , . . . , xk and x1∗ , . . . , xk∗ with various properties we shall need. First, xi ∈ Y when i is odd and xi ∈ Z when i is even. We also know that 0 < xi∗ (xi ) − 1/2 < 1/k for each i. Finally, and perhaps most importantly, the sequence x1∗ , . . . , xk∗ has been carefully chosen to be a special sequence of length k. It follows immediately from the implicit deﬁnition of the norm and from the fact that ran(xi∗ ) = ran(xi ) for each i that k k 1 xi f (k)−1/2 xi∗ (xi ) > kf (k)−1/2 . 2 i=1

i=1

The proof willbe complete if we can ﬁnd a suitable upper bound for the norm of√the alternate sum ki=1 (−1)i−1 xi . For this we apply Lemma 21. The conditions on f ( i ) and on the inclusions of ranges have been taken care of during the construction of the sequences (xi ) and (xi∗ ). It remains to show that |( ki=1 xi∗ )( ki=1 (−1)i Exi )| 16 for every interval E. This follows easily from the fact that xi∗ (xi ) is almost exactly 1/2 for every i; there are possibly two incomplete terms, one at the beginning and one at the end of E, for which we use |xi∗ (xi )| xi 3 (this follows from Lemma 19 applied to the RIS corresponding to xi ). Lemma 21 therefore shows that ki=1 (−1)i−1 xi 45kf (k)−1 .

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We have now constructed two vectors y ∈ Y , the sum of the odd-numbered xi s, and 1/2 y − z. If z ∈ Z, the sum of the even-numbered xi s, such that y + z > (1/90)f (k)√ a is the maximum of y and z, then y/a and z/a have distance < 180/ f (k) < δ/2, and one of them belongs to the unit sphere of Y or Z. It follows that the distance from the unit sphere of Y to the unit sphere of Z is less that δ. Suppose that V = K. The above proof shows that X is HI. We also observe that X is reﬂexive. This follows from Corollary 15. In the vector case, the conclusion is that any two GH-subspaces have distance 0. In particular, we obtain that every subspace Y of X such that all projections Pn , n 0 are strictly singular on Y , is a HI space (because all subspaces of Y are GH-subspaces of X). T HEOREM 22. Let X = X(V) be the space constructed in Section 7 when S = {Id}. Then any two GH-subspaces of X have distance 0. Every subspace Y of X such that all projections Pn , n 0 are strictly singular on Y , is a HI space. When V = K, then X is a reﬂexive HI space.

8. Factorization through a HI space In this section we present some of the results of Argyros and Felouzis [2, Theorems 2.3 and 2.4]. We shall use a variant of the interpolation spaces of Lions and Peetre (see [6] for instance), following the spirit of the exposition of [10] rather than that of interpolation theory. Let W be a bounded symmetric closed convex subset of a Banach space V . Suppose that a = (an ) is a decreasing sequence of positive numbers, such that limn an = 0. For every n 0, we consider the bounded symmetric convex set Cn = 2n W + an BV and we let ja,n be the gauge of Cn . For every n, the gauge ja,n deﬁnes an equivalent norm on V , and we shall call Vn the space V equipped with the equivalent norm ja,n . We shall be interested in the (usually unbounded, possibly inﬁnite) gauge ja = supn0 ja,n on V . It is clear that ja is ﬁnite on W (because ja is less than the gauge jW of W , since ja,n 2−n jW for every n 0). It is also clear that ja is larger than a multiple of the norm of V (because C0 is bounded, say). The classical construction of factorization spaces in [10] uses the 2 1/2 , but we shall instead use a non-standard j “quadratic” gauge qa (x) = ( +∞ n=0 a,n (x) ) way of mixing the norms of the Vn s, namely the construction from the preceding Section 7. We need a notion of thinness of W that guarantees that there is no subspace Y of V on which the ja norm is ﬁnite and equivalent to the V -norm. This was done by Neidinger (see [44,45]) in a similar setting (Neidinger was looking for hereditarily-p interpolation spaces, while we are looking for hereditarily indecomposable interpolation spaces). Let us say that a bounded symmetric closed convex subset W is a-thin in V when there is no inﬁnite-dimensional subspace Y of V such that ja is ﬁnite and bounded on the unit ball BY = Y ∩ BV of Y . This means that the inclusion from V1 to V is strictly singular, where V1 is the space of those v ∈ V such that vV1 = ja (v) < +∞. The set W is thin in Neidinger’s sense when for every subspace Y of V , there exists ε > 0 such that for every C, the unit ball BY of Y is not contained in CW + εBV . If W is thin in Neidinger’s sense, then it is a-thin for every sequence a: there exists ε > 0 such that BY ⊂ 2m W + εBV for every m, in particular for any given , when an < ε2− there exists

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1281

y ∈ BY such that y ∈ / 2n+ W + 2 an BV , thus ja,n (y) 2 and W is a-thin. On the other n hand, if a is the sequence an = e−8 , then W = BL∞ is a-thin in V = L1 = L1 [0, 1] (see [2, Proposition 2.2]), but the inclusion L∞ → L1 is not thin, as mentioned by Neidinger. This shows that the statement of the factorization theorem below is slightly more general when formulated with a-thin instead of thin. We say that a bounded linear operator T : U → V is a-thin if the closure T (BU ) of T (BU ) is a-thin in V . T HEOREM 23. Suppose that T : U → V is a-thin for some a. Then T factors through a HI space. P ROOF. We know that for some a, the set W = T (BU ) is a-thin in V . For every n 0, let Vn denote V with the norm ja,n , and consider the family V = (Vn )n0 . Let X = X(V) be the space constructed in Section 7, and let Z denote the diagonal of this space, in other words Z is the vector subspace of V consisting of those v ∈ V such that v = (v, v, . . .) belongs to X. We deﬁne a norm on Z by vZ = vX . We shall check that T factors through Z, and that Z is a HI space. For every u ∈ U such that u 1, the vector w = T u ∈ V belongs to W , therefore wVn = ja,n (w) 2−n ; the series n w ⊗ en is normally convergent in X and deﬁnes an element T1 (u) ∈ Z. On the other side, the norm of V in X is larger than ja (by the general assumptions about X (V)), thus larger than vV , and this shows that there is a natural inclusion map i from Z to V . We have therefore obtained the factorization T = i ◦ T1 . In order to prove that Z is HI, we have only to check that Z, regarded as a subspace of X via the diagonal map, is such that Pn0 is strictly singular on Z for every n0 0. Given any subspace Y ⊂ Z, it is possible to ﬁnd y ∈ Y such that ja,n (y) < 2−n0 for n = 0, . . . , n0 but yX = 1. This is clear since each of the ﬁrst gauges ja,n is equivalent to the V -norm while ja is unbounded on BY and less than the norm of X. This shows that Pn0 is strictly singular on Z. The result then follows from Theorem 22 (we are a little bit cheating, since we wrote the proof under the additional hypothesis that V is reﬂexive). T HEOREM 24. For every p ∈ (1, +∞), the space p is a quotient of some HI space. This is also true for c0 . It is obvious that this cannot hold for 1 , by the lifting property of 1 . We shall only sketch the case of p , p ∈ (1, +∞); see [2] for a much more general result, but also much more difﬁcult to prove. The strategy for the proof is the following: we shall construct a space V and a symmetric a-thin closed convex subset W ⊂ V which is norming for an q -subspace L of V ∗ . Let U be the Banach space whose unit ball is W , and let us apply the preceding factorization result to the inclusion map T : U → V . Then T = i ◦ T1 , with T1 : U → Y and Y a HI space with an embedding i : Y → V . We need only show that i ∗ induces an isomorphism from L to a subspace of Y ∗ . This is easy since the set T1 (BU ) = W1 , which is smaller than BY , is already norming for the space i ∗ (L). The construction of V uses a tree; (very) roughly speaking, we introduce an inﬁnite branch γ of the tree for every vector z∗ of L, and a sequence wγ ∈ V supported on that branch, which norms z∗ . Next, we deﬁne W to be the symmetric closed convex hull of the set of all elements wγ . For every n 0, let Dn be the subset of [−1, 1] consisting of

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all numbers of the form i2−n−3 , |i| 2n+3 . Let T0 be the set of all ν = (d0 , . . . , dn ), for n 0, such that dj ∈ Dj for j = 0, . . . , n. We say that |ν| = n is the length of ν. We say that ν ν if ν is an initial segment (d0 , . . . , dm ), m n, of the sequence (d0 , . . . , dn ) = ν. We shall restrict our attention to the subtree T ⊂ T0 consisting of those nodes ν such that |ν| p i=0 |di | < 1. On the space c00 (T ) of ﬁnitely supported scalar sequences indexed by T , we consider the p ((k1n )) norm, vV =

+∞

p 1/p |vν |

,

n=0 ν∈Rn

where Rn ⊂ T is the set of nodes ν such that |ν| = n and kn = |Rn |. Let (fν )ν∈T be the natural unit vector basis for V . It is clearly 1-unconditional. For every ν = (d0 , . . . , dn ) ∈ T , we set cν = dn ; if b is a segment in the tree we set x(b) = ν∈b cν fν . Let W be the symmetric closed convex set generated by all vectors x(b); by the deﬁnition of T , we know that x(b)V < 1 for every segment b of T . Let L be the subspace of V ∗ generated by the sequence gn = ν∈Rn fν∗ , for n 0. It is clear that (gn ) is isometrically equivalent to the unit vector basis of q , 1/q + 1/p = 1. It is easy to prove that W is 14 -norming for L. Indeed, given z∗ = n0 vn gn such that z∗ = 1, we choose the branch γ = (ν0 , . . . , νn , . . .), |νn | = n, such that dn = cνn is as close as possible to un = |vn |q−1 sign(vn ): we choose dn ∈ Dn such that sign dn = sign vn , and if |un | 2−n−2 , we may also make sure that 12 |un | |dn | |un |. We get +∞

1 1 cνn vn z∗ x(γ ) = |vn |q : |vn | 2−n−2 . 2 4 n=0

We shall check that W is thin in V . If not, we can ﬁnd a subspace Y of V , such that choosing 0 < ε < 1/4, we have BY ⊂ CW + εBV for some C. By a standard gliding hump argument, we can ﬁnd in Y a normalized sequence (yn ) which is a small perturbation of a sequence supported on disjoint subsets (En ) of the tree, say yn − En yn < ε, where En is the set of all nodes ν such that kn |ν| n , with n < kn+1 for every n 0. We need the following lemma, whose proof is sketched in the appendix. Let us call band in T any subset E consisting of all nodes t such that k |t| , for some k . L EMMA 25. Let α > 0, let (wn ) be a sequence of elements of W and (En ) be a sequence of successive bands in T . There exists an inﬁnite subset M ⊂ N and for every m ∈ M a

, E

, such that: E

w < α, and for every m = m in M, decomposition of Em in Em 1 2 m m m

the nodes in Em1 and Em2 are incomparable. Let us ﬁnish the proof of Theorem 24. We assumed that BY ⊂ CW + εBV , with 0 < ε < 1/4; for every n, there exists wn ∈ W such that yn − Cwn < ε; passing to a subsequence, we may assume that En is decomposed into En and En

satisfying the properties cited in Lemma 25, with α = ε/C. We have En

(yn − Cwn ) < ε and En

(Cwn ) < Cα = ε, therefore En

yn < 2ε and En yn > 1 − 3ε. We may thus ﬁnd a normalized sequence of

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functionals (yn∗ ) in V ∗ , with yn∗ supported on En , and norming for En yn . This sequence (yn∗ ) is isometrically equivalent to the unit vector basis of q . Consider y ∗ = N −1/q (y1∗ + ∗ ); since the supports of the (y ∗ ) are incomparable, every vector x(b) from the · · · + yN j family generating W acts on at most one yi∗ ; it follows that supw∈W |y ∗ (w)| N −1/q . On the other hand, let y = N −1/p (y1 + · · · + yn ) ∈ Y . We have that y ∗ (y) > 1 − 3ε and by assumption, there exists w ∈ W such that y − Cw < ε. This implies that |y ∗ (Cw)| > 1 − 4ε; this is a contradiction when N is large enough.

9. Additional results In this section we present some of the results of [27]. We assume that V = K. Given a semi-group S of spreads, we denote by X = X(S) = X(S, K) the space constructed in Section 7, with V = K. In [27], three examples are given, corresponding to three semigroups of spreads. We shall concentrate here on the example Xs , which is a space with an isometric right shift operator S, on which every bounded operator is a strictly singular perturbation of a normally converging series of powers of S and its adjoint, the left shift on Xs . Before presenting this example, we need to study some general properties of a larger class of examples. Given any set S of spreads containing the identity map, we shall say that it is a ∗semi-group if it is a semi-group closed under taking adjoints. An example of such a set is the collection of all spreads SA,B where A = {m, m + 1, m + 2, . . .} and B = {n, n + 1, n + 2, . . .} for some m, n 0. This is the ∗-semi-group generated by the shift operator. Given any ∗-semi-group S of spreads, we shall say that it is a proper ∗-semi-group if, for every (i, j ) = (k, l), there are only ﬁnitely many spreads S ∈ S for which e∗i (Sej ) = 0 and 0. The ∗-semi-group generated by the shift operator is proper. Let x = vj ej e∗k (Sel ) = ∗ be two elements of c . If S is proper, then except for ﬁnitely many and x ∗ = vi∗ e 00 i S ∈ S, the sum i,j vi∗ vj e∗i (Sej ) = x ∗ (Sx) has only one non-zero term, and is therefore bounded by x∞ x ∗ ∞ . Note that a proper set S of spreads must be countable, and if we write it as {S1 , S2 , . . .} and set Sm = {S1 , . . . , Sm } for every m, then for any x ∈ X(S), x ∗ ∈ X(S)∗ , we have lim sup x ∗ (U x): U ∈ S \ Sm x∞ x ∗ ∞ . m

(P )

Let X = X(S, K) be the space constructed in Section 7, with V = K; by Corollary 15, we know that X is reﬂexive. Let y ∈ X. Recall that for every integer n 1, y(n) = sup

n

Ei y,

i=1

where the supremum is extended to all families E1 < · · · < En of successive intervals. Observe that ei (n) = 1 for every n 1. Given a subspace Y ⊂ X, we will be interested in a seminorm ||| · ||| deﬁned on L(Y, X) as follows. Given T ∈ L(Y, X) let |||T ||| be the supremum of those numbers κ such that for

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every n 1, there exists a vector y ∈ Y such that Pn y 2−n , y(n) 1 and T y > κ. Clearly, |||T ||| T . Let us say the same thing in a slightly different way. The number |||T ||| is the smallest number with the following property: for every ε > 0, there exists an integer n 1 such that, for every y ∈ Y , the conditions Pn y 2−n and y(n) 1 imply that T y |||T ||| + ε. We may also write that for every y ∈ Y and n N(ε),

T y |||T ||| + ε y(n) + 2n Pn y. We say that a bounded sequence (yn ) ⊂ Y is a sequence of almost successive vectors in X if there exists a sequence (xn ) of successive vectors such that limn xn − yn = 0. If (yn ) ⊂ Y is a sequence such that Pn yn 2−n for every n 1, then clearly we may ﬁnd almost successive subsequences (ynk ). Let MY be the set of sequences y = (yn )+∞ n=1 of almost successive vectors in Y such that lim supn yn (n) 1. Now, given T ∈ L(Y, X) it is clear that |||T ||| = sup lim sup T yn . y∈MY

n

L EMMA 26. For every inﬁnite-dimensional subspace Y of X = X(S, K) and every T ∈ L(Y, X), we have (i) if |||T ||| = 0, then T is strictly singular; (ii) if T is compact, then |||T ||| = 0; (iii) if for every z in some inﬁnite-dimensional subspace Z of Y , we have T z z, then |||T ||| 1. P ROOF. By Corollary 10, every subspace Y contains normalized sequences in MY . Hence every subspace of Y contains a norm one vector y such that T y |||T ||| + ε; in particular, if |||T ||| = 0, then T is strictly singular. It is clear that lim T xn = 0 if T is compact and (xn ) almost successive (because X is reﬂexive), hence |||T ||| = 0. Lastly, suppose that T z z for every z in some subspace Z of Y . We know from Corollary 10 that Z contains a normalized sequence (zn ) of almost successive vectors with lim zn (n) = 1. By deﬁnition, |||T ||| limn T zn 1. T HEOREM 27. Let S be a proper ∗-semi-group of spreads. The Banach space X = X(S, K) from Section 7 satisﬁes a lower f -estimate and the following three properties. (i) For every x ∈ X and every SA,B ∈ S, SA,B x x (and therefore SA,B x = x if supp(x) ⊂ A). (ii) If Y is an inﬁnite-dimensional subspace of X, then every operator from Y to X is in the ||| · |||-closure of the set of restrictions to Y of operators in the algebra A generated by S. In particular, all operators on X are ||| · |||-perturbations of operators in A. (iii) The seminorm ||| · ||| on L(X) satisﬁes the algebra inequality |||U V ||| |||U ||||||V |||. Notice a straightforward consequence of this result. If we write G for the ||| · |||completion of A (after quotienting by operators with ||| · ||| zero) then G is a Banach algebra. Given T ∈ L(X), we can ﬁnd by (ii) a ||| · |||-Cauchy sequence (Tn )+∞ n=1 of operators

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in A such that |||T − Tn ||| → 0. Let φ(T ) be the limit of (Tn )+∞ n=1 in G. This map is clearly well-deﬁned. It follows easily from (iii) that it is also a unital algebra homomorphism. The kernel of φ is the set of T such that |||T ||| = 0. We have K(X) ⊂ ker φ ⊂ S(X). The restriction of φ to A is the identity (or more accurately the embedding of A into G). If A is small, then, since the kernel of φ consists of small operators, L(X) is also small. Let us indicate why X does not contain an inﬁnite unconditional sequence. Let Y ⊂ X be a subspace with an unconditional basis (yn ). Let (Mα )α denote an uncountable family of subsets of N, such that any two of them differ by an inﬁnite set. For every α, let Pα denote the projection from Y on to the span of (yn )n∈Mα . For α = β, there exists a subspace Z ⊂ Y such that (Pα − Pβ )z z for every z ∈ Z. This implies that |||Pα − Pβ ||| 1 by Lemma 26, but this contradicts the separability of L(X) in the ||| · |||-norm, that follows from (ii) and the countability of S. L EMMA 28. Let S be a proper ∗-semi-group of spreads, let X = X(S), let Y ⊂ X be an inﬁnite-dimensional subspace and let T be a continuous linear operator from Y to X. Let S= ∞ m=1 Sm be a decomposition of S satisfying condition (P ). Then for every ε > 0 there exists m such that, for every x ∈ Y such that x(m) 1 and Pm x 2−m ,

d T x, m conv λU x: U ∈ Sm , |λ| = 1 ε. P ROOF. Suppose that the result is false. Then, for some ε > 0, we can ﬁnd a se−n such that, setting C = quence (yn )+∞ n n=1 with yn ∈ Y , yn (n) 1 and Pn yn 2 n conv{λUyn : U ∈ Sn , |λ| = 1}, we have d(T yn , Cn ) > 2ε. This yields that (yn ) is bounded away from still denoted 0. We may pass to an almost successive subsequence, (yn ), such that yn − yn < +∞ for some successive sequence (yn ) ⊂ X satisfying yn (n) 1 for every integer n 1. Then for some n0 , (yn )nn0 and (yn )nn0 are equivalent basic sequences (see [39, 1.a.9]), with the additional property that for every α > 0, there exists n1 = n1 (α) n0 such that every norm one vector y = nn1 an yn satisﬁes y − nn1 an yn < α. If we replace Y by the block subspace Y generated by (yn )nn0 and T by T deﬁned on Y by T yn = T yn for n n0 , we still get the conclusion that d(T yn , Cn ) > ε, where Cn is deﬁned from (yn ) as Cn is deﬁned from (yn ), provided n0 was chosen large enough. This argument shows that if the result is false, then it is already false for some block subspace Y and some operator T from Y to X. In this case, it is not hard to show that T can be perturbed (in the operator norm) to an operator whose matrix (with respect to the natural bases of X and Y ) has only ﬁnitely many non-zero entries in each row and column. We may therefore assume that T has this property. We also assume T 1. Since we assumed that the result is false for Y and T , then for some ε > 0, we can ﬁnd a sequence (yn )+∞ n=1 with yn ∈ Y , yn (n) 1 and supp(yn ) > {n} such that d(T yn , Cn ) > ε, and also such that if zn is any one of yn , T yn or Uyn for some U ∈ Sn and zn+1 is any one of yn+1 , T yn+1 or V yn+1 for some V ∈ Sn+1 , then zn < zn+1 . By the Hahn–Banach theorem, for every n 1 there is a norm-one functional yn∗ such that sup yn∗ (x): x ∈ Cn + εB(X) < yn∗ (T yn ).

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It follows that yn∗ (T yn ) > ε and sup |yn∗ (Cn )| 1. Therefore |yn∗ (Uyn )| n−1 for every U ∈ Sn . We may also assume that the support of yn∗ is contained in the smallest interval containing the supports of yn , T yn and Uyn for U ∈ Sn . (The case of complex scalars requires a standard modiﬁcation.) Given ∈ L deﬁne an -pair to be a pair (x, x ∗ ) constructed as follows. Let yn1 , yn2 , . . . , yn be a subsequence of (yn )+∞ n=1 satisfying the RIS condition, which implies 3 2 2 −1 that n1 > e > . Let x = f ()(yn1 +· · ·+yn ) and let x ∗ = f ()−1 (yn∗1 +· · ·+yn∗ ), where the yn∗i are as above. Lemma 19 implies that x 3 and Lemma 20 that x(√) 15. If (x, x ∗ ) is such an -pair, then x ∗ ∈ A∗ (X) and, by our earlier assumptions about supports, x ∗ (T x) = −1

yn∗i (T yni ) > ε.

i=1 −2 for every U ∈ S . Similarly, |x ∗ (U x)| n−1 1 < Let k ∈ K be such that εf (k)1/2 > 45. We now construct sequences x1 , . . . , xk and x1∗ , . . . , xk∗ as follows. Let 1 = j2k and let (x1 , x1∗ ) be an 1 -pair. Let m2 be such that |x1∗ (U x1 )| x1 ∞ x1∗ ∞ if U ∈ S \ Sm2 . The functional x1∗ can be perturbed so that it is in Q and so that ran(x1∗ ) ⊃ ran(x1 ), 2 = σ (x1∗ ) is larger than m2 and √ f ( 2 ) > |ran(x1∗ )|2 +4, while (x1 , x1∗ ) is still an 1 -pair. In general, after x1 , . . . , xi−1 and ∗ have been constructed, let (xi , xi∗ ) be an i -pair such that all of xi , T xi and x1∗ , . . . , xi−1 ∗ ∗ , and then perturb x ∗ in such a way that, xi are supported after all of xi−1 , T xi−1 and xi−1 i ∗ ∗ ∗ setting i+1 = σ (x1 , . . . , xi ), we have |xi (U xi )| xi ∞ xi∗ ∞ whenever U ∈ S \ Si+1 √ and we also have ran(xi∗ ) ⊃ ran(xi ), f ( i+1 ) > |ran( ij =1 xj∗ )|2 + 2i+1 . This yields that ∗ f ( −1 2−i−1 . xi+1 ∞ i+1 ) Now let x = x1 + · · · + xk , let z∗ = x1∗ + · · · + xk∗ and x ∗ = f (k)−1/2 z∗ . Our construction guarantees that x ∗ is a special functional, and therefore of norm at most 1. We have

T x x ∗ (T x) > εkf (k)−1/2. Our aim is now to get an upper bound for x and to deduce an arbitrarily large lower bound for T . For this purpose we use Lemma 21. In order to apply this lemma, it is enough to show that |(U ∗ z∗ )(Ex)| = |z∗ (U Ex)| 16 for any interval E and U ∈ S. We have xi∗ ∈ A∗i for i = 1, . . . , k. Suppose that U ∈ Sm+1 \ Sm , and let t be such that / Si+1 , t m < t +1 . If i > t, then U ∈ Si and |xi∗ (U xi )| < −2 i . If i < t, then U ∈ so |xi∗ (U xi )| xi ∞ xi∗ ∞ 2−i . If i = t, then at least we know that |xi∗ (U xi )| xi 3. Putting all these facts together, we get that |z∗ (U Ex)| 16, as desired (we may have two incomplete terms at both ends of E). Hence, by Lemma 21, x 45kf (k)−1 . It follows that T (ε/45)f (k)1/2 > 1, a contradiction.

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L EMMA 29. Let S, X, Y , T and ε be as in the previous lemma, let m be as given by that lemma and let Am = m conv{λSm : |λ| = 1}. Then there exists U ∈ Am such that |||T − U ||| 64ε. P ROOF. If the statement of the lemma is false, then for every operator U ∈ Am there is a sequence x U = (xn ) ∈ MY of vectors in Y such that limn (T − U )xn > 64ε. We write this symbolically as (T − U )x U > 64ε. This yields that lim inf xn > δ > 0, with δ depending only upon T and ε. At this point, we may argue as in Lemma 28 in order to reduce the situation to the case of a block subspace Y . Since Am is compact in operator norm, we may ﬁnd a ﬁnite set (x α ) ⊂ MY such that for every U ∈ Am , we have (T − U )x α > 64ε for some α. Passing to subsequences, we may assume that the sequences (x α ) can be arranged to be subsequences of a single sequence (yn ) ⊂ Y such that we can ﬁnd a successive sequence that δ yn yn (n) 1 for every n, and such (yn ) in X satisfying yn (n) 1 for every n and yn − yn < +∞. Let n0 be chosen so that (yn )nn0 and (yn )nn0 are equivalent basic sequences. Recall that for every α > 0, there exists n1 = n1 (α) n0 such that every norm one vector y = nn1 an yn satisﬁes y − nn1 an yn < α. Let Y1 ⊂ Y be the subspace generated by the sequence (yn )nn0 ; the conclusion of Lemma 28 is obviously still true for the restriction T1 of T to Y1 . Now, let Y1 be the block subspace generated by the sequence (yn )nn0 ; we may assume that all vectors in Y1 sit after m. Let us deﬁne T1 on Y1 by T1 yn = T yn for every n n0 . For every vector y ∈ Y1 such that y (m) 1, we may ﬁnd – if n0 was chosen large enough – a vector y ∈ Y1 such that T1 y = T y, y − y < ε/2 and y − y (m) 2−m , hence Pm y 2−m and y(m) 3/2. We see that the conclusion of Lemma 28 is still true for T1 , provided we lose an additional ε. Furthermore, for every U ∈ Am , we have (T1 − U )x > 62ε for some x ∈ MY1 . This shows that we may assume that Y is a block subspace such that ∀y ∈ Y,

d T y, {Uy: U ∈ Am } 2εy(m)

(∗)

and that for every U ∈ Am , we have (T − U )x > 62ε for some x ∈ MY . s be a covering of Am by open sets of diameter less than ε in the operator Let Ui=1 norm. For every i = 1, . . . , s, let Ui ∈ Ui and let x i = (xi,n )n be a successive sequence in MY such that (T − Ui )x i > 62ε. By the condition on the diameter of Ui , we have (T − U )x i > 60ε for every U ∈ Ui . As in the last lemma, we can assume that the matrix of T has only ﬁnitely many non-zero entries in each row and column. Our ﬁrst aim is to show that the vectors x U can be chosen continuously in U . (This statement will be made more precise later.) Let (φi )ri=1 be a partition of unity on Am with φi supported inside Ui for each i. Let ∈ L be greater than s and m2 . For each i s, let xi,n1 , . . . , xi,n satisfy the RIS condition and let m < xi,n1 . Let yi = −1 f ()(xi,n1 + · · · + xi,n ). Let this be done in such a way that y1 < · · · < ys and also (T − U )xi,n1 < · · · < (T − U )xi,n for every i and every U ∈ Am . Finally, let the xi,nj be chosen so that (T − U )xi,nj > 60ε for every U ∈ Ui . Now let us consider the vector y(U ) = si=1 φi (U )yi . By Lemma 20 we know that yi (√) 15 for each i = 1, . . . , s, from which it follows by the triangle inequality that

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y(U )(√) 15. We shall show that y(U ) is a “bad” vector for U , by showing that (T − U )y(U ) > 30ε. To do this, let U ∈ Am be ﬁxed and let I = {i: φi (U ) > 0}. Recall that (T − U )xi,nj > ∗ be a norm-one 60ε for every i ∈ I and j = 1, . . . , . For such an i and for j let zi,j ∗ functional such that zi,j ((T − U )xi,nj ) > 60ε. Let these functionals be chosen to be suc ∗ + · · · + z∗ ) and z∗ = ∗ ∗ cessive. Let zi∗ = f ()−1 (zi,1 i∈I zi . Then zi (T − U )yi > 60ε, so i,

z∗ (T − U )y(U ) = z∗ φi (U )(T − U )yi > 60ε. i∈I

However, z∗ f (s)/f () 2, proving our claim. The function U → y(U ) is clearly continuous on Am . The vector y(U ) satisﬁes y(U )(m) 15 and (T − U )y(U ) > 30ε. We now apply a ﬁxed-point theorem. For every U ∈ Am , let Γ (U ) denote the set of V ∈ Am such that (T − V )y(U ) 30ε. Clearly Γ (U ) is a compact convex subset of Am . By property (∗), we know that Γ (U ) is non-empty for every U . The continuity of U → y(U ) gives that Γ is upper semicontinuous, so there exists a point U ∈ Am such that U ∈ Γ (U ). But this is a contradiction. Lemma 29 shows in particular that any operator T : Y → X can be approximated arbitrarily well in the ||| · |||-norm by the restriction of some operator U ∈ A. We have therefore ﬁnished the proof of property (ii). The proof of (iii) is much easier, and will complete the proof of Theorem 27. L EMMA 30. The seminorm ||| · ||| on L(X) satisﬁes the algebra inequality |||U V ||| 75|||U ||||||V |||. P ROOF. In [27], this statement is proved with the constant 1 in place of 75 (see Re−1 mark 13). Pick c > 1 and let (xn )+∞ n=1 ∈ MX be a sequence such that U V xn c |||U V ||| for every n. After suitable perturbations and selections of subsequences we may assume that xn , V xn , U V xn have supports before xn+1 , V xn+1 , U V xn+1 , and that xn+1 , . . . , x2n is a RIS for every n 1. Let u > |||U |||, v > |||V ||| and pick ∈ L large enough so that U x ux(√) ,

V x vx(√)

(∗)

whenever x ∈ X and < x. We consider the vectors xn+1 < · · · < xn+4 , for some n > 4 such that xn+1 , V xn+1 , U V xn+1 have supports after . For every subset A ⊂ {1, . . . , 4 } such that |A| , we know by Lemma 20 applied with r = |A| to the RIS (xn+i )i∈A that i∈A xn+i (√) 15|A|/f (|A|), hence we get by (∗) that i∈A V xn+i √ 15v|A|/f (|A|); by Lemma 16 applied with m = , n = and r = 4 to the successive sequence (κV xn+i )ri=1 , κ = (15v)−1 , this yields r V xn+i 75rf (r)−1 v; √ i=1

( )

Banach spaces with few operators

by (∗) it follows that

r

i=1 U V xn+i 75rf (r)

−1 uv;

1289

but

r U V xn+i c−1 rf (r)−1 |||U V ||| i=1

by the lower f -estimate, and ﬁnally |||U V ||| 75cuv.

9.1. The shift space Let S be the proper ∗-semi-group mentioned earlier, generated by the shift, which we denote by S. That is, S consists of all maps of the form SA,B where A = [m, ∞) and B = [n, ∞). We will write L for the left shift, which is (formally) the adjoint of S. Then every operator in S is of the form S m Ln , because LS = Id. Since SL − Id is of rank one, n every operator in A is a ﬁnite-rank perturbation of an operator of the form N n=0 λn S + N n n=1 μn L , so the difference is of ||| · |||-norm zero. Let Xs denote the space obtained from Theorem 27 in this case. L EMMA 31. Let U =

N

n=0 λn S

U = |||U ||| =

N n=0

n

+

|λn | +

N

n n=1 μn L .

N

Then

|μn |.

n=1

P ROOF. For notational convenience, let λ−n = μn for 1 n N . Clearly it is enough to prove that |||U ||| N n=−N |λn |. In this paper, we did not write almost isometric versions of the basic lemmas, so we will only prove this inequality up to some multiplicative constant. Let m 1 be given. For an integer r ∈ L consider the vector xr = 2r j =r+1 e3j N . Since every unit vector ei satisﬁes ei (n) = 1 for every n, we have xr r/f (r) by Lemma 14; let yr = r −1 f (r)xr ; by Lemma 16, there exists r such that yr (m) 5 and we may choose r as big as we like, in particular r 3N . This shows that the sequence 15 (yr ) has subsequences in M, thus Uyr (5 + ε)|||U ||| for large r. On the other hand, splitting U xr into 3rN singleton pieces from 3rN + 1 to (6r + 1)N gives that U xr rf (3Nr)−1 N n=−N |λn | by the lower f -estimate. This lower bound on U xr gives 10|||U ||| 5f (3Nr)f (r)−1 |||U |||

N n=−N

|λn |.

Let W denote the Wiener convolution algebra 1 (Z). The preceding lemma gives an isometric embedding i from W into L(Xs ). Indeed, since all powers of Sand L have n norm 1, we may associate to any a = (an ) ∈ 1 (Z) the operator i(a) = ∞ n=0 an S + ∞ n n=1 a−n L ∈ L(Xs ). The next result gives, up to a strictly singular perturbation, the converse of this fact. We call Toeplitz operators on Xs the elements from the subspace T = i(W ).

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C OROLLARY 32. There is an algebra homomorphism and projection φ from the space L(Xs ) onto the subspace T consisting of Toeplitz operators with absolutely summable coefﬁcients. If T ∈ L(Xs ) then |||φ(T ) − T ||| = 0. P ROOF. Recall the remark following the statement of Theorem 27. In this case, by Lemma 31, the algebra G, the ||| · |||-completion of A, is the same as the completion in L(Xs ) and also the completion in the operator norm on 1 . Therefore G can be regarded as a subalgebra of L(Xs ) consisting of Toeplitz operators with absolutely summable coefﬁcients. If we do this, then the algebra homomorphism φ deﬁned after Theorem 27 is also a projection. The equation |||φ(T ) − T ||| = 0 follows easily from the deﬁnition of φ. T HEOREM 33. The space Xs is prime. P ROOF. Let P : Xs → Xs be a projection. By the previous corollary the operator φ(P ) is a convolution by some absolutely summable sequence (an )n∈Z . Moreover, φ(P )2 = φ(P ). But the Fourier transform of the sequence (an )n∈Z is a continuous function on the circle squaring to itself. Hence it is constantly zero or one. It follows that a0 is zero or one and all the other an are zero. That is, φ(P ) is zero or the identity. Since P − φ(P ) is strictly singular, it follows that P is of ﬁnite rank or corank. Thus, if P Xs is inﬁnite-dimensional, then it has ﬁnite codimension. Since the shift on Xs is an isometry, it follows that Xs is isometric to its range, namely an hyperplane; using powers of S, we see that Xs and P Xs are isomorphic, which proves the theorem. A simple consequence of Corollary 32 is that, up to strictly singular perturbations, any two operators on Xs commute. Indeed, if V and W are two operators, then φ(V ) and φ(W ) commute, so φ(V W − W V ) = 0, from which it follows that |||V W − W V ||| = 0. For the rest of this section, we assume that Xs has complex scalars. Let ψ : L(Xs ) → C(T) be the composition of φ with the Fourier transform. Then ψ is also a continuous algebra homomorphism. Given an operator T on Xs , let KT be the compact set of μ ∈ C such that μ is inﬁnitely singular for T , which means for us that for every ε > 0 there is an inﬁnite-dimensional subspace Y ⊂ Xs such that T y − μy εy for every y ∈ Y . Since T − φ(T ) is strictly singular, Kφ(T ) = KT . L EMMA 34. The function ψ(T ) takes the value zero at some exp(iθ ) if and only if 0 is inﬁnitely singular for T . P ROOF. If ψ(T ) takes the value zero at exp(iθ ), we can construct an approximate eigenvector for φ(T ) with eigenvalue zero as follows. Suppose that φ(T ) is convolution by the sequence (an )n∈Z , and let ε > 0. We know that ψ(T )(θ ) = n∈Z an exp(inθ ) = 0. Let ∈ L and let 2

2

x = f 2 −2 exp(inθ )en . n=2

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1291

By Lemma 14 we have x = 1, because en (p) = 1 for every p 1. Let U be convolution by the sequence (an )n=− . If is large enough, then U − φ(T ) ε/2, since 2 (an )n∈Z is absolutely summable. Moreover, all but at most 4 of the possible + 2 non2 −2 zero coordinates of U x are equal to f ( ) n=− an exp(inθ ). Taking sufﬁciently large, we can therefore make φ(T ) − U and U x as small as we like. Therefore zero is inﬁnitely singular for φ(T ). Since |||T − φ(T )||| = 0, the same is true for T . Conversely, if ψ(T ) never takes the value zero, then it can be inverted in C(T). A classical result states that the Fourier transform of this inverse will also be in 1 (Z), so in particular φ(T ) has an inverse U which is continuous when considered as an operator on Xs and satisﬁes U = φ(U ). Therefore φ(U T − Id) = 0, so U T − Id is strictly singular and 0 is not inﬁnitely singular for T . C OROLLARY 35. The set KT is the image under ψ(T ) of the unit circle T. P ROOF. This follows from Lemma 34 applied to the operator T − λId.

T HEOREM 36. A subspace Y of Xs is isomorphic to Xs if and only if it has ﬁnite codimension. P ROOF. Let T : Xs → Y be an isomorphism. Then 0 is not inﬁnitely singular for T , so, as in the proof of Lemma 34, we can ﬁnd U such that T U , U T and Id are the same, up to a strictly singular perturbation. Since T U − Id is strictly singular, T U is Fredholm with index zero. In particular codim Y = codim T Xs codim T U Xs < ∞. As we have already mentioned, the if part follows from the existence of the isometric shift.

Appendix Let us check that when g is deﬁned by the formula (F ) from Section 7, then t/g(t) is concave. We have that ln(t ) eM(u) du , k(t) = t/g(t) = exp 1 + eM(u) 0 hence after some computations k

(t) =

k (t) M (ln t) − 1 0 M(ln t ) t (1 + e )

because M is 1-Lipschitz and k (t) 0. P ROOF OF L EMMA 11. We shall explain how to go down from the “big” function f0 at 2 t0 = eu0 to the “small” function f1 at t1 = e4u0 , or equivalently, how to go from the small function M0 (u) = ln(1 + u) at u0 to the large function M1 (u) = ln(3 + 2u) at u1 = 4u20 . More precisely, we need to build a function M ∈ L, that coincides with M0 on [0, u0 ] and

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with M1 on [u1 , +∞), and such that g = gM satisﬁes the conclusions of the lemma, which reduce then to g(t0 ) = f0 (t0 ), g(t1 ) = f1 (t1 ) and f1 g f0 on [t0 , t1 ]. The proof of the other case is similar.

(t)/g (t) = It is clear that M N on [0, u] implies gN gM on [1, eu ], and since tgN N (1 + eN(ln(t )))−1 we see that the inequality M N on some interval [u, v] ⊂ [0, +∞) implies that gM /gN is non-decreasing on the interval [eu , ev ]. We ﬁrst deﬁne a function N ∈ L by letting N(u) = M0 (u) for u u0 and N(u) = M0 (u0 ) + u − u0 = ln(1 + u0 ) + u − u0 for u u0 . There is a unique value v0 > u0 such that N(v0 ) = M1 (v0 ) and a v1 such that N(v1 ) = M1 (u1 ) = ln(3 + 2u1 ), given by v1 = ln(3+2u1 )+u0 −ln(1+u0 ); one can check that v1 < u1 since M1 (u1 )−M0 (u0 ) < u1 −u0 (exercise); it follows that u0 < v0 < v1 < u1 (the reader must draw a picture). Let N0 ∈ L be equal to N on [0, v0 ] and to M1 on [v0 , +∞), and N1 ∈ L be equal to N on [0, v1 ], equal to the constant value M1 (u1 ) on [v1 , u1 ] and equal to M1 on [u1 , +∞). For every s ∈ [0, 1] let Ns = (1 − s)N0 + sN1 , and let gs = gNs ∈ F0 be the corresponding function. Notice that gs (t0 ) = f0 (t0 ) for every s. Our problem is to make sure that for some s = s0 ∈ (0, 1), the value of gs at the other end t1 is what we expect, namely gs (t1 ) = f1 (t1 ). It will follow that gs0 = f1 on [t1 , +∞), and it will only remain to check that f1 gs0 f0 . We shall ﬁrst check that g1 (eu1 ) < f1 (eu1 ), and next that g0 (eu1 ) > f1 (eu1 ): we have v1 v1 g1 (ev1 ) du du = ln = = N(u) e g1 (eu0 ) 1 + (1 + u0 ) eu−u0 1 + u0 u0 +∞ 1 1 1 eu0 −u du < , 1 + u0 u0 1 + u0 6 u1 u1 g1 (eu1 ) du du 1 = ln = < M1 (u1 ) e g1 (ev1 ) 4 + 2u 2 1 + 1 v1 v1

ln

g1 (ev1 ) f0 (eu0 )

so that ﬁnally g1 (eu1 )/f0 (eu0 ) e2/3 < 2. On the other hand 1/2 )

1 1 f1 eu1 /f0 eu0 = 1 + ln t1 1 + ln t0 2 2

1/2 ( u0 1+ = 1 + 2u20 >2 2 (because u0 5) so that f1 (eu1 ) > g1 (eu1 ). Next we see that g0 (eu1 ) > f1 (eu1 ): since N0 = M1 on [v0 , u1 ], the quotient g0 /f1 is constant on the interval [ev0 , eu1 ], thus g0 (eu1 )/f1 (eu1 ) = g0 (ev0 )/f1 (ev0 ) > 1 because N0 < M1 on the interval [0, v0 ). We have that g1 (eu1 ) < f1 (eu1 ) < g0 (eu1 ). By continuity there exists s ∈ (0, 1) such that gs (eu1 ) = f1 (eu1 ). It only remains to check that f1 gs f0 . Since Ns < M1 on [u0 , v0 ) and Ns > M1 on (v0 , u1 ] we know that gs /f1 is increasing on [t0 , ev0 ) and decreasing on (ev0 , t1 ]. Since the quotient is > 1 at t0 and equal to 1 at t1 , it follows that f1 gs on [t0 , t1 ]. Using similar but simpler arguments one can see that gs f0 on the same interval, because Ns M0 on [0, u1].

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P ROOF OF L EMMA 14. Let G(t) = t/g(t) when t 1 and G(t) = t when 0 t 1. This function G is concave and increasing on [0, +∞). For every interval E and every integer 0, let σ (E) =

r

Exi (p ) .

i=1

This expression is increasing with , and σ0 (E) =

r

Exi (1) =

i=1

r

Exi Ex.

i=1

We shall prove by induction on κ, 1 κ r that whenever E is an interval such that Exi = 0 for at most κ indices i, then

Ex G σκ (E) . (∗) Once this is done, we obtain the result for κ = r and E = ran(x), r

r . xi (pr ) G(r) = x G σr ran(x) = G g(r) i=1

Let κ(E) denote the number of indices i ∈ {1, . . . , r} such that Exi = 0 (if κ(E) = 0, then Ex = 0 and this case is obvious). Observe ﬁrst that when Ex 1, we have Ex = G(Ex) G( ri=1 Exi ) G(σ (E)) for every 0. This shows in particular that (∗) is true when κ(E) = 1, since Ex 1 in this case. Assume (∗) true when κ(E) < r, and suppose there exists an interval E such that κ(E) = + 1 and Ex > G(σ+1 (E)); since (∗) is not true for E we qknow that 1 and Ex > 1. By assumption there exists a (q, g)-form x ∗ = g(q)−1 ( j =1 Aj xj∗ ), 2 q p, where Aj = ran(xj∗ ), A1 < · · · < Aq and xj∗ 1, such that

G σ+1 (E) < x ∗ (Ex). Assume ﬁrst that κ(Aj E) for every j = 1, . . . , q. We have Aj Ex G(σ (Aj E)) by the induction hypothesis, and using the concavity of G and the relation q p we obtain ∗ x (Ex)

q 1 1 G σ (Aj E) Aj Ex g(q) g(q) q q

q

j =1

1 q G g(q) q

1 q G g(q) q

q j =1 r i=1

j =1

r q 1 q G σ (Aj E) = Aj Exi (p ) g(q) q

Exi (p+1 )

i=1 j =1

σ+1 (E) q G = . g(q) q

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If σ+1 (E) q, this last expression is equal to

σ+1 (E)/g(q) σ+1 (E)/g σ+1 (E) = G σ+1 (E) , otherwise it is equal to

σ+1 (E) σ+1 (E) = G σ+1 (E) , g(q)g(σ+1 (E)/q) g(σ+1 (E)) so that we have reached a contradiction. In the remaining case there exists j0 ∈ {1, . . . , q} such that Aj0 Exi = 0 for every i such that Exi = 0. Assume, for example, j0 < q (otherwise 1 < j0 deserves a similar treatment). Let m be the last integer i such that Exi = 0. Let Bj0 = Aj0 \ ran(Exm ), Bj 0 +1 = Aj0 ∩ ran(Exm ), Bj

0 +1 = Aj0 +1 , Bj0 +1 = Bj 0 +1 ∪ Bj

0 +1 and Bj = Aj otherwise. We see that Aj0 Ex + Aj0 +1 Ex Bj0 Ex + Bj 0 +1 Exm + Bj

0 +1 Exm Bj0 Ex + Bj0 +1 Exm (2) . Every Bj satisﬁes κ(Bj E) , so that the induction hypothesis applies and since p 2 we obtain q j =1

Aj Ex Bj0 +1 Exm (2) +

Bj Ex

j =j0 +1

G σ (Bj0 +1 E) + G σ (Bj E) , j =j0 +1

and the conclusion follows as before.

P ROOF OF L EMMA 25. Let t be a node in n0 En . If m is the integer such that t ∈ Em , let bt denote the segment consisting of those nodes s ∈ Em such that s t. We let Fm denote the set of nodes t ∈ Em such that x(bt ) α/2 and we let Fm

be the complement . It is clear that b ∩ Fm

is a segment, whenever b is a segment. of Fm in Em of W , with A a ﬁnite family of segments and Let w = b∈A λb x(b) be an element

b∈A |λb | 1. Let us check that Fm w < α/2 for every m. For each b ∈ A, let b de

note the segment b ∩ Fm . If t is the longest node in b , then x(b ) x(bt ) < α/2. Therefore

α F w = λb x b < . m 2 b∈A

For every n let us write En wn = b∈An λb x(b), where An is a ﬁnite set of segments b contained in En , and b∈An |λb | 1. Let Γ be the compact set of inﬁnite branches of the tree T , with the topology of pointwise convergence at nodes. For every b ∈ An , let

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γb be an inﬁnite branch containing the segment b and consider the non-negative measure μn = b∈An |λb |δγb on Γ . Passing to a subsequence we may assume that (μn ) is weak-∗ convergent to a ﬁnite non-negative measure μ on Γ . Let Bm denote the set of γ ∈ Γ that intersect Fm . We want to show that μ(Bm ) tends to 0. If not, we may ﬁnd for every k a branch γ and m1 < · · · < mk such that γ ∈ Bmj for j = 1, . . . , k. Let bj be the segment γ ∩ Emj . It follows from the deﬁnition of Fm j that x(bj )p α/2 forp j = 1, . . . , k. But this is impossible when k is large, because j x(bj ) x(γ ) 1, by the deﬁnition of the space. Suppose that m = m0 satisﬁes μ(Bm ) < α/4. This implies that μn (Bm ) < α/4 when n is large, which means that most of the vectors x(b) used in the construction of En wn sit on branches that do not meet Fm . Taking n > m large enough, we may also have μ(Bn ) < α/8. We let Fn

denote the set of nodes in En that are above some node in Fm . Then Fn

wn <

= F are incomparable. Furthermore, if we α/4 and the nodes in En = Fn \ Fn

and in Em m

let En = Fn ∪ Fn , then En wn < α. We have just explained the beginning of a construction by induction of a sequence (mj )∞ j =0 satisfying the properties asked in Lemma 25. We let m0 = m and m1 = n, where m and n are as in the preceding paragraph. The next value of n, which will be chosen as m2 , must satisfy μn (Bm0 ) < α/4, μn (Bm1 ) < α/8 and μ(Bn ) < α/16. The reader will easily complete the missing steps. References [1] S. Argyros and I. Deliyanni, Examples of asymptotic 1 Banach spaces, Trans. Amer. Math. Soc. 349 (1997), 973–995. [2] S. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000), 243–294. [3] S. Argyros and A. Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Preprint (2001). [4] S. Banach, Théorie des Opérations Linéaires, Warszawa (1932). [5] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Amer. Math. Soc. Colloq. Publ. 48 (1999). [6] J. Bergh and J. Löfström, Interpolation Spaces, Springer (1976). [7] C. Bessaga and A. Pełczy´nski, A generalization of results of R.C. James concerning absolute bases in Banach spaces, Studia Math. 17 (1958), 165–174. [8] J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (1986), 221–226. [9] P.G. Casazza and T. Shura, Tsirelson’s Space, Lecture Notes in Math. 1363, Springer. [10] W.J. Davis, T. Figiel, W.B. Johnson and A. Pełczy´nski, Factoring weakly compact operators, J. Funct. Anal. 17 (1974), 311–327. [11] V. Ferenczi, A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199–225. [12] V. Ferenczi, Operators on subspaces of hereditarily indecomposable Banach spaces, Bull. London Math. Soc. 29 (1997), 338–344. [13] V. Ferenczi, Hereditarily ﬁnitely decomposable Banach spaces, Studia Math. 123 (1997), 135–149. [14] V. Ferenczi, Quotient hereditarily indecomposable Banach spaces, Canad. J. Math. 51 (1999), 566–584. [15] V. Ferenczi and P. Habala, A uniformly convex Banach space whose subspaces fail Gordon–Lewis property, Arch. Math. 71 (1998), 481–492. [16] T. Figiel, An example of inﬁnite dimensional reﬂexive Banach space non isomorphic to its Cartesian square, Studia Math. 42 (1972), 295–306.

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[17] T. Figiel and W.B. Johnson, A uniformly convex Banach space which contains no p , Compositio Math. 29 (1974), 179–190. [18] E. Gluskin, Finite-dimensional analogues of spaces without a basis, Dokl. Akad. Nauk SSSR 261 (1981), 1046–1050; English translation: Soviet Math. Dokl. 24 (3) (1981), 641–644. [19] E. Gluskin, The diameter of the Minkowski compactum is roughly equal to n, Funktsional. Anal. i Prilozhen. 15 (1981), 72–73. English translation: Functional Anal. Appl. 15 (1) (1981), 57–58. [20] W.T. Gowers, Lipschitz functions on classical spaces, European J. Combin. 13 (1992), 141–151. [21] W.T. Gowers, A solution to Banach’s hyperplane problem, Bull. London Math. Soc. 26 (1994), 523–530. [22] W.T. Gowers, A Banach space not containing c0 , 1 or a reﬂexive subspace, Trans. Amer. Math. Soc. 344 (1994), 407–420. [23] W.T. Gowers, A solution to the Schroeder–Bernstein problem for Banach spaces, Bull. London Math. Soc. 28 (1996), 297–304. [24] W.T. Gowers, A new dichotomy for Banach spaces, Geom. Funct. Anal. 6 (1996), 1083–1093. [25] W.T. Gowers, Ramsey methods in Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1071–1097 (this Handbook). [26] W.T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874. [27] W.T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997), 543– 568. [28] P. Habala, A Banach space all of whose subspaces fail the Gordon–Lewis property, Math. Ann. 310 (1998), 197–219. [29] R.C. James, Bases and reﬂexivity of Banach spaces, Ann. of Math. 52 (1950), 518–527. [30] R.C. James, Uniformly non-square Banach spaces, Ann. of Math. 80 (1964), 542–550. [31] R.C. James, A separable somewhat reﬂexive Banach space with nonseparable dual, Bull. Amer. Math. Soc. 80 (1974), 738–743. [32] N. Kalton, The basic sequence problem, Studia Math. 116 (1995), 167–187. [33] N. Kalton and J. Roberts, A rigid subspace of L0 , Trans. Amer. Math. Soc. 266 (1981), 645–654. [34] R. Komorowski and N. Tomczak-Jaegermann, Banach spaces without local unconditional structure, Israel J. Math. 89 (1995), 205–226. [35] R. Komorowski and N. Tomczak-Jaegermann, Erratum to: “Banach spaces without local unconditional structure”, Israel J. Math. 105 (1998), 85–92. [36] J. Lindenstrauss, On complemented subspaces of m, Israel J. Math. 5 (1967), 153–156. [37] J. Lindenstrauss, Some aspects of the theory of Banach spaces, Adv. Math. 5 (1970), 159–180. [38] J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain 1 and whose duals are non-separable, Studia Math. 54 (1975), 81–105. [39] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergebnisse 92, Springer (1977). [40] P. Mankiewicz, A superreﬂexive Banach space X with L(X) admitting a homomorphism onto the Banach algebra C(βN), Israel J. Math. 65 (1989), 1–16. [41] B. Maurey and H.P. Rosenthal, Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61 (1977), 77–98. [42] V.D. Milman, Spectrum of bounded continuous functions on the unit sphere of a Banach space, Functional Anal. Appl. 3 (1969), 67–79. [43] V.D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspekhi Mat. Nauk 26 (1971), 73–149; Russian Math. Surveys 26 (1971), 79–163. [44] R.D. Neidinger, Factoring operators through hereditarily-p spaces, Lecture Notes in Math. 1166, Springer (1985), 116–128. [45] R.D. Neidinger, Concepts in the real interpolation of Banach spaces, Functional Analysis (Austin, TX, 1986–87), Lecture Notes in Math. 1332, Springer (1988), 43–53. [46] E. Odell and T. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281. [47] A. Pełczy´nski, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. [48] C. Read, Different forms of the approximation property, Lecture at the Strobl Conference (1989), and unpublished preprint. [49] T. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), 81–95.

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[50] S. Shelah, A Banach space with few operators, Israel J. Math. 30 (1978), 181–191. [51] S. Shelah and J. Stepr¯ans, A Banach space on which there are few operators, Proc. Amer. Math. Soc. 104 (1988), 101–105. [52] S. Szarek, On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc. 293 (1986), 339–353. [53] S. Szarek, A superreﬂexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (1986), 437–444. [54] B.S. Tsirelson, Not every Banach space contains p or c0 , Functional Anal. Appl. 8 (1974), 138–141 (translated from Russian).

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CHAPTER 30

Type, Cotype and K-Convexity Bernard Maurey Laboratoire d’Analyse et Mathématiques Appliquées, UMR 8050, Université de Marne la Vallée, Boulevard Descartes, Cité Descartes, Champs sur Marne, 77454 Marne la Vallée Cedex 2, France E-mail: [email protected]

Contents 1. The pre-history of type and cotype, as I remember it 2. Super-properties . . . . . . . . . . . . . . . . . . . 3. Ultrapowers and some operator lemmas . . . . . . . 4. Krivine’s theorem . . . . . . . . . . . . . . . . . . . 5. Type, cotype and np s. The MP+K theorem . . . . . 6. K-convexity and Pisier’s theorem . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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1. The pre-history of type and cotype, as I remember it At the end of the sixties, Pietsch [52] promoted the notion of p-summing operators between Banach spaces, which extends to all values of p ∈ [1, +∞) the study of some classes of operators introduced by Grothendieck [20], under different names, for the special values p = 1, 2. In an important paper devoted to p-summing operators, Lindenstrauss and Pełczy´nski [39] gave a second birth to what we know in Banach space theory as the Grothendieck theorem; one formulation of it states that every operator from 1 to 2 is 1-summing; another formulation is the famous Grothendieck’s inequality. Around 1969, L. Schwartz introduced radonifying maps, a notion that turned out to be closely related to p-summing maps. A special case of this notion deals with the Wiener measure and with linear maps from a Hilbert space H to a Banach space X, that transform the canonical cylindrical Gaussian measure of H into a true Radon probability measure on X (see Gross [18,19] for another viewpoint on this subject). L. Schwartz organized a seminar at the Ecole Polytechnique in Paris ([63], 1969–70) about these topics. This is one of the reasons why Paris, and especially the Ecole Polytechnique, became one of the places where the subject of type and cotype was developed. Type and cotype conditions appeared ﬁrst in the framework of p-summing operators, or more precisely in connection with the factorization through Lp , p > 1, of operators with values in L1 (in this paper, operator means bounded linear operator). In the spring of 1972 I saw the preprint of the paper [62] by H. Rosenthal; this paper played an essential role for me; it contains several ideas that I later used and developed in [42]. Two of these ideas taken from [62] are the factorization conditions and the notion of stable type p. By Pietsch’s factorization theorem, which extends some factorization results due to Grothendieck [20], every q-summing operator from C(K) to a Banach space factors through the natural injection C(K) → Lq (K, μ), for some probability measure μ on K. Rosenthal dualizes this fact, and shows that given T : X → L1 linear such that T ∗ is q-summing, then T factors through a multiplication operator Mf : Lp → L1 by a function f ∈ Lq (1/p + 1/q = 1; let us simply write Lr for Lr (K, μ), 0 < r +∞); we have thus T = Mf ◦ T1 , where T1 : X → Lp is bounded and linear. One can give direct conditions on T that guarantee this factorization, with no need to further reference to q-summing maps: if an operator T : X → L1 is such that 1/p 1/p T (xi )p dμ C xi p , i

i

for some C and every ﬁnite sequence (xi ) ⊂ X, then T factors as T = Mf ◦ T1 for some f ∈ Lq . The proof of the factorization theorem is just an application of the Hahn–Banach separation theorem, either directly as in [41], or by going back to Pietsch’s factorization as in [62]. One gets in this way a function f ∈ Lq such that f q 1 and |T (x)/f |p dμ C p xp for every x ∈ X. The above operator T1 is then deﬁned by T1 (x) = T (x)/f ∈ Lp for every x ∈ X. Next, it is shown in [62] that a simple norm condition on X, that happens to be true for X = Ls when 2 s > p > 1, easily implies the above factorization condition,

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as soon as T : X → L1 is bounded (and linear). This condition on a Banach space X is of the form 1/p p dt K f (t)x x , i i i i

i

where K is a constant depending only upon X, (fi ) is a sequence of L1 -normalized p-stable variables, and (xi ) an arbitrary sequence in X. This condition was called stable type p in [41,42]; it was used in [62] (without this name) for the injection of X ⊂ L1 to L1 , and in the general case in [42]. For example, since a Hilbert space has type 2, we obtain in this way that every bounded linear map from a Hilbert space to L1 factors through a multiplication Mf : L2 → L1 , a statement dual to one of the results of [20]: every operator from a C(K)-space to a Hilbert space is 2-summing. By trace duality, this yields that every operator from 1 to 2 is 2-summing; we may call this the easy Grothendieck theorem. The same proof shows that every operator from a C(K)-space, to a space X such that the dual X∗ has type 2, is 2-summing: this result appeared for the ﬁrst time in a paper by Dubinsky, Pełczy´nski and Rosenthal [8]. It is obvious to generalize to operators from X to Lr the condition that gives a factorization through a multiplication operator Lp → Lr (0 < r < p, see [41,42]). In particular, some of the results obtained for 0 < r < 1 are parallel to results obtained earlier by Nikishin [50,51]: since every Banach space X has stable type 1 − ε for every ε > 0, every operator from X to Lr , 0 < r < 1, factors through L1−ε when 1 − ε r. A ﬁrst relation between these topics and ﬁnite-dimensional geometry comes from the paper [62]; there, a delicate quantitative lemma (Lemma 6 from [62]) shows that when the injection from a subspace X ⊂ L1 to L1 does not factor through any Lp , p > 1, then X must contain complemented almost isometric copies of n1 for every n 1, proving thus that every reﬂexive subspace of L1 embeds in some Lp , p > 1 (the main result of [62]). This lemma was extended in [42] to a general Banach space X as follows: when there exists an operator T : X → Lp that does not factor through any Lp+ε , ε > 0, then the injections n1 → np , n ∈ N, uniformly factor through X. In particular, when there exists an operator T : X → L1 that does not factor through any L1+ε , ε > 0, then X contains uniformly isomorphic and complemented copies of n1 , for every n 1. This gives a new (bizarre) proof of Grothendieck’s theorem: since n1 is not uniformly complemented in c0 , the preceding statement implies that every bounded linear map from c0 to L1 factors through L1+ε , and it reduces Grothendieck’s theorem to a much easier variant. It is a model for a list of reduction results, for example, this sort of extension of the Grothendieck theorem: every operator from a cotype 2 space X to any Banach space, which is 2-summing, is already 1-summing (see [42]; as we have just said, when X = L1 , this is the information that one needs in order to pass from the easy Grothendieck theorem to the real one). This line of results displayed interesting connections between some simple ﬁnite-dimensional phenomenons and analytic facts about Banach spaces. In the same years, Hoffman-Jørgensen [22] proved general results about series of vector valued independent random variables, that are in the spirit of Kahane’s inequalities for vector valued Rademacher series; he also deﬁned Rademacher type-p and showed connections to the law of large numbers in [23]. The notion of type 2 (with a different name)

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appeared ﬁrst in [8], and it was shown in this article that stable type 2 and Rademacher type 2 are identical. The results from [22] imply that stable type p and Rademacher type p are closely related for every p ∈ (1, 2]: stable type p implies Rademacher type p, and Rademacher type p implies stable type p − ε for every ε > 0. Later on, it has been universally admitted that Rademacher type is easier to work with, and the notion of stable type p essentially disappeared, except for p = 2, because 2-stable type and cotype express interesting properties of Gaussian probability measures on a Banach space. With Rademacher type p (we say simply type p in what follows), several points are simpliﬁed; it is obvious that type p implies type r for r p, and the opposite for cotype; the results for Lr spaces are easier to formulate, and simple to prove using Khintchine’s inequality: Lr has type r and cotype 2 when 1 r 2 and type 2 and cotype r when 2 r < +∞. Clearly, Lr does not have type r + ε, ε > 0, when 1 r 2, and does not have cotype r − ε when 2 r +∞. This suggested that one could possibly read some geometrical information about X from the limit values of p and q that give type p or cotype q for X. The ﬁrst attempts to relate type, cotype to the fact that X contains almost isometric copies of some classical spaces concerned n∞ and n1 . The ﬁrst result [44] gave the equivalence between non-trivial cotype for X and the fact that X does not contain n∞ uniformly; today, the proof in [44] looks a bit ridiculous by its complication. It was presented at the Conference at Oberwolfach, October 73; at the same meeting, James presented a much deeper result, namely his solution of the “reﬂexive vs B-convex” problem (see below). This was perhaps the beginning of what was later called “Local theory”. For the relation between the absence of n1 s in a Banach space X and other properties of this X, the ﬁrst steps are due to Beck, Giesy and James, several years before this story [1,15,24]; Beck showed the relevance to the law of large numbers in Banach spaces of the fact that X does not contain copies of n1 s. Beck and Giesy deﬁned B-convex Banach spaces as follows: the Banach space X is B-convex if for some n > 1 and ε > 0, and for all norm one vectors (xi )ni=1 in X, at least one choice of signs gives ni=1 ±xi n(1 − ε). Giesy proved several Banach space ﬂavoured results about B-convexity, for example, that X∗ and X∗∗ are B-convex when X is B-convex. James [24] also worked on this class, which he called uniformly non n1 ; in this paper [24], he conjectured that B-convex spaces can be renormed to be uniformly convex, and must therefore be reﬂexive (and he disproved this conjecture in 1973, as we have said above). Shortly after the result for cotype and n∞ , Pisier proved the type and n1 case [53]; he developed the submultiplicativity method for the type constants, which was important for the following paper [45]. Pisier’s result showed that the class of B-convex spaces coincides with the class of spaces X that have type p for some p > 1. Then Pisier and I started to work on the relations between the limit values for the type or cotype of X, and the existence of subspaces of X that look somewhat like np . Our ﬁrst approach to the results of [45] was to strengthen the Dvoretzky–Rogers factorization [11] for a Banach space X, using information on the limits of type and cotype; it just happened that the beautiful result of Krivine [36] (see Section 4) appeared during the preparation of [45] and allowed us to prove a much more satisfactory result. In the ﬁrst version of [45], we proved that when X has type p − ε but not p + ε for every ε > 0, then the injections n1 → np factor almost

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isometrically through a subspace of X for all n 1, which means that we can ﬁnd norm one vectors x1 , . . . , xn in X such that

n i=1

1/p |ai |

p

n n (1 + ε) ai xi (1 + ε) |ai | i=1

i=1

for all scalars (ai ); the second inequality is of course obvious. When p < 2, this is a strengthening of the Dvoretzky–Rogers lemma which says that the above statement holds in every Banach space when p = 2. Krivine’s theorem appeared shortly after the ﬁrst version of [45] was written; fortunately, Studia Math. was so slow to publish at that time that we were able to modify our article in the form which is known as Maurey–Pisier or Maurey–Pisier–Krivine theorem. I will call it here MP+K theorem, to emphasize the fact that these three persons did not work together on this particular paper. Kwapie´n was visiting Paris in 1971 and 72, just before all this started, and he played a signiﬁcant role in the mathematical education of some of the young French; he gave several seminar talks that had a serious impact on us; he read and found the mistakes in several false “new proofs” that I had for the Grothendieck theorem, and he was the ﬁrst person who checked the eventually correct proof that I gave in [42]. His result in [37] had a great inﬂuence on the subject of type and cotype; it appeared actually before the deﬁnitions of type and cotype were given, but it is nice to formulate it as follows: if X has both type 2 and cotype 2, then X is isomorphic to a Hilbert space. This is one of the ﬁrst isomorphic characterizations of the Hilbert space. Some time later, I used in [43] a small modiﬁcation of Kwapie´n’s argument and showed that every bounded linear operator from a subspace X0 of a type 2 space X to a cotype 2 space Y factors through a Hilbert space, and extends to an operator from the whole space X to Y . In particular, every cotype 2 subspace X0 of a type 2 space X is Hilbertian and complemented in X. This was a generalization of a well known result due to Kadets and Pełczy´nski [32], that Hilbertian subspaces of Lp , 2 p < +∞, are complemented. Super-properties appeared in the work of James on super-reﬂexivity (see [25] and [26], and Section 2 below); ultraproduct methods [7] give more insight on super-properties: a property is a super-property when it passes to ultrapowers. Super-reﬂexivity is obviously a super-property, and B-convexity is another super-property; James showed that superreﬂexive spaces are B-convex. Deciding whether B-convex and super-reﬂexive spaces are the same class, as was conjectured by James in [24], remained a difﬁcult problem for some time, and was ﬁnally solved by James, who constructed a non-reﬂexive B-convex space ([27], improved in [28]); before this, Brunel and Sucheston [4,5] had tried to prove that B-convex spaces were reﬂexive, and a part of their attempt introduced an important concept, that of spreading model, which will be used here in Sections 4 and 5. From this point on, there were two clearly distinct settings: super-reﬂexive spaces are those that can be renormed to be uniformly convex (Enﬂo [12]); they have martingale type p (the basis for Pisier’s renorming theorem [54]), and the class of B-convex or type-p spaces, p > 1, is strictly larger. However, contrary to the general case, type and uniform convexity are strongly related for lattices (see Johnson [30], and [40, 1.f]). In a lattice X with non-trivial cotype, it is possible to prove Khintchine-type inequalities. Given (xi )ni=1 in X, these inequalities permit to replace the estimate of a Rademacher average ni=1 εi (t)xi in L2 (X)

Type, cotype and K-convexity

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by an estimate of the square function ( ni=1 |xi |2 )1/2 in X. This kind of “functional calculus” for lattices was developed by Krivine in [35], where he obtained interesting formulations of the Grothendieck theorem, relating operators between lattices and the square function (see also [40, 1.f.14]). Early signs of a tendency to move from abstract Banach spaces to the study of C ∗ algebras and operator spaces also came in this framework. Tomczak-Jaegermann [66] proved that the Schatten classes have the same type or cotype properties than the Lp spaces. Pisier [55] generalized Grothendieck’s theorem to C ∗ -algebras; the result was revisited by Haagerup [21] and was the start for many further exchanges between them. Several other factorization results related to Grothendieck’s theorem were proved in those years, see [60]. The ﬁrst really striking application of cotype as a classiﬁcation tool appears in the results of Figiel, Lindenstrauss and Milman [13]. They showed that Dvoretzky’s theorem takes a very strong form in cotype 2 spaces: if X has cotype 2, there exists a constant c > 0 such that for every integer n, every n-dimensional subspace of X contains a further subspace X0 such that dim X0 = m cn and d(X0 , m 2 ) 2. This result makes use of a certain fundamental formula

k = η(τ )nMr2 /b2 proved in [13, Theorem 2.6], relating the dimension k of (1 + τ )-spherical sections of an n-dimensional normed space to some integral invariant Mr . This formula appears already – with a different normalization – as Eq. (14) in [46]. It gives the spectacular consequence above when using cotype 2 in an appropriate way; actually, [13] quantiﬁes the dimension of spherical sections in terms of the cotype q property, for every q 2, and the previous result for cotype 2 is a special case. Another approach to the problem of spherical sections, the notion of volume ratio developed by Szarek and Tomczak-Jaegermann [65], also singles out the special behaviour of cotype 2 spaces. This approach is based on the work of Szarek [64], who introduced volume arguments in a new proof of the results of Kashin [34] about n1 ; of course Szarek need not mention cotype 2 when working with the explicit norm of n1 ! The fact that cotype 2 spaces have a uniformly bounded volume ratio was proved later by Bourgain and Milman [3], and this motivated the introduction of weak cotype 2 by Milman and Pisier ([48], see also Chapter 10 of Pisier’s book [61]). Type is a nice tool for estimating the behaviour of the entropy of a convex hull; a simple observation of mine, written in [57], was used in entropy problems by Carl [6]. This observation states that in a Banach space X with type p > 1, every point x from the convex hull of a subset A of the unit ball BX can be approximated by a convex combination of n points of A, with an error of order n−1/q (with q conjugate to p). Lemma 9 below is in the spirit of this result. Type and cotype have some simple stability properties; for example, the dual of a type p space has cotype q for the conjugate exponent, but the converse is false as shown by the pair (1 , ∞ ). The two young and ignorant authors of [45] left open a nice intriguing conjecture: is it possible to dualize cotype when we have some non-trivial type? It is clear that what is needed is the boundedness of the Rademacher projection on L2 (X). Spaces such that the Rademacher projection is bounded were called K-convex in [45] (was it because K was

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the ﬁrst available letter after J for J -convex, a notion due to James and named by Brunel and Sucheston [5], or to acknowledge the importance of Kwapie´n’s work on Rademacher averages)? It was conjectured in [45] that every space with type r > 1 should be K-convex, which would imply that the dual X∗ of a space X with cotype q and some non-trivial type should be of type p, with 1/p + 1/q = 1. Six years later, Pisier proved what I consider the most beautiful result in this area, making use of Kato’s theorem on holomorphic semigroups (see [58] and Section 6 of this article): every B-convex space is K-convex. Although very beautiful, the preceding theorem is not the one that has been most useful for local theory. The most useful is another result obtained earlier by Pisier [56], on the way to the general theorem above. This result asserts that the K-convexity constant of X is bounded by C(1 + ln dX ), where dX is the distance from X to the Hilbert space of the appropriate dimension (see Theorem 13 below). In particular, the K-convexity constant is bounded by C(1 + ln n) for any n-dimensional normed space. The quantitative ﬁnitedimensional K-convexity, together with the notion of -norm, leads to a powerful tool for geometric estimates (Theorem 3.11 in [61]; this theorem appeared ﬁrst in [14]). These results play an important role in the QS-theorem of Milman ([47], see also [61]).

2. Super-properties Several of the properties P that are deﬁned for a Banach space X are expressed in the following way: suppose that a number NP (E) is associated to every ﬁnite-dimensional normed space E, in such a way that NP (F ) tends to NP (E) when the Banach–Mazur distance d(F, E) between F and E tends to 1; the most common such dependence is when NP (F ) d(F, E)NP (E). We then say that the Banach space X satisﬁes property P when NP (X) = supE NP (E) < +∞, where the supremum is extended to all ﬁnite-dimensional subspaces E of X. Clearly, the fact that such a property P holds for X only depends upon the family F (X) of all ﬁnite-dimensional normed spaces F such that for every ε > 0, there exists E ⊂ X for which d(F, E) < 1 + ε. After James [25], we say that Y is ﬁnitely representable in X when F (Y ) ⊂ F (X); for instance, Lp is ﬁnitely representable in p , and it is known that X∗∗ is ﬁnitely representable in X for every Banach space X (local reﬂexivity). A property P of Banach spaces is called super-property if we know that whenever a Banach space X has P , then every Banach space Y ﬁnitely representable in X has P . Clearly, every property P expressed by NP (X) < +∞ as above is a super-property. Super-properties were deﬁned by James in [25]. Type and cotype are such properties. Let us recall a few deﬁnitions and facts that are developed in [31]. Let (εi )+∞ i=1 denote the sequence of Rademacher functions on [0, 1], or any independent sequence of centered Bernoulli random variables. Let p ∈ [1, +∞). We say that X has type p when there exists a constant T such that 2 1/2 n 1/p n 1 p εi (t)xi dt T xi , 0 i=1

i=1

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1307

for every n 1 and every sequence (xi )ni=1 ⊂ X; we denote by Tp (X) the smallest constant T with this property; obviously, every normed space X has type 1 with T1 (X) = 1. On the other hand, it follows from Khintchine’s inequalities that no non-zero normed space has type p when p > 2. Saying that X has type p is obviously equivalent to the fact that the family of ﬁnite-dimensional subspaces E of X satisﬁes supE Tp (E) < +∞, thus having type p is a super-property. We say that X has cotype q when there exists a constant Cq (X) such that n

1/q xi

q

i=1

2 1/2 n 1 Cq (X) εi (t)xi dt 0 i=1

for every n 1 and every sequence (xi )ni=1 ⊂ X; again, this is equivalent to the fact that supE Cq (E) < +∞, and cotype is therefore another super-property. In both deﬁnitions of type and cotype, the choice of the L2 norm for the Rademacher averages is irrelevant (except for the exact value of the constants); this follows from Kahane’s inequalities (see [33, Chapter II, Th. 4]), which state that for every q < ∞, there exists a constant Kq such that q 1/q n 1 n 1 εi (t)xi dt Kq εi (t)xi dt 0 0 i=1

i=1

for every n 1 and every family (xi )ni=1 of vectors in a Banach space. It is easy to show that when X has type or cotype, then the same holds for the space L2 (X) of X-valued square integrable functions. This fact is used below in Sections 5 and 6.

3. Ultrapowers and some operator lemmas In the next section about Krivine’s theorem, we use a classical fact for operators on a complex Banach space X: if λ is a boundary point of the spectrum Sp(T ) of T ∈ L(X), then λ is an approximate eigenvalue for T , which means that there exists a sequence (xn ) ⊂ X of norm one vectors such that limn (T (xn ) − λxn ) = 0. We shall need a slightly less classical fact about commuting operators, which is very easy to obtain using the notion of ultrapower (Lemma 1 below). We shall ﬁrst recall a few facts about ultrapower techniques. These techniques became popular in Banach space theory after the paper by DacunhaCastelle and Krivine [7]; approximately at the same time, similar objects were introduced for C ∗ -algebras [29]. The limit spaces used by James [25] in his study of super-reﬂexivity, the spreading models of Brunel and Sucheston [4], belong to the same family of tools which make possible to construct an abstract space from different pieces taken at different places. Suppose that U is a non-trivial ultraﬁlter on N. If X is a Banach space, we consider in X∞ := ∞ (X) the closed subspace KU of all sequences y = (yn ) ∈ X∞ such that limn→U yn = 0, and we let XU be the quotient space X∞ /KU . Let πU denote the quotient map from X∞ to XU . If x = (xn ) and ξ = πU (x), then ξ = limn→U xn . We have

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B. Maurey

a canonical isometry iX,U from X to XU that sends x ∈ X to the class of the constant sequence x = (xn ) where xn = x for every n. Using this isometric embedding we shall consider that X ⊂ XU . The crucial fact is here: suppose that η1 , . . . , η ∈ XU are represented by sequences yj = (yj,n )n0 ∈ X∞ , for j = 1, . . . , , and that we have a ﬁnite number of inequality relations bi,j ηj < ci , i = 1, . . . , k, (R) ai < xi + j =1

where ai , ci ∈ R, xi ∈ X, (bi,j ) is a matrix of scalars. Let us say that a property depending upon n ∈ N is true when n is U -large if the set A ⊂ N of those n for which the property holds belongs to U ; then we can say that when n is U -large, we have in X bi,j yj,n < ci , ai < xi +

i = 1, . . . , k.

(Rn )

j =1

This implies that XU is ﬁnitely representable in X (and slightly more: if E is any ﬁnitedimensional subspace of XU , we can ﬁnd a (1 + ε)-isomorphism T from E into X such that T (x) = x for every x ∈ E ∩ X). We see that F belongs to F (X) if and only if F is isometric to a subspace of XU . Every super-property of X passes to XU , for example, type or cotype. Suppose now that T is a bounded linear operator on X. We deﬁne T∞ on X∞ in the obvious way,

T∞ (x) = T (xn ) , whenever x = (xn ) ∈ X∞ . It is clear that KU is stable under T∞ , so that T∞ induces a bounded linear map TU on XU . It is easy to check that T → TU is an isometric homomorphism of unital Banach algebras from L(X) to L(XU ). Using the above principle (R) ⇒ (Rn ), we see that if x = (xn ) ∈ X∞ and if ξ = π(x) ∈ XU , then this vector ξ satisﬁes TU (ξ ) = λξ if and only if limn→U (T (xn ) − λxn ) = 0; in particular, if X is complex, for every boundary point λ of the spectrum of T we can ﬁnd a sequence (xn ) ⊂ X of norm one vectors such that limn (T (xn ) − λxn ) = 0, which shows that the eigenspace ker(TU − λI ) is not trivial. L EMMA 1. Suppose that X is a complex Banach space, and that S, T are commuting bounded linear operators on X. If (xn ) ⊂ X is a sequence of norm one vectors such that T (xn ) − λxn tends to 0, we can ﬁnd μ ∈ C and a norm one vector x ∈ X such that T (x) ∼ λx and S(x) ∼ μx. P ROOF. We know that Xλ = ker(TU − λI ) is not {0}, and SU commutes with TU , therefore Xλ is stable under SU . If μ is a boundary point of the spectrum of the restriction of SU to Xλ , we can ﬁnd a norm one vector ξ in Xλ such that SU (ξ ) ∼ μξ . Bringing back ξ to X –

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1309

using (R) ⇒ (Rn ), with η1 = ξ , η2 = TU (ξ ) and η3 = SU (ξ ) – we obtain for every ε > 0 a norm one vector x ∈ X such that T (x) − λx < ε and S(x) − μx < ε. Let X be a complex Banach space, and let T be an into isomorphism from X into X, with x CT (x) for every x ∈ X. For every integer n 1, we may deﬁne Kn as the smallest constant for which x Kn T n (x) 1/n

for every x ∈ X. It is clear that Km+n Km Kn , so that r = limn Kn exists by a standard lemma. Also, Kn C n and Kn T n (x) Kn T n x yield that 0 < T −1 r C. L EMMA 2. There exists λ ∈ C with |λ| = r and a sequence (xn ) of norm one vectors in X such that limn (T (xn ) − λ−1 xn ) = 0. P ROOF. We introduce an operator S of which r will be the spectral radius; this S acts as a sort of inverse for TU . For every x ∈ X, let N(x) denote the supremum of k such that x belongs to the range of T k (this value N(x) may be +∞). Let Z0 be the subspace of XU consisting of all ξ that have a representative x = (xn ) such that limU N(xn ) = +∞. It is obvious that Z0 is stable under TU ; let Z be the closure in XU of Z0 , and let TZ denote the restriction of TU to Z. When ξ ∈ Z0 , we see that ξ = TZ (η) for some (unique) η: indeed, if x = (xn ) belongs to the class of ξ and limU N(xn ) = +∞, we have that N(xn ) 1 when n is U -large, which means that A = {n: N(xn ) 1} ∈ U ; hence for every n ∈ A we have xn = T (yn ) for some yn ∈ X; if we let yn = 0 for n ∈ / A, then y = (yn ) satisﬁes limn→U N(yn ) = +∞ (because N(yn ) N(xn ) − 1); if η = π(y), then η belongs to Z and TZ (η) = ξ ; clearly η Cξ . This shows that TZ is invertible in L(Z). Let S = TZ−1 . It is quite clear that S n Kn , so that the spectral radius ρ(S) = limk S k 1/ k of S satisﬁes ρ(S) r; we shall see that r = ρ(S). Let us ﬁx k 2 and ε > 0. For n large, we know that Knk > (r − ε)nk , thus we can ﬁnd a vector xn ∈ X such that xn √> (r − ε)nk T nk (xn ). Let h be a large integer, but small compared to n, say h − 1 < n h, for example. If we had jk

T (xn ) (r − 2ε)k T j k+k (xn ) = (r − 2ε)k T k T j k (xn ) for every j = h, . . . , n − 1, it would follow that (r − ε)nk T nk (xn ) < xn C hk (r − 2ε)nk−hk T nk (xn ), √ which is impossible when n is large. For every n n0 , and for some j such that n j < n, we may thus ﬁnd a vector yn =√αT j k (xn ) such that 1 = yn > (r − 2ε)k T k (yn ), and this vector satisﬁes N(yn ) k n. If y = (yn ) and η = π(y) we get η ∈ Z and S k (η) > (r − 2ε)k η. It follows that the spectral radius of S is larger than r − 2ε, hence equal to r.

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Let λ ∈ Sp(S) be such that |λ| = r. It follows from the “boundary of the spectrum lemma” that we can ﬁnd a norm one vector ξ ∈ Z such that S(ξ ) ∼ λξ , or TZ (ξ ) ∼ λ−1 ξ ; bringing back ξ to X in the usual way gives a norm one vector x for which T (x) ∼ λ−1 x, as was to be proved.

4. Krivine’s theorem See [49, Chapter 12] or [2, Chapter 12] for a more precise presentation of the results of this section. I prefer here to tell a pleasant story, rather than being too technical. Roughly speaking, Krivine’s theorem says that every Banach space X contains (1 + ε)-isomorphs of np , for some p ∈ [1, +∞] and every n 1, or in other words it says that some p (or c0 , when p = +∞) is ﬁnitely representable in X. More precise statements tell us that, given a basic sequence in X, or simply a sequence (xn ) with no Cauchy subsequence, there exists p ∈ [1, +∞] such that for every n 1 and ε > 0, we can ﬁnd blocks of the given sequence that are (1 + ε)-equivalent to the unit vector basis of np . It is sometimes useful to be more speciﬁc, and to predict what values of p can be realized, starting from some norm invariants of the sequence (xn ). This will be the case in the next section about type, cotype and the MP+K theorem. The proofs of Krivine’s theorem are usually divided into two steps: the ﬁrst step replaces the given sequence by one that has some minimal regularity; this step uses only subsequences, or just differences of two vectors from the original sequence (as opposed to the second step, that requires clever long blockings). The argument is due to Brunel and Sucheston: given a sequence with no Cauchy subsequence, and using Ramsey’s theorem, we may ﬁnd a subsequence which is asymptotically invariant under spreading, see [4], and also [17]; alternatively, this can be achieved by general abstract arguments involving iterated ultrapowers, usual in model theory where a somewhat parent notion of indiscernible sequence is deﬁned. Given a Banach space X and a space Y of scalar sequences, we say that Y is a spreading model for X if there exists a normalized sequence (xn ) ⊂ X, with no Cauchy subsequence, such that k k aj ej = lim aj xnj j =1

Y

j =1

X

for every k 1 and all scalars (aj )kj =1 ; the limit is taken when n1 → ∞ and n1 < n2 < · · · < nk , and (ej ) denotes the standard unit vector basis for the space of scalar sequences. The second part of this ﬁrst step, also due to Brunel and Sucheston, is to observe that the differences (e2j +1 − e2j ) are suppression-unconditional in Y (see below for a deﬁnition); further, the differences are bounded away from zero because the sequence (xn ) had no Cauchy subsequence; this implies that we can ﬁnd 2-unconditional ﬁnite sequences (zi )ki=1 in X, with k as large as we wish, whose vectors zi are differences zi = xn2i − xn2i−1 of two suitable vectors from the given sequence (xn ). The spreading model Y is ﬁnitely representable in X, in a special way: any ﬁnite sequence (yk ) of blocks of the basis in Y can be sent to blocks from the sequence (xn ) in X. We shall therefore present the rest of

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the proof of Krivine’s theorem assuming that we start from this situation, replacing the original space X by a spreading model X , which is (block) ﬁnitely representable in X and has a nice basis. The real thing is to prove Krivine’s theorem for X . Let X be a Banach space with a basis (en )n0 ; we say that this basis is a suppressionunconditional basis when for every x ∈ X, the norm does not increase if we replace one of the non-zero coordinates of x by 0; this yields that the basis is unconditional, with unconditionality constant 2 (in the real case). Let X be a Banach space with a suppressionunconditional basis (en )n0 ; we say that the norm is invariant under spreading if for every integer k 0 and all n0 < n1 < · · · < nk , k k aj enj = a j ej j =0

j =0

for all scalars (aj ). Let x = kj =0 aj ej be a vector with ﬁnite support in X; we say that y is a copy of x if y = kj =0 aj enj for some n0 < n1 < · · · < nk . If x = aj ej and y = bj ej , we write x < y when all non-zero coordinates of x appear before those of y, that is max{j : aj = 0} < min{j : bj = 0}. We say that x1 , . . . , xn are successive vectors if x1 < x2 < · · · < xn . After the preliminary work has been done, the heart of Krivine’s result is the following Theorem 3. The arguments of Brunel and Sucheston imply that for every Banach space X, we can ﬁnd a space X0 with a suppression-unconditional basis, invariant under spreading, such that X0 is ﬁnitely representable in X; if X0 contains kp , then X will also. We shall therefore assume that X is a Banach space with a suppression-unconditional basis (en )n0 , and a norm invariant under spreading. For every integer n 1, let Rn be the smallest constant and Sn be the largest constant such that for every x ∈ X, we have n Sn x xi Rn x i=1

whenever x1 < x2 < · · · < xn are successive copies of x. T HEOREM 3. Let X be a Banach space with a suppression-unconditional basis (en )n0 , and a norm invariant under spreading; suppose that p 1 is deﬁned by the equation (a) 21/p = lim supn (R2n )1/n or (b) 21/p = lim infn (S2n )1/n . For every k 1 and ε > 0 it is possible to ﬁnd k successive blocks x1 < · · · < xk in X that are (1 + ε)-equivalent to the unit vector basis of kp , and that are copies of some norm one vector x ∈ X.

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P ROOF. Let I be the set of rational numbers r such that 0 r < 1, let (fr )r∈I be the standard unit vector basis for R(I ) , and let us deﬁne a norm on the linear span Y0 of (fr )r∈I as follows: if r0 < r1 < · · · < rk , let k k aj frj = a j ej j =0

Y

j =0

X

for all scalar coefﬁcients (aj ). If Y0 is real, we complexify it in any reasonable way, for example, x + iy = supsin(θ )x + cos(θ )y , θ

which preserves invariance under spreading and unconditionality. Let Y be the completion of Y0 ; it is clear that (fr ) is a suppression-unconditional basis for Y , invariant under spreading in the new context. We say that y ∈ Y is a copy of y = r∈I ar fr if y = r∈I ar fφ(r) for some increasing map φ from I into itself. What we mean by successive copies of a given vector in Y is clear. It is also clear that Y is ﬁnitely representable in X, and a ﬁnitedimensional subspace of Y generated by successive copies of some vector in Y can be approximated by a subspace of X, generated by successive copies of some vector in X. We can now relate the behaviour of sums of copies of vectors in X to the properties of some linear operators deﬁned on this space Y . Indeed, we may deﬁne a doubling operator D on Y by the formula ∀y ∈ Y,

D(y) =

0r<1/2

y(2r)fr +

y(2r − 1)fr ,

1/2r<1

or D(y)(r) = y(2r mod 1), considering y as a function I → C. For every y ∈ Y0 , the vector D(y) is the sum of two copies y1 < y2 of y, hence y D(y) 2y. It is clear that the constant R2n for the initial space X is equal to the norm of D n , therefore in case (a), we see that 21/p is the spectral radius of D. We may thus ﬁnd λ ∈ C with |λ| = 21/p and a norm one vector z ∈ Y0 such that D(z) ∼ λz. In case (b), the constant S2n appears to be the reciprocal of the constant Kn associated to the into isomorphism D (see before Lemma 2), 1/n therefore if 21/p = limn S2n , we know by Lemma 2 that we can again ﬁnd λ ∈ C with |λ| = 21/p and a norm one vector z ∈ Y0 such that D(z) ∼ λz. Using unconditionality, we get D(|z|) ∼ |λ||z|. In both cases (a) and (b) we found a norm one vector y = α|z| ∈ Y0 (with 1/2 α 2) such that D(y) ∼ 21/p y. Reproducing y in X gives a norm one vector x ∈ X such that, when x1 < x2 are copies of x, then x1 + x2 is very close to some copy x of 21/p x. I like to call such a vector x a Krivine vector. Suppose that x1 < x2 < · · · < xk are copies of this vector x. If n 1 is given and if D(y) − 21/p y has norm smaller than some εn > 0, we deduce that k p k p aj xj ∼ aj , j =1

j =1

(K)

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provided all coefﬁcients are of the form aj = 2−kj /p , for some integer kj such that 0 p kj n, and j aj = 1 (if K = max kj , replace each aj xj by 2K−kj copies of 2−K/p x; this gives 2K copies of 2−K/p x, which we may group two by two again, obtaining after K steps a single copy of the vector x). This is not quite enough, and we also introduce an operator T on Y which reproduces three times every vector y ∈ Y , T (y) =

y(3r)fr +

0r<1/3

y(3r − 1)fr +

1/3r<2/3

y(3r − 2)fr .

2/3r<1

It is clear that DT = T D is the operator that replaces every vector x by six copies of x; the commutation property and Lemma 1 enable us to ﬁnd a norm one vector z such that D(z) ∼ 21/p z and T (z) ∼ μz; then T (|z|) ∼ |μ||z|, so that we may assume that z and μ are real and 0. Some simple lattice arguments (involving comparisons of the norms of sums of respectively 2h , 3i and 2j copies of z when 2h < 3i < 2j ) show that necessarily μ = 31/p . If D(z) − 21/p z and T (z) − 31/p z are small enough, and if z1 < z2 < · · · < zk are copies of this vector z, we may try to extend relation (K) to coefﬁcients (aj ) such that aj = 2j 3mj for some j , mj ∈ Z; since these values are dense in [0, ∞), we are in a good position. However, dealing with the error terms is painful, and we may instead pass to the ultrapower YU , which is still a lattice, with a linear ordering deﬁned in this way: we say that ξ < η if ξ and η have representatives (xn ) and (yn ) with xn < yn for every n, and we say that η is a copy of ξ if ξ and η have representatives (xn ) and (yn ) such that yn is a copy of xn for every n; in YU we can ﬁnd a norm one vector η such that DU (η) = 21/p η and TU (η) = 31/p η; to get this, we take for η the class of a normalized sequence (zn ) in Y with D(zn ) − 21/p zn → 0 and T (zn ) − 31/p zn → 0. In this framework where we have equalities, this vector η and when the it is easy to prove that when η1 , . . . , ηk are successive copies of p coefﬁcients (aj ) satisfy aj = 2j 3mj , with j , mj ∈ Z, then kj =1 aj ηj p = kj =1 aj ; next, we extend this by density to all non-negative scalars. Going back to X, and using the special form of the vectors η1 , . . . , ηk , we can ﬁnd successive copies x1 , . . . , xk of some norm one vector x in X such that k p k k p p −p/2 aj aj xj (1 + ε)p/2 aj , (K ) (1 + ε) j =1

j =1

j =1

for all non-negative scalars (aj ). Everything would be ﬁne if the basis in X was 1-unconditional, but it is not so: what we get so far is a sequence x1 , . . . , xk which is 2(1 + ε)-equivalent to the kp -basis in the real case, and 4(1 + ε) in the complex case, for every k 2: if v = aj xj , the p -norm of the coefﬁcients is dominated by (v + p + v − p )1/p 21/p v, using ﬁrst (K ) then suppression unconditionality; in the other direction use v 21−1/p (v + p + v − p )1/p . Suppose p < ∞ for simplicity; if k = m2 and if we form new blocks y1 , . . . , ym in m −1/p j X of the form yi = m j =1 (−1) xm(i−1)+j , then (y1 , . . . , ym ) is still a sequence of successive copies of some y ∈ X, hence invariant under spreading, 5-equivalent to the m p

1314

B. Maurey

basis (say), but the unconditional constant is improved to something arbitrarily close to 1 as m grows (in the complex case, a similar trick using a primitive root of unity does the (m) ])m , by setting required job). We build a limit space X1 from the sequence ([y1(m) , . . . , ym n a j ej j =1

X1

n∧m (m) = lim aj yj m→U j =1

for every n 1 and all scalars (aj ). This space X1 is ﬁnitely representable in X, with a 1-unconditional basis, invariant under spreading, and 5-equivalent to the p basis. In X1 we have clearly 21/p = lim supn (R2n )1/n . Applying the above construction to X1 gives new blocks x1 , . . . , xk that satisfy (K ) in X1 : this ﬁnishes the proof, since the basis in X1 is 1-unconditional. The proof above is due to Lemberg [38], who was Krivine’s PhD student in the years ’80. The fundamental facts are still the same as in the original paper [36], but the details in [36] are harder to follow. Combining the arguments of Brunel and Sucheston and the preceding theorem, we obtain one of the usual forms of Krivine’s theorem. C OROLLARY 4. Suppose that X is a real or complex Banach space, and (xn ) a bounded sequence in X with no Cauchy subsequence. For some p ∈ [1, +∞], for every k 1 and ε > 0 it is possible to ﬁnd k successive blocks of the given sequence that are (1 + ε)equivalent to the unit vector basis of kp . Our next corollary is expressed in a slightly unnatural way, but suitable for the next section. C OROLLARY 5. Suppose r, s 1 are given. If for some κ > 0 and for every n 2, a Banach space X contains a normalized suppression-unconditional sequence y(n) = (y1(n) , . . . , yn(n) ) such that (n) 1/r yi κ|C| i∈C

for every subset C ⊂ {1, . . . , n}, or such that (n) 1/s y i κ|C| i∈C

for every subset C ⊂ {1, . . . , n}, then for some p r (or p s) and for every k 1, ε > 0, it is possible when n N(k, ε) to form k successive blocks of the given sequence y(n) that are (1 + ε)-equivalent to the unit vector basis of kp . P ROOF. We construct as we did before a limit space X from the long sequences as fol(n) lows. Using Brunel–Sucheston principle, we may select from our long sequences (yi )

Type, cotype and K-convexity (n)

1315

(n)

some (ﬁnite) subsequences z1 , . . . , zkn that are almost indiscernible, and have a length kn tending to ∞ with n; then we deﬁne a norm on c00 (the space of ﬁnitely supported scalar sequences) by m ci ei

X

i=1

m∧k n (n) = lim ci zi , n→U i=1

where (ei )i0 denotes the unit vector basis of c00 . Notice that when n is U -large, the length kn exceeds m; this yields that (ei ) is normalized in X . We obtain a space X with a normalized suppression-unconditional basis and a norm invariant under spreading. In the ﬁrst case, we get for every n 1 1/r

κn

n ei i=1

Rn e1 X = Rn ,

X

and similarly in the second case we obtain that Sn κn1/s . We know from Theorem 3 that we may get kp in X , with p such that 21/p = limn (R2n )1/n or 21/p = limn (S2n )1/n , thus p r in the ﬁrst case and p s in the second.

5. Type, cotype and np s. The MP+K theorem Let X be a Banach space. We denote by pX the supremum of all p such that X has type p, and by qX we denote the inﬁmum of all q such that X has cotype q. It is clear using Khintchine’s inequality that pX 2 qX , already when X = R. T HEOREM 6. Let X be an inﬁnite-dimensional Banach space; for every integer k 1 and ε > 0, the space X contains (1 + ε)-isomorphs of kpX and of kqX . For the type case and 1 < p < 2, there exists a quantitative estimate due to Pisier [59], see also [49, Theorem 13.12]. The dimension k of a good isomorph in X of kp is given there as a function of the stable type p constant STp (X) of the normed space X. P ROOF. If pX = 2 we may use Dvoretzky’s theorem [10]. Assume pX < 2 and choose r such that pX < r < 2. For each n 1, let ϕ(n) denote the smallest constant such that r 1 n n r εi (t)xi dt ϕ(n) xi r 0 i=1

i=1

1316

B. Maurey

for every family x1 , . . . , xn of n vectors in X. It is clear that ϕ is non-decreasing, and tends to X does not have type r. Suppose that x1 , . . . , xn are chosen in X so that n+∞ since r = 1 and x i i=1 r 1 n 1999 ϕ(n)r . εi (t)xi dt > 2000 0 i=1

We shall use an exhaustion argument inspired by Nikishin’s paper [51]. Let (Bα )α∈I be a maximal family of disjoint subsets of {1, . . . , n} such that r 1 1 ε (t)x xi r . i i dt < 2000 0 i∈Bα

i∈Bα

If B denotes the union of these sets Bα , and m denotes the cardinality of I (notice that m < n because |Bα | > 1), we get r 1 ε (t)x dt = i i 0

i∈B

r ε (s) ε (t)x α i i ds dt α∈I

i∈Bα

r r 1 r dt ϕ(m) ϕ(m) ε (t)x xi r i i 2000 0 α∈I

n ϕ(n)r

2000

i∈Bα

xi r =

i=1

α∈I i∈Bα

ϕ(n)r . 2000

Let A denote the complement of B and for every j 0 let Aj = k ∈ A: 2−j −1 < xk 2−j . Observe that xk 1 for every k because ri=1 xi r = 1, so that the sets (Aj )j 0 cover the set A. Let N = maxj |Aj | denote the maximal cardinality of the sets (Aj )j 0 . Then r 1/r +∞ 1 dt ε (t)x i i 0

j =0

i∈A

N

+∞

1 0

r 1/r εi (t)xi dt

i∈Aj

2−j = 2N.

j =0

We obtain

1999 2000

1/r

r 1/r n 1 ϕ(n) ϕ(n) < εi (t)xi dt + 2N 1/r 2000 0 i=1

Type, cotype and K-convexity

1317

which shows that N is big when ϕ(n) is big. Let j0 be such that |Aj0 | = N . By maximality of B we obtain for every non-empty subset C of Aj0 r 1 2−(j0 +1)r r dt 1 |C|. ε (t)x x i i i 2000 2000 0 i∈C

i∈C

Replacing the vectors (xi )i∈Aj0 by normalized vectors (yi ), we obtain a normalized sequence (y1 , y2 , . . . , ym ), as long as we wish, such that r 1/r 1 ε (t)y κ|C|1/r i i dt 0

i∈C

for every subset C of {1, . . . , m} (with κ = 12 2000−1/r ). This inequality remains true if we replace the Lr (X) norm by the norm of L1 (X) and κ by some κ > 0 (use Kahane’s inequalities). For every n 1, we may thus ﬁnd an unconditional normalized sequence (n) in L1 (X), of the form (εj yj )nj=1 , with the above property, and since r < 2 it implies that for some c = c(r, κ ) > 0, we have nj=1 aj εj yj(n) L1 (X) c( nj=1 |aj |2 )1/2 for all scalars. From Corollary 5 follows that for every integer m, we can, when n is large (n) enough, get blocks z1 , . . . , zm ∈ L1 (X) of (εj yj )nj=1 that are (1 +ε)-equivalent to the unit m vector basis of p for some p r, and the 2 -norm of the coefﬁcients in each block zi is bounded by c(r, κ )−1 . By Kahane’s inequalities again, all Ls (X) norms are equivalent on (n) the span of (εj yj )nj=1 , hence the sequence (z1 , . . . , zm ) considered in L2 (X) is uniformly equivalent to the unit vector basis of m p ; since L2 (X) has type s whenever X has type s, we have for every s < pX and for some constants K, Ks 1 m K −1 m1/p εi (t)zi 0 i=1

dt Ks m1/s L2 (X)

for every m 1. This yields that s p, for every s < pX , hence pX p. Starting with a long enough sequence (zi )m i=1 and blocking again in the p -sense we may ﬁnd three blocks b1 , b2 , b3 ∈ L1 (X) of some sequence (εj yj ), supported on three disjoint intervals J1 , J2 , J3 and such that, letting ω = (εj ) and bi (ω) =

aj εj yj ,

i = 1, 2, 3,

j ∈Ji

then the three functions b1 , b2 , b3 are (1 + ε)-equivalent to the unit vector basis of 3p in the norm of L1 (X), and the coefﬁcients satisfy j ∈Ji |aj |2 < τ 2 /12 for i = 1, 2, 3 and a small τ > 0 (use p < 2). For every ﬁxed triple (c1 , c2 , c3 ) of scalars, this implies by Azuma’s inequality (see [49, 7.4]) a strong concentration for the set of ω such that

c1 b1 (ω) + c2 b2 (ω) + c3 b3 (ω) ∼ |c1 |p + |c2 |p + |c3 |p 1/p ,

1318

B. Maurey

and allows us to select a choice of ω = (εj ) that works for all (ci )3i=1 , by a standard δ-net argument on the unit sphere of 3p ; this shows that for most of the choices ω of signs, the vectors b1 (ω), b2 (ω), b3 (ω) in X form a nice copy of the unit vector basis of 3p . We may choose r close enough to pX so that 3pX is almost isometric to 3p , since pX p r, and this ends the proof in this case k = 3. The reader will easily pass from 3 to an arbitrary integer k. Let us be more speciﬁc about the use of Azuma’s inequality. On the space Ω = {−1, 1}n , we deﬁne for every c in the unit sphere of 3p the function 3 fc (ω) = fc (ε1 , . . . , εn ) = ci aj εj yj i=1

j ∈Ji

and we consider the ﬁnite martingale Mj (ε1 , . . . , εj ) =

fc (ε1 , . . . , εn ) dεj +1 · · · dεn ,

for j = 0, . . . , n. The differences (dj )nj=0 of this martingale satisfy |dj +1 | = |Mj +1 − Mj | |aj +1 |, hence S 2 = nj=1 |dj |2 τ 2 /4. Azuma’s inequality gives P

ω ∈ Ω: fc (ω) − M0 t 2 exp −t 2 / 4S 2 2 exp −t 2 /τ 2

for every t > 0, where M0 = M0 (c) is equal to the norm of c1 b1 + c2 b2 + c3 b3 in L1 (X), which is (1 + ε)-equivalent to the 3p -norm of c, namely 1. If Λ is a δ-net on the unit sphere of 3p and if τ was so small that 2|Λ| < exp(δ 2 /τ 2 ), we may ﬁnd ω such that c1 b1 (ω) + c2 b2 (ω) + c3 b3 (ω) − M0 (c) δ for every c ∈ Λ, from which the result follows. Let us pass to the cotype case. If qX = 2 we may use Dvoretzky’s theorem. Assume qX > 2. We choose s such that qX > s > 2. Let ψ(n) denote the smallest constant such that s 1 n n xi s ψ(n)s εi (t)xi dt 0 i=1

i=1

for every family x1 , . . . , xn of n vectors in X. It is clear that ψ is non-decreasing, and tends to +∞ since X does not have cotype s. Suppose that x1 , . . . , xn are chosen in X so that n s i=1 xi = 1 and 1999 ψ(n)s 1> 2000

s 1 n εi (t)xi dt. 0 i=1

Type, cotype and K-convexity

1319

Let (Bα )α∈I be a maximal family of mutually disjoint non-empty subsets of {1, . . . , n} such that s 1 1 s xi εi (t)xi dt. 2000 0 i∈Bα

i∈Bα

If B denotes the union of these sets Bα , and m < n denotes the cardinality of the index set I , we get s 1 1 xi s = xi s ε (t)x i i dt 2000 0 i∈B

α∈I i∈Bα

α∈I

i∈Bα

s ψ(m)s ε (s) ε (t)x α i i ds dt 2000 α∈I

i∈Bα

s n ψ(n)s 1 εi (t)xi dt. 2000 0 i=1

Let A denote the complement of B and for every j 0 let Aj = k ∈ A: 2−j −1 < xk 2−j . We have i∈A

1998 ψ(n)s xi > 2000 s

s 1 n εi (t)xi dt. 0 i=1

Let j1 be the smallest j 0 such that Aj is not empty. If N = |Aj0 | is the largest cardinality of the sets Aj , then N

+∞

2−j s

j =j1

1998 ψ(n)s 2−j1 s−s 2000

which shows that N is large when ψ(n) is large. By maximality of B, s 1 1 s xi > εi (t)xi dt 2000 0 i∈C

i∈C

for every non-empty subset C ⊂ Aj0 . We change the (xi )i∈Aj0 to normalized vectors, and go to a limit space X , ﬁnitely representable in X and containing a normalized sequence (yi )i0 such that for some κ0 , κ0 |C|

1/s

s 1/s 1 εi (t)yi dt 0

i∈C

1320

B. Maurey

for every ﬁnite subset C. But this sequence (yi )i0 can’t have any Cauchy subsequence, or else the above property would be true with yi ∼ y, in other words, true in a one-dimensional setting; in this case, Khintchine’s inequality tells us that the integral is larger than |C|1/2 > κ0 |C|1/s , which is impossible when |C| is large. By Brunel and Sucheston, we can pass to differences (ym − yn ) in order to get a suppression-unconditional sequence invariant under spreading (with a poor normalization). We have s 1/s 1 ε (t)(y − y ) 2κ0 |C|1/s i 2i+1 2i dt 0

i∈C

for every ﬁnite subset C, but we may now get rid of the signs (εi (t)) since the sequence of differences is 2-unconditional. We obtain therefore in X a normalized suppressionunconditional sequence (xi ) such that

1/s x i κ |C| i∈C

for every ﬁnite subset C. We end by applying the second case of Corollary 5.

6. K-convexity and Pisier’s theorem When X is a type p space, then the dual X∗ has cotype q for the conjugate exponent (1/p + 1/q = 1); this is very easy: if (xi∗ )ni=1 is given in X∗ , we can ﬁnd (xi )ni=1 ⊂ X such that ni=1 xi∗ (xi ) > ( ni=1 xi∗ q )1/q − ε and ni=1 xi p = 1; then, by orthogonality of the functions (εi )

n ∗ q x i i=1

1/q −ε <

n

xi∗ (xi ) =

i=1

1 n 0

i=1

εi (t)xi∗

n

εj (t)xj dt

j =1

2 1/2 n 2 1/2 n 1 1 εj (t)xj dt εi (t)xi∗ dt 0 0 j =1

i=1

2 1/2 n 1 ∗ Tp (X) εi (t)xi dt , 0 i=1

therefore Cq (X∗ ) Tp (X). Obviously the converse is false since 1 , dual of c0 , has cotype 2, while c0 has no non-trivial type. However, this does not happen when X∗ has cotype q and non-trivial type: then, X has type p. This fact was conjectured in [45] (although the authors had little evidence that supported this conjecture at the time), and proved by Pisier six years later [58]. Using local reﬂexivity, and since type and cotype are superproperties, the preceding claim is equivalent to saying that when a Banach space Y has

Type, cotype and K-convexity

1321

non-trivial type and cotype q, then the dual Y ∗ has type p with 1/p + 1/q = 1. This will follow from the easy Lemma 7 below and from the main result of this section, Theorem 12. Let us consider the group G = {1, −1} and let μ denote the invariant probability measure m, denote the i-th on Gm , that gives measure 2−m to every atom. On Gm , let εi , i = 1, . . . ,* coordinate function, εi (g1 , . . . , gm ) = gi . If α ⊂ {1, . . . , m} let wα = i∈α εi ; using the standard convention, we get the constant function 1 on Gm when α = ∅. Let |α| denote the cardinality of the set α. This family of functions (wα ) is the Walsh system; it is the family of characters of the Abelian group Gm . Every function f from Gm to a Banach space X can be expressed as ∀ω ∈ Gm ,

f (ω) =

wα (ω)xα ,

α

for some family (xα ) ⊂ X. Given a function f = α wα xα , the part of the expansion corresponding to sets α with |α| = 1 is the Rademacher projection RX (f ) = |α|=1 wα xα of the function f (we have wα = εi when α = {i}). L EMMA 7. If the Rademacher projection RX is bounded on L2 (Gm , μ, X) by some constant K, uniformly in m 1, then the cotype q property of X dualizes to the type p property of X∗ , and

Tp X∗ KCq (X) (1/p + 1/q = 1). m Suppose that f ∈ L2 (G P ROOF. , X); the Rademacher projection of f is of the form m RX f = i=1 εi xi , where xi = εi (ω)f (ω) dμ(ω). The cotype q property and the boundm m edness of RX imply that the map f → (xi )m i=1 is bounded from L2 (G , X) to q (X). It m ∗ m ∗ follows that the adjoint map is bounded from p (X ) to L2 (G , X ), and this adjoint map m ∗ is the map that sends (xi∗ )m i=1 εi xi . We get therefore i=1 to

2 1/2 m 1/p m 1 p ∗ ∗ x εi (t)xi dt RX Cq (X) . i 0 i=1

i=1

K such that for every D EFINITION 8. We say that X is K-convex if there exists a constant m 1 and every function f ∈ L2 (Gm , X), expressed as f = α wα xα , we have RX f L2

= wα xα |α|=1

Kf L2 ,

L2

which means that RX L(L2 (X)) K. The smallest possible constant K is the K-convexity constant of X. It is equal to the supremum of RX , when the number m of Rademacher functions tends to inﬁnity. When this supremum is ﬁnite, we may directly deﬁne RX on the inﬁnite product GN , and the K-convexity constant is the norm of RX on L2 (GN , X). It is clear that K-convexity

1322

B. Maurey

is a super-property, and it passes to the dual X∗ with the same constant. It follows from Kahane’s inequality that the projection is also bounded in Lq (X) for 2 q < +∞, and using duality we see that RX is then bounded in Lp (X) for all p such that 1 < p < +∞. It follows from Lemma 7 that the K-convexity constant of L1 (Gm ) tends to inﬁnity with m (because L1 has cotype 2 while its dual L∞ does not have type 2), but it is instructive to give a concrete estimate. Let gˆ = (ˆε1 , . . . , εˆ m ) ∈ Gm be ﬁxed; the function fgˆ , equal to 2m at gˆ and to 0 elsewhere, has norm one in L1 (Gm ), and its expansion is fgˆ =

wα (g)w ˆ α.

α

It follows that the function f from Gm to L1 (Gm ) deﬁned by

f g, g = wα (g)wα g α m has norm one in L2 (Gm √, L1 (G )), but its Rademacher projection (Rf )(g) = m ε (g)εj has norm m/2 by Khintchine’s inequality. Observe that we get KX √j =1 j m c log dim √ X, with X = L1 (G ). It is known that for any Banach lattice X we have KX C 1 + log dim X (see [56]), and the preceding simple example shows that this result is precise for lattices. For general Banach spaces, see Theorem 13 below. Let us describe the semi-group approach: on the multiplicative group G = {1, −1} we (1) consider for −1 c 1 the probability measure μc deﬁned by

μ(1) c =

1+c 1−c δ1 + δ−1 , 2 2 (1)

where δg denotes the unit mass at g ∈ G. Using δ−1 ∗ ε1 = −ε1 we get that μc ∗ ε1 = (1) (1) (m) m cε1 . Also μ(1) b ∗ μc = μbc . On G we consider the m-fold tensor product μc = μc |α| of m copies of μ(1) bc . Given a function c . We see that μc ∗ wα = c wα and μb ∗ μc= μ|α| m f ∈ L2 (G , X), expressed as α wα xα , we see that μc ∗ f = α c wα xα . Since μc is a probability measure, convolution with μc is a norm 1 operator on L2 (Gm , X), for every real or complex Banach space X and every c ∈ [−1, 1]. In order to pass to the classical semi-group setting, we shall perform the following change of variable. For t 0, let νt = μe−t . We get that νt ∗ νs = νs+t . Given a function f = α wα xα ∈ L2 (Gm , X) we set Tt f = νt ∗ f =

e−|α|t wα xα

(W )

α

and we call (Tt )t 0 the Walsh semi-group. We noticed that each Tt is a contraction on L2 (Gm , X). Let Pi , i = 1, . . . , m, denote the projection on L2 (Gm , X) deﬁned by (Pi f )(ε1 , . . . , εm ) =

f (ε1 , . . . , εi−1 , ε, εi+1 , . . . , εm ) dε.

Type, cotype and K-convexity

1323

It is clear that Pi is a norm one projection, and Pi Pj = Pj Pi for all i, j = 1, . . . , m. Let α Q *i = I − Pi . We have Pi εi = 0, Pi εj = εj for j = i. For every α ⊂ {1, . . . , m} let P = i∈α Pi . We see by checking the action on every wα that Tt =

m m $

$

Pi + e−t Qi = 1 − e−t Pi + e−t I . i=1

i=1

It follows, by expanding the last product, that Tt is a convex combination of commuting norm one projections of the form P α . For the next lemma it is natural to quantify the type-p property of a Banach space X in a way close to the deﬁnition of B-convex Banach spaces. We let N(X) denote the smallest integer n 1 such that 1 n εi (t)xi dt n/16 0 i=1

for every family x1 , . . . , xn of vectors in X such that xi 1 for each i. Of course, if X has type p > 1, then we have N(X) (16Tp (X))q , where q < +∞ is the number conjugate to p > 1. We let N(X) = +∞ when X is not B-convex. L EMMA 9. Suppose that X is a B-convex Banach space, and assume that M is a convex combination of contractive commuting projections on X. Then n+1 M − M n 1/4 when n max(N(X), 256). P ROOF. Let M=

cα Pα ,

α

where cα 0, α cα = 1, and where the (Pα )s are commuting projections on X, such that Pα 1; we get in particular that M 1. Let ξ be a random variable on some probability space Ω, with values in the space of operators on X and with P (ξ = Pα ) = cα for every α. Then Eξ = M, and if ξ1 , ξ2 are two independent copies of ξ , then Eξ1 ξ2 = M 2 . Let ξ1 , . . . , ξn be independent copies of ξ , with n max(N(X), 256). Suppose that x ∈ X, x = 1 and let us consider, for a ﬁxed choice ε of εi = ±1, the random variable Zε on Ω deﬁned by Zε (ω) = ni=1 εi ξi (ω)x. Let B = {i: εi = 1} and C = {i: εi = −1}, k = |B| and* = |C|. Assume that k . Then, letting ξ B denote the (random) operator equal to j ∈B ξj , and noting that ξ B (ω) 1 for every ω, n n B B B εi ξi x ξ εi ξi x = ξ x− ξ ξi x . i=1

i=1

i∈B

i∈C

1324

B. Maurey

Taking expectation on Ω, n B B E εi ξi x Eξ x − Eξ ξi x i=1 i∈B i∈C = kM k x − M k+1 x M k x − M k+1 x − | − k| n M n x − M n+1 x − | − k| 2 (if k, we replace ξ B by ξ C ). On the other hand we get, taking the expectation E over all signs, noting that | − k| = | ni=1 εi | and since n N(X) n √ n n n+1

M x − M x − n EE εi ξi x n/16 2 i=1

so that, using n 256 n M x − M n+1 x 1/8 + 2n−1/2 1/4.

R EMARK 10. Suppose that X is a type-p Banach space, with p > 1 and type-p constant Tp . Assume that M is a convex combination of contractive commuting projections on X. Then x + M(x) (4Tp )−q x for every x ∈ X (q is the exponent conjugate to p). It follows that I + M is invertible and that (I + M)−1 (4Tp )q . It is well known to experts that the uniform invertibility of I + Tt is precisely what is needed in Kato’s theorem for proving that a semi-group (Tt )t 0 is holomorphic. The proof of the remark is a slight modiﬁcation of the proof of the preceding lemma. Suppose that x = 1 and x + Mx < ε. It follows that M k x + M k+1 x < ε for every k 0 since M 1, and M k x x − kε = 1 − kε by the triangle inequality. Taking expectations as before, n εi ξi x nM k x − ε n(1 − kε) − ε n − (n + 1)2 ε/2. E i=1

Taking the expectation E over all signs n εi ξi x Tp n1/p . n − n ε EE 2

i=1

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If we choose n such that 1/4 < nε 1/2, then n/2 Tp n1/p , thus 4n (4Tp )q since q 2, and we get ε (4Tp )−q . If we want to see why things can go wrong when X contains n1 s, we may modify the example showing that the K-convexity constant of L1 (Gm ) is large. We shall only sketch the idea. Let us consider the function f0 from [0, 1]m to the space of measures on [0, 1]m such that f0 (x) is the Dirac mass at x for every x ∈ [0, 1]m (this function is not Bochner measurable; a genuine example should correct this fact). If Pi is deﬁned for every g ∈ L2 ([0, 1]m , X) by (Pi g)(x1 , . . . , xm ) =

1

g(x1 , . . . , xi−1 , y, xi+1 , . . . , xm ) dy 0

for i = 1, . . . , m, then the (Pi ) are commuting norm one projections. For every α ⊂ {1, . . . , m}, the vector value (P α f0 )(x) is the Lebesgue measure on some |α|-dimensional unit cube. When x varies, these probability measures are pairwise and this is the * disjoint, −t e ((1 − )Pi + e−t I ) source of all the problems. The corresponding semi-group St = m i=1 −mt behaves very badly. In particular, the inequality I − St 2(1 − e ) shows that the hypothesis for Kato’s theorem is not satisﬁed uniformly in m in this example, where X = M (the space of measures). We are ready to begin the proof that B-convexity implies K-convexity, using the Walsh semi-group (Tt )t 0 deﬁned by relation (W ). Recall that each operator Tt is a convex combination of commuting norm one projections on L2 (X). If X is B-convex, then L2 (X) is also B-convex; it follows from Lemma 9 that Tnt − T(n+1)t 1/4 for every t > 0, when n max(N(L2 (X)), 256). For the rest of the paper we assume that X is a complex Banach space. We strongly recommend reading [49, Chapter 14] (and Appendix IV about Kato’s theorem for semi-groups). For the lazy reader who does not want to hear about general semi-groups, we shall sketch a proof of Kato’s theorem in the simpliﬁed setting which is needed here. We consider m Rademacher functions ε1 , . . . , εm *and the corresponding 2m Walsh functions (wα ) that are deﬁned by the formula wα = i∈α εi , where α ranges over the 2m subsets of {1, . . . , m}; next we ﬁx 2m vectors (yα ) in X, and we let E be the 2m -dimensional complex subspace of L2 (Gm , X) generated by the algebraic basis (wα yα ). Our operators (Tt )t 0 act diagonally on this basis of E, since Tt (wα yα ) = e−t |α| wα yα for every α. Deﬁning the complex extension Tz of Tt on E is straightforward: we simply say that Tz acts on E by Tz (wα yα ) = e−z|α| wα yα for every α, but of course the problem is to ﬁnd bounds for the norm of Tz , independent of the particular subspace E ⊂ L2 (X). We see that Tt = e−t A , where A is represented in the basis (wα yα ) by a diagonal matrix with entries in {0, 1, . . . , m}, namely A(wα yα ) = |α|wα yα . The Rademacher projection corresponds to the matrix B obtained by replacing in A all diagonal entries = 1 by zero entries. For the proof of Theorem 12 below, we shall keep m and the 2m -dimensional subspace E ⊂ L2 (Gm , X) ﬁxed. Our aim is to ﬁnd a bound for the norm of the matrix B, acting on

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B. Maurey

this subspace E by B(wα yα ) = wα yα if |α| = 1 and B(wα yα ) = 0 otherwise; we are looking for a bound K independent of m and of the particular subspace E. From the nature of the problem it is clear that such a bound K will be a bound for the norm of the Rademacher projection RX acting on L2 (GN , X), that is to say a bound for the K-convexity constant of X. The control of the complex extension of the semi-group begins with a standard exercise in functions of one complex variable. Consider η = v + iπ , v > 0, and the two conjugate rays R = R+ η and R = R+ η, ¯ symmetric with respect to the real axis, contained in the half plane !z > 0. Let ξ = π + iu, with |u| < v, and consider the holomorphic function f (z) = e−ξ z . Then for every real a 0, we have e−ξ a =

1 2iπ

e−ξ z (z − a)−1 dz, Γ

where Γ is essentially the path given by these two rays, except for a little detour to avoid z = 0 (this is needed in the case a = 0; see the ﬁgure in [49, Appendix IV]). We have −ξ z e−π(1−|u|/v)!z e for every z in the convex cone limited by R and R, therefore the integral is convergent since |u| < v. It is a standard exercise to show that the integral over Γ is indeed equal to e−ξ a (approximate Γ by the integral over a bounded closed contour that uses part of the two rays and part of a large circle centered at 0, and apply Cauchy’s formula). In our (ﬁnite-dimensional) vector situation, the generator A of the semi-group is expressed by a diagonal matrix with non-negative real diagonal, so that the next equation is by no means harder to prove than the scalar case, e

−ξ A

1 = 2iπ

e−ξ z (zI − A)−1 dz. Γ

This can be done not only for ξ = π + iu, |u| < v, but as well for any ξ = α + iβ with α > 0 and π|β| < vα, in other words for every ξ in a sector of angle θ around the positive real axis, where π tan θ = v. The above formula, extended to these values of ξ , deﬁnes the complex extension of the semi-group. It is clear (and standard) that we can bound the complex extension of the semi-group, acting on the ﬁxed ﬁnite-dimensional subspace E, if we have a suitable bound for the norm of the resolvent (zI − A)−1 on the two rays R and R (again, this norm is understood as norm of an operator from E to E). L EMMA 11. Let E ⊂ L2 (Gm , X) be as above. Assume that X is a B-convex Banach space, let n max(N(L2 (X)), 256) and let v be such that 0 < v 1/n. For every complex number z belonging to the ray R = R+ (v + iπ) or to the conjugate ray R, we have (zI − A)−1 36πn/|'z|.

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P ROOF. Let λ = v ± iπ , and suppose that ε > 0 is chosen in such a way that (A − λI )−1 > 1/ε; we can ﬁnd a norm one vector x ∈ E such that Ax − λx < ε. The function ϕ(t) = Tt (x) = e−t A x satisﬁes the differential equation ϕ (t) = −Aϕ(t) = −Tt (Ax). Since Tt is a contraction semi-group, we deduce that for every t > 0 ϕ (t) + λϕ(t) = Tt (Ax − λx) ε. If we write this as ϕ (t) + λϕ(t) = g(t) with g(t) ε and solve the differential equation, we get t −λt λs e g(s) ds , ϕ(t) = e x+ 0

which implies that ϕ(t) − e−λt x εt. Let n max(N(L2 (X)), 256). By Lemma 9, we know that for every s > 0, we have T(n+1)s −Tns 1/4, since Ts is a convex combination of commuting norm one projections. We shall use this fact with s = 1; when s = 1, we get ϕ(n + 1) − ϕ(n) 1/4 and e−λs = e−λ = − e−v since eiπ = −1. We have

1/3 < e−1 < e−vn 1 + e−v = e−λ(n+1) x − e−λn x . By the triangle inequality, 1/3 < e−λ(n+1) x − e−λn x ϕ(n + 1) − e−λ(n+1) x + ϕ(n) − e−λn x + 1/4, hence 1/12 ϕ(n + 1) − e−λ(n+1) x + ϕ(n) − e−λn x (2n + 1)ε 3nε. It follows that (A − λI )−1 36n. We may apply the same proof to the generator As = s −1 A, for every s > 0; obviously, this As also generates a semi-group consisting of convex combinations of commuting contractive projections, and this implies as above that (λI − As )−1 36n or (sλI − A)−1 36n/s hence (zI − A)−1 36πn/|'z| when z belongs to the rays R = R+ (v + iπ) or R.

T HEOREM 12. Let X be a B-convex Banach space. Then the Rademacher projection RX is bounded on L2 (GN , X), and RX eκ max(N(L2 (X)),256) for some universal constant κ.

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P ROOF. Let us consider again the 2m -dimensional subspace E of L2 (Gm , X). We may deduce from Lemma 11 that the semi-group (Tt ) acting on E has a nicely bounded complex extension to the sector mentioned before, but since we are mainly interested in the Rademacher projection we are going to take a shortcut. Let N = max(N(L2 (X)), 256) and v = 1/N ; consider the path Γ consisting of the ray R = R+ (v + iπ) and its conjugate R. Provided that the integral makes sense, 1 ϕ(z)(zI − A)−1 dz 2iπ Γ represents, when ϕ is holomorphic on C, the diagonal matrix where each diagonal entry k of A is replaced by ϕ(k). Recall that the diagonal entries of A are integers. In order to get the matrix B of the Rademacher projection, we naturally introduce ϕ1 (z) = sin(πz)/(1 −z) that kills all entries = 1 in A. We need to multiply this ϕ1 by a suitable exponential that guarantees that the integral converges and that the Cauchy formula applies to the unbounded contour Γ . Let us consider the matrix sin(πz) −π 2 z/v 1 e C= (zI − A)−1 dz. 2iπ Γ π(1 − z) The sin function eliminates the problem at 0. Also, one can check that the integral is absolutely convergent. It is easy to see that this matrix C is a multiple of the Rademacher 2 projection B, namely C = e−π /v B and using the bound from Lemma 11 we can show that C κ1 N, 2

where κ1 is an universal constant. It follows that B κ1 N eπ N eκN . Let us detail the preceding computation. We have N = max(N(L2 (X)), 256) and v = 2 1/N , thus 0 < v < 1/2. Let z0 = v ± iπ . If z = sz0 , s > 0, then |sin(πz)| eπ s and 2 2 |cos(πz)| eπ s , therefore |sin(πz) e−π z/v | 1. We also have |1 − z| πs and |1 − z| 1 − sv 1/2 when 0 < s < 1. Next, we use sin(πz) π|z| max cos(uπz) πs|z0 | eπ 2 s 0
for 0 < s < 1, so that C

1 π

1 0

2πs|z0 | 36N 1 |z0 | ds + π s π

72N|z0 π

|2

+

36N|z0 | 500N. π3

∞ 1

1 36N |z0 | ds π 2s s

We ﬁnish with another result of Pisier, that has been very useful for local theory. We did not try to optimize the constant, but to give an argument as simple as possible (essentially identical to Pisier’s proof).

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T HEOREM 13. Let dX denote the Banach–Mazur distance from X to the Hilbert space of the same dimension. Then the norm of the Rademacher projection in L2 (X) is bounded by 4 ln dX when dX e (and by dX in any case). P ROOF. The proof is comparatively simple. Let f = to X. Assume that f L2 (X) = 1. For z ∈ C, let P (z) =

α wα xα

be a function from Gm

z|α| wα xα .

α

This deﬁnes a holomorphic function (a polynomial) from C to L2 (X). The Rademacher projection RX (f ) of f is the derivative P (0) of P at z = 0. When X = H is a Hilbert space, we get by orthogonality, for every z in the closed unit disc D in C P (z)2 L

2 (H )

=

α

|z|2|α| xα 2

xα 2 = f 2L2 (H ) ,

α

thus P (z) 1 for every z ∈ D in this case. If the distance from X to some Hilbert space is d, then clearly P (z) d, therefore P (z) dX for every z ∈ D. On the other hand, we have seen that P (x)L2 (X) f L2 (X) = 1 when x is real and |x| 1, because P (x) = μx ∗ f in that case, with μx a probability measure. For convenience, we transfer the problem to the closed strip S = {z: |'z| 1}: the mapping ϕ(z) = tanh(πz/4) maps S to the closed unit disc, and sends the line 'z = 0 to the segment −1 x 1. The L2 (X)-valued function q(z) = P (ϕ(z)) is bounded on S, holomorphic on the open strip S0 , bounded by dX on S and by 1 on the line 'z = 0. The result follows then from q (0) = πP (0)/4 and from the following lemma. When dX e we get |P (0)| = 4π −1 |g (0)| 4π −1 e ln dX < 4 ln dX . L EMMA 14. Let g be a bounded and continuous function, deﬁned on the closed strip S = {z: |'z| 1}, holomorphic on the open strip S0 , with values in a Banach space Y ; assume that g is bounded by C e on S and bounded by 1 on the line 'z = 0. Then |g (0)| e ln C. P ROOF. Let g1 (z) = (g(z) − g(−z))/2; then g1 obeys the same bounds as does g, and g1 (0) = g (0); furthermore, g1 (0) = 0. Let 0 < θ 1. Since g1 is bounded, |g1 | 1 on the line 'z = 0 and |g1 | C on the line 'z = 1, the three lines lemma implies that |g1 | is bounded by 11−θ C θ = C θ on the line 'z = θ ; the same argument applies to the line 'z = −θ ; now k(z) = g1 (z)/z (with k(0) = g1 (0)) is bounded on the strip S, and bounded by θ −1 C θ on the two lines 'z = ±θ , therefore |g (0)| = |g1 (0)| = |k(0)| θ −1 C θ . The optimal choice of θ in (0, 1] is θ = (ln C)−1 , which is licit because ln C 1.

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CHAPTER 31

Distortion and Asymptotic Structure Edward Odell∗ Department of Mathematics, The University of Texas, 1 University Station C1200, Austin, TX 78712, USA E-mail: [email protected]

Th. Schlumprecht∗ Department of Mathematics, Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Contents 1. Distortion in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335 2. Asymptotic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359

* Research supported by NSF.

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The distortion problem for Hilbert space 2 may be stated as follows. Do there exist sets A, B ⊆ S2 with inf{a − b: a ∈ A, b ∈ B} > 0 and such that A ∩ X = ∅, B ∩ X = ∅ for all inﬁnite-dimensional closed subspaces X ⊆ 2 ? We shall see that the answer is yes. But how does one choose such sets? What criteria can be used when sorting through the elements of S2 to determine which vectors go into A or B? Any x ∈ S2 can be the ﬁrst element of an orthonormal basis for 2 . S2 viewed by standing at a point x looks no different if you move to point y. The approach we use to distort 2 is indirect. It seems impossible to distort Hilbert space by only working within the category of Hilbert space. We need to expand to the category of Banach spaces and to use some deep analysis of the structure of a certain recently discovered Banach space which has come to be called S [50]. We then will infer the distortion of 2 via a certain non-linear transfer. The Banach space S is a descendent of Tsirelson’s famous space T . The key to the marvelous properties of T (and S) is that the norm is deﬁned implicitly as opposed to explicitly. One states a certain norm equation and shows that a solution exists. As concerns the distortion of 2 one is ultimately led to the discovery of a remarkable collection of sets An ⊆ S2 which are “large” (inf{x − a: a ∈ A, x ∈ X} = 0 for all inﬁnite-dimensional subspaces X ⊆ 2 ) and nearly biorthogonal (|an , am | < εmin(n,m) for an ∈ An , am ∈ Am , m = n for some εi ↓ 0). In Section 1 we focus on the distortion problem and explain how distortion enters into Banach space theory. The distortion problem for 2 is equivalent to the following. If f : S2 → R is Lipschitz, is f nearly constant (up to an arbitrary ε > 0) on some inﬁnitedimensional subspace of any given inﬁnite-dimensional subspace of 2 ? We discuss both the distortion problem and the Lipschitz function stabilization problem in the broader context of general Banach spaces. Section 2 concerns asymptotic structure. This is a notion which lies between the ﬁniteand inﬁnite-dimensional theory.

1. Distortion in Banach spaces The distortion problem in Banach spaces arose from work of James [19] and Milman [32] in the 1960’s. James proved that every isomorph of 1 (respectively, c0 ) contains a subspace almost isometric to 1 (respectively, c0 ). Equivalently (see Deﬁnition 3 below) 1 and c0 are not distortable. Milman showed that if has X no distortable subspace then X contains an almost isometric copy of c0 or p for some 1 p < ∞ and asked if a distortable space could exist. A few years later Tsirelson [54] produced his famous space T which does not contain any isomorph of c0 or p (1 p < ∞). Hence there exists a distortable Banach space. This left what came to be called “the distortion problem”: is p distortable for 1 < p < ∞? X, Y, Z, . . . shall denote separable inﬁnite-dimensional real Banach spaces. X ⊆ Y shall mean that X is a closed linear subspace of Y . F, G, . . . shall denote ﬁnite-dimensional real Banach spaces. SX denotes the unit sphere of X and BX is the closed unit ball of X. We begin with the result of James cited above, giving the proof in the 1 case. Recall that X contains almost isometric copies of 1 if for all ε > 0 there exists Y ⊆ X with d(Y, 1 ) < 1 + ε. The idea behind the proof is to choose a block basis (xi ) of the unit

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vector basis for 1 which gives essentially the worst possible equivalence between the 1 norm w.r.t. (xi ) and the equivalent norm and then note that this block basis itself must be nearly isometric to the unit vector basis of 1 . T HEOREM 1 ([19]). If X is isomorphic to 1 (respectively, c0 ) then X contains almost isometric copies of 1 (respectively, c0 ). P ROOF. Let ||| · ||| be an equivalent norm on 1 and let (ei ) be the unit vector basis for 1 . Let ε > 0. It sufﬁces to prove that there exists a ||| · |||-normalized block basis (xi ) of (ei ) with ||| ai xi ||| (1 − ε) |ai | for all (ai ) ∈ c00 . Let (yi ) be any ||| · |||-normalized block basis of (ei ). For n ∈ N let ∞ i=n ai yi ∞ cn := inf : 0 = (ai ) ∈ c00 . i=n |ai | Clearly cn - c for some c > 0. Choose δ smallenough and n0 ∈ N large enough so that cn0 /(c + δ) > 1 − ε, and choose for i ∈ N, xi = j ∈Fi bj yj such that n0 F1 < F2 < · · · , |||xi ||| = 1, and j ∈Fi |bj | > 1/(c + δ). Thus we conclude for any (ai ) ∈ c00 that ∞ ∞ ∞ ai xi = ai bj yj cn0 |ai | |bj | i=1

i=1

cn0 c+δ

j ∈Fi

∞

|ai | (1 − ε)

i=1

j ∈Fi

|ai |,

i=1

which implies the claim and ﬁnishes the proof.

The same method can be used to produce, for any equivalent norm on p (1 < p < ∞) or c0 , a normalized block basis with a very tight upper p estimate or tight lower p estimate but not both simultaneously. The triangle inequality in 1 gives us the upper estimate for free. If we produce a tight lower c0 estimate in an isomorph of c0 then a tight upper estimate automatically ensues (see, e.g., [25]) which is how Theorem 1 is proved for c0 . (Alternatively, one could use that for every normalized weakly null sequence (yi ) in any Banach space and ε > 0 some subsequence (zi ) of (yi ) satisﬁes ai zi (1 − ε)(ai )c0 for all scalars (ai ) (see, e.g., [36]).) Thus 1 and c0 are not distortable by the nature of their extreme positions (the largest and smallest norm) amongst Banach spaces. There is also the following ﬁnite-dimensional version of James’ blocking argument. It shows how one can improve the 1 -constant of a ﬁnite basic sequence if one is willing to decrease its length. P ROPOSITION 2. Let N, k ∈ N, C > 0 and assume that x1 , x2 , . . . , xN k is a normalized basic sequence for which Nk Nk ai xi C |ai |, i=1

i=1

for all a1 , a2 , . . . , aN k ∈ R.

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Then there is a normalized block basis (yi )N i=1 of (xi ) so that N N ai yi C 1/ k |ai |, i=1

for all a1 , a2 , . . . , aN ∈ R.

i=1

P ROOF. We will prove the proposition by induction on k ∈ N. For k = 1 there is nothing to k prove. Assuming the claim is true for k − 1, and given that (xi )N i=1 satisﬁes the assumption we are in one of the following two situations. It may be that for all j = 1, . . . , N N k−1 N k−1 ai xi+(j −1)N k−1 : |ai | = 1 C 1−1/ k . Cj = min i=1

i=1

k−1 (j ) N k−1 (j ) In this case we choose yj = N i=1 ai xi+(j −1)N k−1 with yj = 1 and i=1 |ai | 1/ k−1 and deduce from the assumptions that for a1 , . . . , an ∈ R C N k−1 N N N N (j ) 1/ k a CC 1/ k−1 aj yj C |aj | |a | = C |aj |, j i j =1

j =1

j =1

i=1

j =1

which completes the proof. Otherwise there is a j ∈ {1, 2, . . . , N} with Cj C 1−1/ k , and we can apply the induck−1 1−1/ k in order to get a normalized block tion hypothesis to (xi+(j −1)N k−1 )N i=1 and C = C (yi )N i=1 for which N N N

1−1/ k 1/(k−1) 1/ k ai yi C |ai | = C |ai |. i=1

i=1

i=1

D EFINITION 3. Let λ > 1. X is λ-distortable if there exists an equivalent norm | · | on X so that for all Y ⊆ X, |y1 | sup : y1 , y2 ∈ S(Y,·) λ. |y2 | X is distortable if X is λ-distortable for some λ > 1. X is arbitrarily distortable if X is λ-distortable for all λ > 1. D EFINITION 4. Let f : SX → R. f stabilizes if for all Y ⊆ X and ε > 0 there exists Z ⊆ Y so that osc(f, SZ ) ≡ sup f (z1 ) − f (z2 ): z1 , z2 ∈ SZ < ε.

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Clearly X does not contain a distortable subspace iff every equivalent norm on X stabilizes. One can enlarge the question as to whether a given X contains a distortable subspace to whether every reasonable (e.g., Lipschitz or uniformly continuous, it makes no difference) f : SX → R stabilizes. Some insight to the connection between these questions is provided by the following simple proposition. A set A ⊆ SX is asymptotic if for all Y ⊆ X, Y ∩ A = ∅. A is nearly asymptotic (or large) in X if d(A, Y ) = 0 for all Y ⊆ X. Sets A, B ⊆ SX are separated if the minimum distance between them is positive, i.e., if md(A, B) ≡ inf{a − b: a ∈ A , b ∈ B} > 0. P ROPOSITION 5. (a) There exists a Lipschitz f : SX → R which does not stabilize iff there exist Y ⊆ X and separated sets A, B ⊆ SY which are (nearly) asymptotic in Y . (b) If X is uniformly convex then X contains a distortable subspace iff there exists a Lipschitz f : SX → R which does not stabilize. (c) Let 1 < p < ∞. Then p is distortable iff there exist separated asymptotic sets in Sp . S KETCH OF PROOF. (a) If A, B are separated asymptotic (or nearly asymptotic) for Y then f (x) ≡ inf{x − a: a ∈ A} is Lipschitz but does not stabilize. Conversely if f is Lipschitz and does not stabilize then there exist Y ⊆ X and c < d so that {x ∈ SY : f (x) < c} and {x ∈ SY : f (x) > d} are separated and asymptotic in Y . (b) If Y ⊆ X is distorted by | · | then | · | can be extended to an equivalent norm on X and | · | : SX → R is Lipschitz but does not stabilize. (This implication does not require X to be uniformly convex.) If A and B are separated asymptotic sets for Y ⊆ X then c¯0 (A ∪ −A) is the unit ball for a norm | · | on some Z ⊆ Y and | · | distorts (Z, · ). (c) This follows from (a), (b) and the fact that every subspace of p contains almost isometric copies of p . We have seen that in a special case (X isomorphic to c0 or 1 ) all equivalent norms stabilize. Finite dimensionally things work out nicely; there are good stabilization results. We state two such theorems. The ﬁrst was observed by Milman (see [33], p. 6) in connection with Dvoretsky’s famous theorem that one ﬁnds almost isometric copies of n2 for all n in any X. T HEOREM 6 (First stabilization principle). For all C > 0, ε > 0 and k ∈ N there exists n = n(C, ε, k) so that if dim E = n and f : SE → R is C-Lipschitz (|f (x) − f (y)| Cx − y) then there exists F ⊆ E, dim F = k with osc(f, SF ) < ε. The second principle is a reworking of the ﬁrst in the setting of ﬁnite-dimensional spaces with bases and subspaces spanned by block bases. T HEOREM 7 (Second stabilization principle). Given C > 0, ε > 0 and k ∈ N there exists n = n(C, ε, k) ∈ N so that if dim E = n and E has a basis (xi )n1 whose basis constant does not exceed C and f : SE → R is C-Lipschitz then there is a normalized block basis (yi )k1 of (xi )k1 so that osc(f, Syi k ) < ε. 1

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The proof of the second stabilization principle is given in [38]. It relies mostly on Lemberg’s [24] proof of Krivine’s theorem. The exception is the case where F = n∞ (see [38] for a proof in this case). T HEOREM 8 (Krivine’s theorem [23]). Given C > 0, ε > 0 and k ∈ N there exists n = n(C, ε, k) ∈ N so that if (ei )ni=1 is C-basic then there exist p ∈ [1, ∞] and a block basis (xi )k1 of (ei )n1 which is (1 + ε)-equivalent to the unit vector basis of kp . Next we present a proof of Milman’s theorem [18]. T HEOREM 9. If X does not contain almost isometric copies of c0 or p for some 1 p < ∞ then X contains a distortable subspace. First we set some notation. For x ∈ X the type on X generated by x is the function τx : X → R given by τx (y) = x + y. The norm on X generated by x is the function · x given by yx =

1 1 τyx (y) + τyx (−y) = yx + y + yx − y . 2 2

It is not difﬁcult to see that · x is an equivalent norm on X [41]. A normalized block basis (xi ) ⊆ X is said to doubly generate an p type over X [49] if for all b ∈ X and (α, β) ∈ S2p , lim lim b + αxi + βxj = lim b + xi .

i→∞ j →∞

i→∞

It is easy to show that if (xi ) doubly generates an p type over X then for all ε > 0 there exists a subsequence of (xi ) which is (1 + ε)-equivalent to the unit vector basis of p . A similar deﬁnition and result can be made for p replaced by c0 . P ROOF OF T HEOREM 9. We shall prove that if · x stabilizes on X for all x ∈ X then there exists a sequence (xi ) in X which doubly generates an p type over X for some 1 p < ∞ or which doubly generates a c0 type over X. We break the proof into three steps. Let (yi ) be a normalized basic sequence in X. A normalized basic sequence (xi ) in X generates a spreading model E = [(ei )] if there exist εn ↓ 0 so that for all n, (αi )n1 ⊆ [−1, 1]n , n k1 < · · · < kn , n j1 < · · · < jn , n n αi xki − αi xji < εn . 1

i=1

Under these circumstances we may deﬁne n n αi ei = lim · · · lim αi xki . k1 →∞ kn →∞ i=1

i=1

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(ei )∞ i=1 then becomes a normalized basis for E ≡ [(ei )]. (xi ) generates a spreading model X ⊕ E over X if for all x ∈ X, n n αi ei ≡ lim · · · lim x + αi xki x + k1 →∞ kn →∞ i=1

i=1

exists. Every normalized basic sequence can be seen via Ramsey theory to yield a subsequence which generates a spreading model over X [6]. Rosenthal [48], cleverly using the Borsuk–Ulam theorem, showed that every basic sequence in X admits a normalized block basis (xi ) whichgenerates a spreading model E which is 1-unconditional over X. This means that x + n1 αi ei = x + n1 εi αi ei for all x ∈ X, (αi )n1 ⊆ R and εi = ±1. Actually we only really need below that x + e = x − e for all x ∈ X, e ∈ E. Step 1. There exists a normalized block basis (zi ) of (yi ) which generates a 1-unconditional spreading model E = [(ei )] over X so that x + e = x + e for all x ∈ X, e, e ∈ E with e = e .

(1)

Note that a consequence of this is that for all e ∈ [ei ]∞ i=2 and α ∈ R x + αe1 + e = lim x + αzn + e = lim x + αzn + ee2 n→∞

n→∞

= x + αe1 + ee2 .

(2)

∞

More generally it follows that for any m ∈ N and a ∈ ei m i=1 , e, e ∈ [ei ]i=m+1 with e =

e we have

x + a + e = x + a + e .

(3)

P ROOF OF S TEP 1. Let εn ↓ 0 and let (di )∞ 1 ⊆ X be dense in X. Since · di stabilizes (n) for all i we can recursively choose for each n ∈ N a normalized block basis (yi ) of (yi ) so that (a) (yi(n) ) is a block basis of (yi(n−1) ) (take (yi(0)) = (yi )), (b) | ydi − zdi | < εn if y, z ∈ S[y (n) ]∞ and i n. j

j=1

(n)

Set zn = yn . Then for all i n ∈ N yd − zd < εn , i i

whenever y, z ∈ S[zi ]∞ . i=n

In particular since (dj ) is dense in X for all normalized block bases (wj ) of (zj ) and x ∈ X, limj wj x exists and the limit depends solely upon x (not the particular sequence (wj )). Using [48] we may assume in addition that (zj ) generates a 1-unconditional spreading model E = [(ei )] over X.

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Let (wj ) be any normalized block basis of (zj ). Then for x ∈ X

1 τx (zj ) + τx (−zj ) j →∞ j →∞ 2 = lim zj x = lim wj x = lim x + wj

x + e1 = lim τx (zj ) = lim j

j →∞

j →∞

and (1) follows. Step 2. There is a subsequence (wj ) of (zj ) so that for k, m ∈ N, b ∈ mBdi ,wi m1 , m < n1 < · · · < nk k k k (4) αi wni − b + αi ei < εm if αi ei 1. b + i=1

1

i=1

Note that if k were replaced by m this would merely become the fact that E = [(ei )] is a spreading model of (wj ) over X. Taking b = 0 we see that (wi ) and (ei ) are equivalent. (4) follows easily by a diagonal argument from the following claim. Given x ∈ X and ε > 0 there exists a subsequence (˜zj ) of (zj ) so that k k αj z˜ j − x + αj ej < ε x + j =1

j =1

whenever kj =1 αj ej 1. Moreover this holds for all subsequences of (˜zj ). Given x ∈ X and ε > 0, we inductively choose (˜zj ) ⊆ (zj ) so that i i αj z˜ j + β z˜ i+1 + γ e2 − x + αj z˜ j + βe1 + γ e2 < ε2−(i+1) x + j =1

j =1

for all (αj )i1 ⊆ [−1, 1], γ , β ∈ [−1, 1] and i ∈ N. Thus if ki=1 αi ei 1 (which implies that |αi | 1 for i k) k k αj z˜ j − x + αj ej x + j =1 j =1 k i k i−1 k αj z˜ j + αj ej − x + αj z˜ j + αj ej x + i=1 j =1 j =i+1 j =1 j =i k i−1 k αj z˜ j + αi z˜ i + αj ej e2 = x + i=1 j =1 j =i+1 i−1 k αj z˜ j + αi e1 + αj ej e2 − x + j =1

k i=1

2−i ε < ε.

j =i+1

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The equality in the calculation comes via (1) and (3). The same estimates remain valid for any subsequence of (˜zj ). This completes Step 2. The next step combined with Step 2 completes the proof. Step 3. There exists a normalized block basis (ai ) of (ei ) which doubly generates a c0 type or an p type over X ⊕ E for some 1 p < ∞. Given δ > 0, a ﬁnite set S ⊆ X ⊕ E and k ∈ N by Theorem 7 there exists a normalized block basis (ci )k1 of (ei ) with | x + c − x + c | < δ for c, c ∈ Sci k and x ∈ S. By 1 Theorem 8 if k is sufﬁciently large, there exists p ∈ [1, ∞] and a normalized block basis (a, b) of (ci )k1 with

αa + βb − |α|p + |β|p 1/p < δ

if |α|, |β| 1.

Using this inductively, and passing to a convergent subsequence in [1, ∞] of the p’s thus produced we obtain a normalized block basis (a1 , b1 , a2 , b2 , . . .) of (ei ) and p ∈ [1, ∞] so that

x + αam + βbm − x + |α|p + |β|p 1/p bm < 2−m

(5)

if m ∈ N, x ∈ mB(di )m1 ∪(ei )m1 and |α|, |β| 1. As usual we let (|α|p + |β|p )1/p = max(|α|, |β|) if p = ∞. Let m ∈ N, x ∈ mB(di )m1 ∪(ei )m1 , (α, β) ∈ S2p and m < n1 < n2 . Then by (3) x + αan1 + βan2 = x + αan1 + βbn1 . By virtue of (5) this yields that (ai ) doubly generates an p type (c0 type if p = ∞) over X ⊕ E. Tsirelson’s space T was the ﬁrst example of a Banach space that did not contain an isomorph of c0 or p for 1 p < ∞. We present the description of T (actually the dual space of the example in [54]) due to Figiel and Johnson [13]. By virtue of Theorem 9, T must contain a distortable subspace. In fact T is itself distortable and we sketch this argument below. First we set some notation. Let (ei ) be the unit vector basis of c00 . For E, F ⊆ N, “E < F ” if E or F is empty or if max E < min F . If x, y ∈ c00 , “x < y” if supp x < supp y as subsets of N, (Ei )n1 , subsets /E of N, are admissible if {n} E1 < · · · < En . For E ⊆ N, x ∈ c00 we let Ex(i) = 0 if i ∈ and x(i) otherwise. We shall choose a certain norm · on c00 and let T be the completion of c00 under this norm. L EMMA 10. There exists a norm · on c00 satisfying

n 1 n x = max x∞ , sup Ei x: n ∈ N and (Ei )1 is admissible . 2 i=1

(∗)

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This is the norm that deﬁnes T . It is easy to check that (∗) holds for all x ∈ T if “max” is replaced by “sup”. T was the ﬁrst “non-classical” Banach space. Previously norms had been deﬁned by more explicit formulas. The formula in (∗) is implicit. The norm is the solution of (∗) which of course must be shown to exist. Since [54] many variations and extensions of Tsirelson’s space have been constructed. Their norms can be described by an implicit equation similar to (∗) and their existence follows from the following general principle. We let N be the class of all norms · on c00 for which (ei ) is a normalized monotone basis for (c00 , · ) satisfying ai ei maxi |ai | for all (ai ) ∈ c00 . If · , | · | ∈ N we write · | · | if x |x| for all x ∈ c00 . P ROPOSITION 11. Let P : N → N satisfy P · P | · | whenever · | · |. Then P admits a smallest ﬁxed point. The proposition is proved in [43] via transﬁnite induction. Lemma 10 follows by taking (P · )(x) = max(x∞ , sup{ 12 ni=1 Ei x: n ∈ N, (Ei )n1 is admissible}). Recall that if X has a basis (ei ) then X is asymptotic 1 (w.r.t. (ei )) if there exists λ > 0 so that n1 xi λ n1 xi whenever (xi )n1 is admissible (i.e., (supp xi )n1 is admissible). T HEOREM 12. T is a reﬂexive Banach space having a normalized 1-unconditional basis (ei ). T is asymptotic 1 (w.r.t. (ei )) and is 2 − ε distortable for all ε > 0. T does not contain any subspace isomorphic to c0 or p (1 p < ∞). S KETCH OF PROOF. From (∗) it follows easily that (ei ) is normalized and a 1-unconditional basis for T . Also T is asymptotic 1 (with λ = 1/2). Hence it is easy to see that the only possible c0 or p which T could contain is 1 . Once we show that T is distortable then by Theorem 1 it will follow that T cannot contain 1 . Thus by James’ theorem that a non-reﬂexive space with an unconditional basis must contain an isomorph of c0 or 1 it follows that T is reﬂexive. Let ε > 0. For n ∈ N and x ∈ T set xn = sup{ 12 ni=1 Ei x: E1 < · · · < En }. Let (yi ) be any block basis of (ei ). We shall show that for n sufﬁciently large there exist y, z ∈ S(yi ) with yn < 1/2 + ε and zn > 1 − ε. This result yields that T is 2 − ε distortable for all positive ε. From Krivine’s theorem (or by Proposition 2) we have that for any m ∈ N there exists ∞ to the unit vector baa normalized block basis (xi )m 1 of (yi )m which is 1 + ε¯ equivalent m sis of 1 (¯ε = ε¯ (ε) to be speciﬁed). Let n/m < ε¯ and set x = m1 m 1 xi ; x is called an m -average with constant 1 + ε ¯ . Clearly 1 x 1/(1 + ε ¯ ). Choose E1 < · · · < En so 1 1 n that xn = 2 j =1 Ej x. Set F = {i m: |{j : Ej xi = 0}| > 1}. Then |F | < n and so n n 1 1 1 Ej 1 Ej x xi xi + Ej 2 2 m m j =1

j =1

i ∈F /

i∈F

1 1 1 1 n 1 xi xi < + < + ε¯ . + 2 m m m 2 2 i∈F

i∈ /F

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Let y = x/x. Then y ∈ S(yi ) and yn < (1/2 + ε¯ )(1 + ε¯ ) < 1/2 + ε for ε¯ sufﬁciently small. i Next let x = n2 ni=1 xi where yn < x1 < · · · < xn and each xi is an m 1 average with constant ε¯ where mi is large depending upon ε¯ and max supp xi−1 . In [16], (xi )n1 is called an RIS for rapidly increasing sequence of 1 averages. Clearly xn 1/(1 + ε¯ ). Let x = 1 k k i=1 Ei x where (Ei )1 is admissible. Let i0 be minimal so that Ei xi0 = 0 for some i. 2 Then there are relatively few Ei ’s relative to the length mj of the average xj for j > i0 . An argument much like the one above yields 2 1 2 x 1+n + ε¯ = 1 + + 2¯ε. n 2 n The claim follows taking z = x/x (with ε¯ small and n large).

Distortion for T was achieved by working with two types of vectors known to exist in all block subspaces: m 1 -averages and averages of RIS sequences. This idea plays a key role in the work of Gowers and Maurey [16] and in our discussion of S below. The same proof yields a more general statement. P ROPOSITION 13. Let (ei ) be a basis for a space X not containing 1 . Let Pn be the basis projection onto ei ni=1 and for x ∈ X set |x| = supn (Pn x + (I − Pn )x). If 1 is block ﬁnitely representable in every block basis of (ei ), then | · | is a distortion of some subspace of X. Following the discovery of T many variants appeared which solved a number of problems over the next 15 years. But the distortion problem for p , the unconditional basic sequence space and other like famous problems remained unsolved. The breakthrough came with the construction of the Tsirelson type space S in 1989 [50]. S was the ﬁrst arbitrarily distortable Banach space. Moreover it satisﬁes a stronger type of distortion criterion; it is biorthogonally distortable. D EFINITION 14. X is biorthogonally distortable if there exist sets (An , A∗n )n∈N with An ⊆ SX , A∗n ⊆ BX∗ and λ > 0, εi ↓ 0 satisfying (a) An is asymptotic in X for n ∈ N, (b) sup{x ∗ (x): x ∗ ∈ A∗n } λ for each x ∈ An for n ∈ N, (c) For n = m, sup{|x ∗ (x)|: x ∗ ∈ A∗m , x ∈ An } εmin(n,m) . It is easy to see that if X is biorthogonally distortable then X is arbitrarily distortable via the collection of norms |x|n ≡ n1 x + sup{|x ∗ (x)|: x ∗ ∈ A∗n }. That the space S, which we are about to deﬁne, is biorthogonally distortable was ﬁrst noted in [16]. It was shown to be arbitrarily distortable in [50]. Set f (n) = log2 (n + 1) for n 1. S is the completion of c00 under the implicit norm (whose existence follows from Proposition 11) ∞ 1 x = max x∞ , sup Ei x: 2 and E1 < · · · < E . f () i=1

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The unit vector basis (ei ) is a 1-unconditional 1-subsymmetric basis for S. Thus ai ei = ±ai en(i) for all choices of sign and n(1) < n(2) < · · ·. The admissibility criterion necessary in T to avoid 1 is no longer needed due to the damping factors f ()−1 . As in the case of T once we show that S is (biorthogonally) distortable it follows that S is reﬂexive and does not contain any p or c0 . (Further results on S can be found in [3] and [4].) T HEOREM 15. S is biorthogonally distortable. 1 S KETCH OF PROOF. For ∈ N and x ∈ S we set x = sup{ f () i=1 Ei x: E1 < · · · < E }. It sufﬁces to prove the following claim: Given εk ↓ 0 there is a sequence kn ↑ 0 so that for all n ∈ N and all inﬁnite-dimensional subspaces Y there is a y ∈ Y , y = 1 so that

ykn > 1 − εn

and ykm < εmin(m,n)

for all m = n.

(1)

The theorem follows taking An to be all such y’s and A∗n

=

kn 1 ∗ ∗ ∗ ∗ xi : x1 < · · · < xkn , xi ∈ BS ∗ for i kn . f (kn ) i=1

By x1∗ < x2∗ < · · · we mean w.r.t. (ei∗ ), the sequence of biorthogonal functionals of (ei ). In order to show the claim we will proceed as in the proof of Theorem 3 in [50] in which it is shown that for ∈ N and each subspace Z there is a z ∈ SZ so that z ≈ 1/f (). Moreover a block sequence in S consisting of increasing 1 -averages has a spreading model isometric to the basis (ei ) in S. Actually in [51] it was shown that there are subsequences of such sequences which are isomorphically equivalent to (ei ). If x1 < · · · < x in S then by the deﬁnition of the norm i=1 xi f 1() i=1 xi . It follows then from Proposition 2 that 1 is block ﬁnitely representable in every block basis of (ei ). In particular if (zi ) is a block basis of (ei ) we can ﬁnd m 1 averages with constant 1 + ε for all m ∈ N and ε > 0. Thus we may choose a block basis (yn ) of (zi ) so that given εn ↓ 0 and integers mn ↑ ∞ n each yn is an m 1 -average with constant 1 + εn . The following two key observations appear in [50]. The ﬁrst one follows from the fact that the triangle inequality is an equality when applied to blocks in 1 and can be shown in a similar way to the proof of the distortability of T . lim yn =

n→∞

1 f ()

for ∈ N.

(2)

This in turn implies that for x ∈ S and 0 ∈ N lim sup x + yn max 1, sup x +

n→∞

0

0

1 . f (0 )

(3)

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Iterating (3) we obtain by induction for every k ∈ N k f (k) k f (k) + . yni lim · · · lim sup n1 →∞ nk →∞ k k f ( 0) 0 i=1

(4)

The limit in (4) may be presumed to exist via Ramsey theory by ﬁrst passing to a subsequence of (yn ) if necessary. The second key observation requires a more extensive proof which we omit (see Lemmas 4 and 6 in [50]). k k lim · · · lim yni = lim · · · lim yni n1 →∞ nk →∞ n1 →∞ nk →∞ i=1

i=1

k

k k = = ei for all k ∈ N. f (k)

(5)

i=1

For < k in N we obtain from (5), k f (k) lim · · · lim yni n1 →∞ nk →∞ k i=1

1 f (k) lim · · · lim f () k n1 →∞ nk →∞ K i × max ynj : 1 = K0 K1 · · · K = k i=1 j =Ki−1

k i 1 f (k) = max ei : k1 + · · · + k + k f () k

i=1 j =1

1 f (k) ki max : k1 + · · · + k + k = f () k f (ki )

i=1

1 f (k) +k . f () f ((k + )/) k

(6)

(Here the concavity of f is used as in the proof of Lemma 4 in [50].) Given εn ↓ 0 choose kn ↑ ∞ with f (kn ) kn + km < εm kn +km · kn f (km )f km f (km ) f (km ) + < εm , km f (kn )

and

whenever m < n.

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Using this choice for kn it now follows from (4) and (6) that every inﬁnite-dimensional subspace Y contains a normalized vector y satisfying (1). Let us return to the distortion problem: is 2 (or p ) distortable? By Proposition 5 this is equivalent to ﬁnding asymptotic sets A, B ⊆ S2 with md(A, B) > 0. Distortion can be transferred between one p space and another via the Mazur map Mp : S1 → Sp deﬁned by Mp (x)(i) = sign x(i)|x(i)|1/p . Mp is a uniform homeomorphism [47] and moreover preserves block bases of (ei ). Thus if A is asymptotic in 1 (respectively, p ) then Mp (A) is nearly asymptotic in p (respectively, Mp−1 (A) is nearly asymptotic in 1 ). Thus p is distortable for some (or all) p iff S1 admits separated asymptotic sets iff there exists a Lipschitz function f : S1 → R which does not stabilize (Proposition 5). This equivalence does not solve the problem but it does suggest that one might try to ﬁnd separated asymptotic sets in the sphere of some space and then ﬁnd a generalized Mazur map to transfer the distortion to p . This turns out to work.

The generalized Mazur map ([26,14]) Let X have a 1-unconditional normalized basis (ei ). If x = The entropy map

ai ei we set |x| =

|ai |ei .

E : (1 ∩ c00 ) × X → [−∞, ∞) is deﬁned by

E(h, x) = E |h|, |x| = |hi | |log xi | i

under the convention 0 log 0 ≡ 0, for x = xi ei and h = (hi ). FX : S1 ∩ c00 → X is deﬁned as follows. For h = (hi ) ∈ S1 ∩ c00 , FX (h) is the unique x = xi ei ∈ X satisfying (i) E(h, x) E(h, y) for all y ∈ SX , (ii) supp h = supp x ≡ B, and (iii) sign xi = sign hi for i ∈ B. Of course one must observe that such an element x exists, which is easy. The uniqueness of x follows from the strict convexity of the log function: if supp x = supp y = B and x = y then E(h, 12 |x| + 12 |y|) > 12 E(h, |x|) + 12 E(h, |y|). The map FX is uniformly continuous if X is uniformly convex ([39], Proposition 2.4) and agrees with the Mazur map Mp if X = p ([39], Proposition 2.5). When X is in addition uniformly smooth, FX extends to a uniform homeomorphism between S1 and SX ([39], Proposition 2.6). Using some renorming tricks one has T HEOREM 16. Let X be a Banach space with an unconditional basis. Then SX and S1 are uniformly homeomorphic iff X does not contain n∞ ’s uniformly in n.

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The “only if” direction is due to Enﬂo [12]. This theorem has been extended to Banach lattices [11] and via the complex interpolation method to other spaces ([7], Chapter 12). Using this generalized Mazur map we ultimately obtain T HEOREM 17. p is biorthogonally distortable for 1 < p < ∞. We sketch the proof. Our proof that S was biorthogonally distortable yielded the following P ROPOSITION 18. Let εi ↓ 0. There exist sets An ⊂ SS and A∗n ⊂ BS satisfying the properties (a), (b) and (c) (cf. Deﬁnition 14) (a) |xk∗ (x )| < εmin(k,) if k = , xk∗ ∈ A∗k , x ∈ A . (b) For k ∈ N and x ∈ Ak there exists x ∗ ∈ A∗k with x ∗ (x) > 1 − εk . (c) Ak is nearly asymptotic in S, i.e., d(X, Ak ) = 0 for all X ⊆ S. We deﬁne (x ∗ ◦ x)(i) = x ∗ (i)x(i) for x ∗ =

x ∗ (i)ei∗ ∈ S ∗ and x =

x(i)ei ∈ S. Set

∗

∗ xk∗ ◦ xk ∗ ∗ : x ∈ Ak , xk ∈ Ak and xk |xk | = xk ◦ xk 1 − εk . Bk = 1 |xk∗ |(|xk |) k

The sets Bk ⊆ 1 are easilyseen to be unconditional (x = (xi ) ∈ B ⇔ (±xi ) ∈ B) and spreading (x = (xi ) ∈ B ⇒ xi eni ∈ B for all n1 < n2 < · · ·) by the proof of Theorem 15. P ROPOSITION 19. Bk is nearly asymptotic in 1 . We omit the technical proof which depends on the map FS ∗ (see [39], Theorem 3.4). Theorem 17 follows for p = 2 by taking Ck = M2 (Bk ). Ck is nearly asymptotic and moreover if vk ∈ Ck , v ∈ C with k = let

|vk |2 = xk∗ ◦ xk /xk∗ |xk | and |v |2 = x∗ ◦ x /x∗ |x | , where xk∗ , xk and x∗ , x are as in the deﬁnition of Bk and B . Letting λ = (1 − ε1 )−1 ,

x ∗ (j )xk (j )x ∗ (j )x (j )2 |vk |, |v | λ k j

1/2 ∗ λ xk (j )x (j ) j

×

1/2

x∗ (j )xk (j )

(by Cauchy–Schwarz)

j

! "1/2 ! ∗ " x , |xk | 1/2 λεmin(k,) = λ xk∗ , |x |

(by Proposition 18).

For p = 2 a similar argument works. (Ck , Dk ) biorthogonally distort p where Ck = Mp (Bk ) and Dk = Mq (Bk ), 1/p + 1/q = 1.

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Combining Theorems 9 and 17 we obtain T HEOREM 20. If X does not contain a distortable subspace then every subspace of X contains an isomorph of c0 or 1 . Gowers [15] proved that every Lipschitz function f : Sc0 → R stabilizes. Thus combining our above work with this result we have. T HEOREM 21. Suppose that every Lipschitz f : SX → R stabilizes. Then every subspace of X contains an isomorph of c0 . The fact that 2 is biorthogonally distortable leads to some interesting renormings. One can prove that given k ∈ N and ε > 0 there exists a renorming | · | of 2 so that for all X ⊆ 2 there exists E ⊆ X with d(E, k∞ ) < 1 + ε. Thus every inﬁnite-dimensional subspace contains k-cubes as k-dimensional slices of the unit sphere (up to ε). More generally we have [39] T HEOREM 22. Let X be a biorthogonally distortable Banach space with basis (ei ). For k ∈ N and ε > 0 there exists an equivalent norm | · | on X so that if (wi )k1 is a normalized k monotone basis and (xi )∞ 1 is a block basis of (ei ) then there exists a block basis (bi )1 of k (xi ) which is (1 + ε)-equivalent to (wi )1 . This is proved by a renorming trick that derives from [31] and was exploited by Gowers and Maurey [16] to show that for all K < ∞ such a space could be renormed so that no block basis of (ei ) is K-unconditional. One can easily prove that every X contains a basic sequence (ei ) with basis projections Pn satisfying limn Pn = 1 ((ei ) is asymptotically monotone). It was open as to whether one could also obtain lim I − Pn = 1 ((ei ) is asymptotically bimonotone). Theorem 22 yields that this is false even for isomorphs by considering the summing of 2 . This follows basis (si ) whose norm is given by ai si = supk | ki=1 ai |. This monotone basis has s1 − 2s2 = 1 yet (I − P1 )(s1 − 2s2 ) = 2s2 = 2. The relations between the notions X is distortable, X is arbitrarily distortable and X is biorthogonally distortable remain unclear. If is unknown if a distortable space contains an arbitrarily distortable subspace or if an arbitrarily distortable space contains a biorthogonally distortable subspace. It seems to be quite an interesting question as to whether there exists a distortable space of bounded distortion. D EFINITION 23. X is of λ-bounded distortion if X is distortable and for all equivalent norms | · | on X and all Y ⊆ X there exists Z ⊆ Y with |y| sup : y, z ∈ SZ,· λ. |z| At this point the prime candidate for such a space is T , the ﬁrst known distortable space. Theorem 12 gives the best current estimate: it is not even known if T is 2-distortable. We do know something about the structure of spaces of bounded distortion, should they exist.

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T HEOREM 24. Let X be a space of λ-bounded distortion. Then X contains a basic sequence (ei ) which is (a) ([34]) asymptotically c0 or p for some 1 p < ∞ and (b) ([53]) unconditional. We sketch the proof following the argument in [28] for (a). First by Krivine’s theorem it is easy to see that if (ei ) is a basic sequence then K(ei ) ≡ p ∈ [1, ∞]: np is block ﬁnitely representable in (ei ) for all n ∈ N is a closed non-empty subset of [1, ∞]. Since K(fi ) ⊆ K(gi ) if (fi ) and (gi ) are block ∞ bases of (ei ) with (fi )∞ n a block basis of (gi )1 for some n it follows that one can ﬁnd a block basis (fi ) of (ei ) with K(fi ) = K(gi ) for all block bases (gi ) of (fi ). In other words, the set of Krivine p’s can be stabilized by passing to a block basis. (a) We may choose a basic sequence (ei ) in X so that the Krivine p’s are stabilized. First we note that K(ei ) = {p} for some unique p. Indeed if there exist p < q in K(ei ) then as in the proof of Theorem 12 one obtains that the class of norms ( · )n∈N arbitrarily distort [(ei )] where xn = sup

n

1/p Ei xp

: E1 < · · · < En .

i=1

Secondly using that X is of λ-bounded distortion one obtains, using the same family of norms, that there exists C = C(λ) and a block basis (xi ) of (ei ) so that if (yi )n1 is admissible w.r.t. (xi ) then n n 1/p C yi yi . 1

1

It remains to produce uniform asymptotic upper p estimates. To accomplish this one passes to another normalized block basis, (yi ) which has essentially stabilized asymptotic upper p estimates. Say n n 1/p p zi Cn zi 1

1

whenever yn z1 < · · · < zn w.r.t. (yi ) where Cn is the essentially smallest constant that can be achieved. If (Cn ) is unbounded choose m with Cm large. Choose N / m. Set B=

m i=1

zi /Cm : z1 < · · · < zm w.r.t. (yi ) and

m i=1

zi 1 p

Distortion and asymptotic structure

1351

and A∗N = N −1/q

N

!

" ∗

∗ xi∗ : x1∗ < · · · < xN are in the unit ball of yi

.

i=1

One shows (again by a similar argument) that |x ∗ (b)| C(λ)/Cm for x ∗ ∈ A∗N and b ∈ B. ∗ Also for all block subspaces of (yi ) one can ﬁnd an N p vector b on which x (b) > 1/2 ∗ ∗ ∗ ∗ ∗ say for some x ∈ AN . It follows that the norms |x|N = sup{|x (x)|: x ∈ AN } arbitrarily distort [(yi )]. (b) The technique used to show that X must contain an unconditional basic sequence is indirect. One shows that X contains basic sequences of unbounded order in terms of their unconditional structure. Given K < ∞ let T (X, K) be the set of all normalized ﬁnite basic sequences (xi )n1 ⊆ X which are K-unconditional. Then T (X, K) is naturally a tree under (xi )n1 (yi )m 1 if n m and xi = yi for i n. If X does not contain an unconditional basic sequence then T (X, K) is well founded (the tree has no inﬁnite branches). Set T (0) = T (X, K), T (α+1) = {(xi )n1 : there exists xn+1 so that (xi )n+1 ∈ T (α) } and 1 (β) (α) T = α<β T if β is a limit ordinal. Since X is separable and T (X, K) is closed in the product topology, it follows that o(T ) ≡ inf{α < w1 : T (α) = φ} < w1 (see [9]). One way to prove (b) is to use (a) to produce for a certain K = K(λ) for all α < w1 a normalized basic sequence (xiα ) so that the tree of subsets of N, {F ⊆ N: (xiα )i∈F is K-unconditional}, has order wα (see [37]; for a direct proof avoiding the use of (a) but following the same tree complexity idea see the original proof [53]). The order for this tree of subsets is by extension. The Schreier classes (Sα )α<w1 [1] are deﬁned as follows: S0 = {n}: n ∈ N ∪ φ, n Sα+1 = Ei : n ∈ N, {n} E1 < · · · < En and Ei ∈ Sα for i n . 1

If β is a limit ordinal choose βn ↑ β and set Sβ = {E: for some n ∈ N, {n} E ∈ Sβn }. (Ei )n1 is α-admissible if E1 < · · · < En and (min Ei )ni=1 ∈ Sα . It is easy to see that Sα is a well founded tree of sets with o(Sα ) = wα . (xi ) is said to be α-unconditional with constant C if (xi )i∈F is C-unconditional for all F ∈ Sα . Let (xi0 )∞ i=1 be an asymptotic p basis in X (here we have used a)) for some p. Assume α (xi ) has been chosen to be α-unconditional. Deﬁne an equivalent norm on xiα by |x| = n sup{ i=1 ±Ei x: (Ei )n1 is (α + 1)-admissible and the Ei ’s are intervals of integers}. | · | is an equivalent norm since (xiα ) isasymptotic p . One shows that if (Ei )n1 is an α-admissible family of intervals then | n1 ±Ei x| 4|x| for x ∈ xiα . This requires a combinatorial argument given in [53]. Then, using that X is of λ-bounded distortion it follows that there exists a block basis (xiα+1 ) of (xi ) so that (xiα+1 ) is (α +1)-unconditional with constant K(λ). A diagonal argument is used at limit ordinals. Maurey [28] has extended Theorem 17. He has shown that the argument used to biorthogonally distort p (1 < p < ∞) can be generalized to the case of an asymptotic p

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space not containing n1 ’s uniformly with an unconditional basis. Thus one obtains, since every X must contain either an arbitrarily distortable subspace or a subspace of bounded distortion, T HEOREM 25 ([28,53]). If X does not contain n1 ’s uniformly then X contains an arbitrarily distortable subspace. There do exist asymptotic 1 spaces which are arbitrarily distortable (and even asymptotic 1 spaces not containing an unconditional basic sequence) [5]. These are certain mixed Tsirelson spaces. For example (see [2]), the space X = T (Sn , 1/(n + 1))n∈N whose norm is given by the implicit equation: for x ∈ c00 , k 1 k Ei x: (Ei )1 is Sn -admissible x = max x∞ , sup sup n+1 n1

i=1

is arbitrarily distortable. It is easy to see that T = T (Sn , 2−n )n∈N is also a mixed Tsirelson space. One has [2] that 1/n if X = T (Sn , On )n∈N with O ≡ lim On and On /O n → 0 then X is arbitrarily distortable. These results present evidence why T is a prime candidate for a space of bounded distortion. Further evidence is given in [45] and [46]. Maurey [29] has shown that the sets (An ) which yield a biorthogonal distortion of 2 can in addition to the properties of being unconditional and spreading can be taken to be symmetric (x = (x(i)) ∈ An iff (xπ(i)) ∈ An for all permutations π of N. TomczakJaegermann [52] has shown that the Schatten classes Cp are arbitrarily distortable for 1 < p < ∞.

2. Asymptotic structure In Section 1 we saw that a Banach space X need not contain c0 or p and Lipschitz functions on SX need not stabilize. Yet the ﬁnite-dimensional analogues are valid. One is left with the task of deﬁning a structural framework for an arbitrary X which bridges the ﬁniteand inﬁnite-dimensional structure. There are two such structures that have been deﬁned. One, mentioned in the ﬁrst section, is that of spreading models E or more generally spreading models over X, X ⊕ E. While extremely useful, spreading models have certain shortfalls. For example, the relationship is not transitive; if F is a spreading model of E and E is a spreading model of X then F need not be a spreading model of X [6]. Secondly spreading models do not satisfy our desire to ﬁnd an inﬁnite-dimensional extension of Krivine’s theorem; indeed there exists a space X so that no spreading model is isomorphic to c0 or p [40]. Perhaps the strongest stabilization result involving spreading models is the following result. The proof uses the second stabilization principle. T HEOREM 26 ([38]). Let (Fn ) be a sequence of ﬁnite-dimensional subspaces of a space X with dim Fn → ∞. There exist Gn ⊆ Fkn for some k1 < k2 < · · · with dim Gn → ∞ so

Distortion and asymptotic structure

1353

that all sequences (xn ) with xn ∈ SGn have the same spreading model E = [(ei )] over X. In particular (ei ) is 1-unconditional over X. Moreover Gn can be chosen so that for all ε > 0, k ∈ N and x ∈ X there exists k0 ∈ N so that if k0 n1 < n2 < · · · < nk then k k αi ei − x + αi xi < ε x + i=1

i=1

whenever xi ∈ SGni and |αi | 1 for i k. Recently a second ﬁnite- inﬁnite-dimensional bridge structure was deﬁned ([34,30]). We ﬁrst examine the simplest version of this notion. Suppose that X has a basis or more n generally an FDD (ﬁnite-dimensional decomposition) (Ei )∞ i=1 . For n ∈ N we say (xi )1 ∈ n {X, (Ei )}n , the n-th asymptotic structure of X w.r.t. (Ei ), if (xi )1 is a normalized basic sequence with the property that: ∀ε > 0 ∀k1 ∃y1 ∈ SEi ∞ ∀k2 ∃y2 ∈ SEi ∞ · · · ∀kn ∃yn ∈ SEi ∞ k k k 1

2

n

with (xi )n1 (1 + ε)-equivalent to (yi )n1 . Thus (xi )n1 can be found (up to ε) in X arbitrarily far out and arbitrarily separated w.r.t. n m (Ei ). Note that if (zi )m 1 is a normalized block basis of (xi )1 ∈ {X, (Ei )}n then (zi )1 ∈ {X, (Ei )}m . It follows by the existence of spreading models and Krivine’s theorem that there exists p ∈ [1, ∞] so that the unit vector basis of np belongs to {X, (Ei )}n for n ∈ N. P ROPOSITION 27. Let (Ei ) be an FDD for X and suppose that |{X, (Ei )}2 | = 1. Then there exists p ∈ [1, ∞] so that for all n if (xi )n1 ∈ {X, (Ei )}n then (xi )n1 is 1-equivalent to the unit vector basis of np . Moreover X contains almost isometric copies of p , if 1 p < ∞ or of c0 if p = ∞. The ﬁrst part is easy given our previous remarks. The “moreover” statement is also not difﬁcult to prove directly (see [30]) but will in fact follows from Theorem 30 below. Asymptotic structure can also be understood in terms of trees on X. Let Tk = {(n1 , . . . , nk ): ni ∈ N}. τ ∈ Tk (X, (Ei )) if τ = {(xn1 ,...,ni ): (n1 , . . . , ni ) ∈ Tk } ⊆ SX , j ∞ (xn )∞ n=1 is a block basis of (Ei ) and for every 1 j < k and (ni )1 ∈ Tk , (xn1 ,...,nj ,n )i=1 is a normalized block basis of (Ei ). We shall say that τ ∈ Tk (X, (Ei )) converges to (xi )k1 if (xi )k1 is a normalized basic sequence and for some εi ↓ 0 for all (n1 , . . . , nk ) ∈ Tk , (xn1 ,...,ni )ki=1 is (1 + εn1 )-equivalent to (xi )k1 . P ROPOSITION 28. Let (Ei ) be an FDD for X. Let k ∈ N. Then (xi )k1 ∈ {X, (Ei )}k iff there exists a tree τ ∈ Tk (X, (Ei )) which converges to (xi )k1 . The proposition follows easily from the relevant deﬁnitions. The asymptotic structure of X may also be characterized in terms of trees as follows.

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P ROPOSITION 29. {X, (Ei )}k is the smallest class C of normalized bases of length k having the property that if τ ∈ Tk (X, (Ei )) and ε > 0 then some branch of τ is (1 + ε)equivalent to a member of C. This follows from the fact that if τ ∈ Tk (X, (Ei )) then there exists a convergent subtree τ ⊆ τ , τ ∈ Tk (X, (Ei )). This latter fact can be proved using Ramsey theory (see [22]). Another interpretation of asymptotic structure is given by the next theorem. Recall that kn+1 for all n and some sequence of integers (Fi ) is a blocking of (Ei ) if Fn = Ei i=k n +1 0 = k1 < k2 < · · ·. (Fi ) is a skipped blocking of (Ei ) if there exist integers 1 p1 q1 < qn q1 + 1 < p2 q2 < q2 + 1 < p3 · · · so that Fn = Ei i=p for n ∈ N. n T HEOREM 30. Let εi ↓ 0. There exists a blocking (Hi ) of (Ei ) so that for all k if (Fi )k1 is k any skipped blocking of (Hi )∞ k and xi ∈ SFi for i k then (xi )1 is (1 + εk )-equivalent to an element of {X, (Ei )}k . P ROOF. By a diagonal argument it sufﬁces to produce for a ﬁxed k ∈ N and ε > 0 a blocking (Hi ) of (Ei ) so that any normalized block basis (xi )k1 relative to any skipped blocking (Fi )k1 of (Hi )∞ 2 is (1 + ε)-equivalent to an element of {X, (Ei )}k . By a standard compactness argument one need only show the validity of the following sentence. ∃N2 ∀x2 ∈ SEi ∞ · · · ∃Nk ∀xi ∈ SEi ∞ , (xi )k1 is (1 + ε)-equivalent to “∃N1 ∀x1 ∈ SEi ∞ N1 N2 Nk an element in {X, (Ei )}k .” If false by formally negating the sentence one easily constructs a tree τ ∈ Tk (X, (Ei )) so that no branch of τ is (1 + ε)-equivalent to any element of {X, (Ei )}k . Proposition 29 then yields a contradiction. Let (Ei ) be an FDD for X. We shall say that X is asymptotic p w.r.t. (Ei ) if there exists K < ∞ so that for all k and (xi )k1 ∈ {X, (Ei )}k , d(xi k1 , kp ) K. This is formally weaker than assuming (xi )k1 is K-equivalent to the unit vector basis of kp but as observed in [30] the weaker assumption implies the stronger at least if 1 < p < ∞ (the case p = 1 or ∞ remains open). T HEOREM 31. If X is asymptotic p w.r.t. the FDD (Ei ) with 1 < p < ∞ then there exists K < ∞ so that for all k, (xi )k1 is K-equivalent to the unit vector basis of kp for all (xi )k1 ∈ {X, (Ei )}k . Also one has a nice duality result. Let us note that the asymptotic structure we have discussed w.r.t. an FDD can be and is indeed done in a more general context in [30], e.g., with respect to fundamental, total minimal systems and the next theorem is valid in that broader context. T HEOREM 32. Let 1 p ∞ and let X be a reﬂexive asymptotic p space w.r.t. the FDD(Ei ). Then X∗ is asymptotic q w.r.t. (Ei∗ ) where 1/p + 1/q = 1.

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One can also consider inﬁnite-dimensional spaces which reﬂect the asymptotic structure of X. A space Y with a normalized basis (yi ) is an asymptotic version of X if (yi )n1 ∈ {X, (Ei )}n for all n. This includes the class of all spreading models of normalized block bases of (Ei ). Moreover one can show [30] that there exists such a Y which satisﬁes {Y, (yi )}k = {X, (Ei )}k for all k ∈ N. (Y is called a universal asymptotic version of X.) The asymptotic structure of a space can be stabilized. P ROPOSITION 33. Let (Ei ) be an FDD for X. There exists a normalized block basis (yi ) of (Ei ) so that if Y = [(yi )] and Z = [(Hi )] where (Hi ) is any FDD obtained by blocking a block basis of (yi ) then for all k Y, (yi ) k = Z, (Hi ) k . One may ask, how small must this stabilized asymptotic structure be? The answer is not very. T HEOREM 34 ([43]). There exists a normalized monotone basis (ei ) for a reﬂexive space X with the following property. For all k, all normalized monotone bases (xi )k1 and all Y = [(yi )] where (yi )∞ 1 is a block basis of (ei ), (xi )k1 ∈ Y, (yi ) k . In particular X cannot contain an unconditional basic sequence. The example is technically difﬁcult. We will not present the argument but shall present the norm. This gives the ﬂavor of both the construction and of the possibilities afforded by generalizations of conditional Tsirelson-type norms. It is worth noting that a somewhat simpler example is given in [40]. In this case the basis (ei ) is unconditional and the unit vector basis of kp belongs to {X, (ei )}k for all k and p ∈ [1, ∞]. H ⊆ c00 ∩ B∞ is taken to be a countable set of non-zero vectors with three properties: (i) H is dense in B∞ ∩ c00 w.r.t. · 1 . (ii) ∀a ∈ H and intervals I of integers, Ia ∈ H if I a = 0. n 1 n 1 (iii) If a1 < · · · < an in H then ni=1 ai , f (n) i=1 ai and n i=1 ai all belong to H . ∞ A subsequence M = (Mn )n=1 ⊆ N is taken with M1 = 2 and we let σ : (a1 , . . . , an ): n ∈ N, a1 < · · · < an , ai ∈ H for i n → N be an injection satisfying 4 more properties. (iv) If a1 < · · · < an belong to H and I is an interval in N and [j1 , j2 ] = {i: I ai = 0, i = n} = ∅, then σ (I aj1 , . . . , I aj2 ) σ (a1 , . . . , an ). (v) If a1 < · · · < an belong to H then max supp an1< σ (a1 , . . . , an ). < ∞ where as before f (n) = (vi) If Im (σ ) is the range of σ then n∈Im (σ ) f (n) log2 (n + 1).

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¯¯ are the predecessor and successor of m in Im (σ ) then for (vii) If m ∈ Im (σ ) and m, ¯ m ¯¯ ∞) ∈ [1, m] ¯ ∪ [m, ' 3 <m 1 f (m) min(, m) f (m) f () , + + 3f (m) 2 m f () m f () m , > m. &

If · ∈ N and X = (c00 , · ) for m 2 we set m 1 ∗ ∗ ∗ ∗ = ai : ai ∈ H ∩ BX∗ for i m and a1 < · · · < am , m i=1 X ∞ X X m A = AX A AX for m 2 and AX = A . m, m= n m=2

AX m

m2

nm

m ∗ X ={ 1 ∗ ∗ X X Let for m 2, Bm i=1 ai : (a1 , . . . , am ) ⊆ A is (A , M, σ )-admisf (m) sible}. ∗ ) is ( AX , M, σ )-admissible means that a1∗ < The statement that (a1∗ , . . . , am ∗ X ∗ ∗ X ∗ · · · < am , a1 ∈ iMm Ai and ai+1 ∈ A σ (a1 ,...,ai∗ ) for 1 i < m. We take ∞ X X X X ∞ B = n=2 Bn and B = (Bn )n=2 . For m 1 set X Cm

=

m X

1 ∗ ∗ ∗ X ai : a1 , . . . , am ⊆ B is B , M, σ -admissible and f (m) i=1

CX =

∞

CnX .

n=1

Then one uses Proposition 11 to show that there exists · ∈ N so that X = (c00 , · ) satisﬁes for all x ∈ c00 , (viii) x = max(x∞ , sup{|a ∗ (x)|: a ∗ ∈ C X }. This is the space which yields Theorem 34. Asymptotic structure has been generalized in several ways. Milman and Wagner have extended the notion to operators [35]. Also Wagner [55] has given a higher-order ordinal notion in terms of certain α-games for α < w1 . The deﬁnition of {X, (Ei )}k yields this game for α = k. And of course one need not assume that the space X has an FDD. Indeed [30] consider the broader forum where the tail spaces of an FDD are replaced by ﬁnite codimensional subspaces in any non-trivial ﬁltration on X. Γ ⊆ cof(X), the set of all ﬁnite codimensional subspaces of X, is called a ﬁltration on X if for all Y, Z ∈ Γ there exists W ∈ Γ with W ⊆ Y ∩ Z. One then has (ei )k1 ∈ {X, Γ }k if ∀ε > 0 ∀Y1 ∈ Γ ∃y1 ∈ SY1 · · · ∀Yk ∈ Γ ∃yk ∈ SY1 so that (yi )k1 is the (1 + ε)-equivalent to (ei )k1 .

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As noted in [30] this can be expressed in terms of a game where Player I chooses Y ∈ Γ and Player II chooses y ∈ SY with each player making k alternate plays starting with I. Thus (ei )k1 ∈ {X, Γ }k iff for all ε > 0 Player II has a winning strategy for the set of all normalized bases (1 + ε)-equivalent to (ei )k1 . By regarding X ⊆ [En ], some FDD, and Γ = {X ∩ [Ei ]∞ n : n ∈ N} one obtains a relativized notion of asymptotic structure w.r.t. an FDD and the relativized versions of the previous structural results remain valid [22]. Working with this Γ is nice because one has a coordinate system. What happens however for Γ = cof(X)? And is there an inﬁnite version of asymptotic structure? These are addressed in [44] and we now discuss some of the results contained therein. We consider the two-player game where Player I chooses Y1 ∈ cof(X) and then Player II chooses y1 ∈ SY1 and then Player I chooses Y2 ∈ cof(X) and so on. Player I wins the ω ≡ {(x )∞ : x ∈ S for all i} if (y ) ∈ A. One can deﬁne in A-game for a given A ⊆ SX i 1 i X i a natural way what it means to say that Player I has a winning strategy for A and we denote this by WI (A). For ε > 0 we let Aε = {(yi ) ⊆ SX : there exists (xi ) ∈ A with ω , given the xi − yi < ε/2i for all i ∈ N} and we let Aε be the closure of this set in SX product topology of the discrete topology on SX . ω . There exists a space Z with an FDD (E ) so that X ⊆ Z and T HEOREM 35. Let A ⊆ SX i such that the following are equivalent. (a) For all ε > 0, (WI (Aε )). ω (b) For all ε > 0 there exists a blocking (Gi ) of (Ei ) and δi ↓ 0 so that: if (xn ) ∈ SX and there exist integers 1 = k0 < k1 < · · · with

Id −P kn −1 (xn ) < δn [G ] j j=k n−1 +1

for all n then (xn ) ∈ Aε . Moreover if X∗ is separable, (En ) can be chosen to be shrinking and if X is reﬂexive, Z can be chosen to be reﬂexive. In these cases (a) is equivalent to (c) Every weakly null tree T ∈ Tω (X) has a branch in Aε . The hypothesis on T in (c) means that T = (x(n1 ,...,nk ) : n1 < · · · < nk are positive integers) ⊆ SX and the successors of every node, including φ, form a weakly null sequence. ω : (x ) is K-equivalent to the unit vector This theorem can be applied to A = {(xi ) ∈ SX i basis of p } to yield the following.

T HEOREM 36. Let X be reﬂexive and let 1 0 there exists a ﬁnite codimensional subspace X0 of X which (C 2 + ε)-embeds into ( Fi )p for some sequence (Fi ) of ﬁnite-dimensional spaces. This theorem generalizes results of [21]. A similar theorem for the p = ∞ case was proved by Kalton [20]: if X does not contain 1 and for some K < ∞ every weakly null tree T ∈ Tω (X) admits a branch K-equivalent to the unit vector basis of c0 , then X embeds into c0 .

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The proofs of these theorems use Ramsey theory and Martin’s theorem that Borel games are determined [27]. Unlike the ﬁnite asymptotic structure case there is in general no smallest closed set A of normalized bases so that every weakly null tree T ∈ Tω (X) has a branch nearly in A. So there is no unique notion of inﬁnite asymptotic structure, but Theorem 35 does allow one to say something useful. Finally what can be said if the asymptotic structure is as small as possible, either in the [30] sense or in the sense of spreading models? Proposition 28 yielded some information but using the above results one can say more. Recall that {X, cof(X)}2 denotes the asymptotic structure (of length 2) w.r.t. ﬁltration of all ﬁnite codimensional subspaces of X. T HEOREM 37. Let X be reﬂexive and let |{X, cof(X)}2 | = 1. Then there exists 1 0, some ﬁnite codimensional subspace of X (1 + ε)-embeds into the p sum of ﬁnite-dimensional spaces. If X has a basis (xi ) and there is only one spreading model (ei ) that can be obtained as a spreading model of a block basis of (ei ) then, by the proof of Krivine’s theorem, one obtains that (ei ) is 1-equivalent to the unit vector basis of c0 or p for some 1 p < ∞. T HEOREM 38. Let (xi ) be a basis for X and assume that all spreading models of a normalized block basis of (xi ) are 1-equivalent to the unit vector basis of 1 (respectively, c0 ). Then X contains an isomorph of 1 (respectively, c0 ). It is still open if the theorem extends to p (1 < p < ∞). There is no isomorphic version of this theorem. For example, all spreading models of T are 2-equivalent to the unit vector basis of 1 yet T does not contain 1 . Most recently a third notion has been constructed which generalizes spreading models [17]. Spreading models are generated by basic sequences. Suppose that (xin )n,i∈N is a normalized array in a Banach space X so that for some K < ∞ each row (xin )i∈N is j K-basic and for all n i1 < · · · < in , (xij )nj=1 is K-basic. Then given ε ↓ 0 one can pron for some p(1) < p(2) < · · ·, so that for all n, duce a subarray (yin ), given by yin = xp(i) n i1 < · · · < in and n 1 < · · · < n n n j j aj yij − aj yj < εn , j =1

j =1

if (aj )n1 ⊆ [−1, 1]. The proof uses Ramsey’s theorem. It follows that n n j aj xij ≡ a j ej lim · · · lim i1 →∞ in →∞ j =1

j =1

exists. (ej ) is called an asymptotic model of X. We refer the reader to [17] for a discussion on asymptotic models.

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References [1] D. Alspach and S. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992), 1–44. [2] G. Androulakis and E. Odell, Distorting mixed Tsirelson spaces, Israel J. Math. 109 (1999), 125–149. [3] G. Androulakis and Th. Schlumprecht, The Banach space S is complementably minimal and subsequentially prime, Preprint. [4] G. Androulakis and Th. Schlumprecht, On the subsymmetric sequences in S, Preprint. [5] S. Argyros and I. Deliyanni, Examples of asymptotically 1 Banach spaces, Trans. Amer. Math. Soc. 349 (1997), 973–995. [6] B. Beauzamy and J.-T. Lapresté, Modèles Étalés des Espace de Banach, Travaux en Cours, Herman, Paris (1984). [7] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Amer. Math. Soc. Colloq. Publ. 48 (2000). [8] J. Bourgain, The Szlenk index and operators on C(K)-spaces, Bull. Soc. Math. Belg. Sér. B 31 (1979), 87–117. [9] J. Bourgain, On convergent sequences of continuous functions, Bull. Soc. Math. Belg. 32 (1980), 235–249. [10] J. Bourgain, On ﬁnite-dimensional homogeneous Banach spaces, GAFA Israel Seminar 1986–97, J. Lindenstrauss and V. Milman, eds, Lecture Notes in Math. 1317, Springer (1988), 232–239. [11] F. Chaatit, On the uniform homeomorphisms of the unit spheres of certain Banach lattices, Paciﬁc J. Math. 168 (1995), 11–31. [12] P. Enﬂo, On a problem of Smirnov, Ark. Mat. 8 (1969), 107–109. [13] T. Figiel and W.B. Johnson, A uniformly convex Banach space which contains no p , Compositio Math. 29 (1974), 179–190. [14] T.A. Gillespie, Factorization in Banach function spaces, Indag. Math. 43 (1981), 287–300. [15] W.T. Gowers, Lipschitz functions on classical spaces, European J. Combin. 13 (1992), 141–151. [16] W.T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851–874. [17] L. Halbeisen and E. Odell, Asymptotic models in Banach spaces, Preprint. [18] R. Haydon, E. Odell, H. Rosenthal and Th. Schlumprecht, On distorted norms in Banach spaces and the existence of p types, Unpublished manuscript. [19] R.C. James, Uniformly nonsquare Banach spaces, Ann. of Math. (2) 80 (1964), 542–550. [20] N.J. Kalton, On subspaces of c0 and extensions of operators into C(K)-spaces, Oxford Quart. J. 52 (2001), 313–328. [21] N.J. Kalton and D. Werner, Property (M), M-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137–178. [22] H. Knaust, E. Odell and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach space, Positivity 3 (1999), 173–199. [23] J.L. Krivine, Sous espaces de dimension ﬁnie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976), 1–29. [24] H. Lemberg, Nouvelle démonstration d’un théorème de J.L. Krivine sur la ﬁnie représentation de p dans un espaces de Banach, Israel J. Math. 39 (1981), 341–348. [25] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, New York (1977). [26] G.Ya. Lozanovsky, On some Banach lattices, Siberian Math. J. 10 (1969), 584–599. [27] D.A. Martin, Borel determinancy, Ann. of Math. 102 (1975), 363–371. [28] B. Maurey, A remark about distortion, Oper. Theory Adv. Appl. 77 (1995), 131–142. [29] B. Maurey, Symmetric distortion in 2 , Oper. Theory Adv. Appl. 77 (1995), 143–147. [30] B. Maurey, V.D. Milman and N. Tomczak-Jaegermann, Asymptotic inﬁnite-dimensional theory of Banach spaces, Oper. Theory Adv. Appl. 77 (1994), 149–175. [31] B. Maurey and H. Rosenthal, Normalized weakly null sequences with no unconditional subsequences, Studia Math. 61 (1971), 77–98. [32] V.D. Milman, Geometric theory of Banach spaces II, geometry of the unit sphere, Russian Math. Surveys 26 (1971), 79–163.

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[33] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, Berlin (1986). [34] V.D. Milman and N. Tomczak-Jaegermann, Asymptotic p spaces and bounded distortions, Contemp. Math. 144 (1993), 173–195. [35] V.D. Milman and R. Wagner, Asymptotic versions of operators and operator ideals, Convex Geometric Analysis (Berkeley, CA, 1996), Cambridge Univ. Press (1999), 165–169. [36] E. Odell, On Schreier unconditional sequences, Contemp. Math. 144 (1993), 197–201. [37] E. Odell, On subspaces, asymptotic structure and distortions of Banach spaces; connections with logic, Analysis and Logic, C. Finet and C. Michaux, eds, London Math. Soc. Lecture Notes Ser. 262, to appear. [38] E. Odell, H. Rosenthal and Th. Schlumprecht, On weakly null FDD’s in Banach spaces, Israel J. Math. 84 (1993), 333–351. [39] E. Odell and Th. Schlumprecht, The distortion problem, Acta Math. 173 (1994), 259–281. [40] E. Odell and Th. Schlumprecht, On the richness of the set of p’s in Krivine’s theorem, Oper. Theory Adv. Appl. 77 (1995), 177–198. [41] E. Odell and Th. Schlumprecht, Asymptotic properties of Banach spaces under renormings, J. Amer. Math. Soc. 11 (1998), 175–188. [42] E. Odell and Th. Schlumprecht, A problem on spreading models, J. Funct. Anal. 153 (1998), 249–261. [43] E. Odell and Th. Schlumprecht, A Banach space block ﬁnitely universal for monotone bases, Trans. Amer. Math. Soc. 352 (4) (1999), 1859–1888. [44] E. Odell and Th. Schlumprecht, Trees and branches in Banach spaces, Trans. Amer. Math. Soc. 354 (10) (2002), 4085–4108. [45] E. Odell and N. Tomczak-Jaegermann, On certain equivalent norms on Tsirelson’s space, Illinois J. Math. 44 (2000), 51–71. [46] E. Odell, N. Tomczak-Jaegermann and R. Wagner, Proximity to 1 and distortion in asymptotic 1 spaces, J. Funct. Anal. 150 (1997), 101–145. [47] M. Ribe, Existence of separable uniformly homeomorphic non isomorphic Banach spaces, Israel J. Math. 48 (1984), 139–147. [48] H. Rosenthal, Some remarks concerning unconditional basic sequences, Longhorn Notes: Texas Functional Analysis Seminar 1982–83, University of Texas, Austin, 15–48. [49] H. Rosenthal, Double dual types and the Maurey characterization of Banach spaces containing 1 , Longhorn Notes: Texas Functional Analysis Seminar 1983–84, University of Texas, Austin, 1–37. [50] Th. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1991), 81–95. [51] Th. Schlumprecht, A complementably minimal Banach space not containing c0 or p , Seminar Notes in Functional Analysis and Partial Differential Equations, Baton Rouge, Louisiana (1992). [52] N. Tomczak-Jaegermann, Distortions on Schatten classes Cp , Oper. Theory Adv. Appl. 77 (1995), 327– 334. [53] N. Tomczak-Jaegermann, Banach spaces of type p have arbitrarily distortable subspaces, Geom. Funct. Anal. 6 (1996), 1074–1082. [54] B.S. Tsirelson, Not every Banach space contains p or c0 , Functional Anal. Appl. 8 (1974), 138–141. [55] R. Wagner, Finite higher-order games and an inductive approach towards Gowers’ dichotomy, Ann. Pure Appl. Logic 111 (2001), 39–60.

CHAPTER 32

Sobolev Spaces∗ Aleksander Pełczy´nski and Michał Wojciechowski ´ Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, P.O. Box 21, 00-956 Warsaw, Poland E-mail: [email protected]; [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Classical Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The canonical embedding and the Sobolev projection . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Isomorphisms of Sobolev spaces. Linear extension theorems . . . . . . . . . . . . . . . . . . . . . . 4. Non-isomorphism of non-reﬂexive Sobolev spaces of several variables with classical Banach spaces 5. Properties of C(Q) spaces shared by C (k) (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Embedding theorems of Sobolev type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Interpolation in Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Anisotropic Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Partially supported by the KBN Grant 2 P03A 03614.

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Sobolev spaces

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Introduction Sobolev spaces were introduced by S.L. Sobolev in the late thirties of the 20th century. They and their relatives play an important role in various branches of mathematics: partial differential equations, potential theory, differential geometry, approximation theory, analysis on Euclidean spaces and on Lie groups. For comprehensive information the reader might consult the monographs [62,90–92,30,7,95]. The limited volume of this survey forced us to restrict ourselves to the simplest Sobolev spaces. We do not discuss the important relatives of classical Sobolev spaces like weighted Sobolev spaces, Besov spaces, spaces of Bessel potentials, Sobolev–Liouville spaces, Sobolev spaces of functions with fractional derivatives, etc. The reader is referred to the monographs mentioned above and the monographs [94,1] and [70] for study of these objects. Sobolev spaces are Banach spaces of smooth functions of one and several variables with conditions imposed on a few ﬁrst (distributional) partial derivatives. In the classical case one requires that the derivatives up to a prescribed order belong to some Lp -space. The aim of this survey is to review the results on Banach space isomorphic properties of the simplest Sobolev spaces. Fixing the order of derivatives and a domain Ω on which the functions are deﬁned, for varying p we get the scale of Sobolev spaces. The important feature of this scale is that it naturally embeds into the scale Lp (Ω, E) of E-valued functions where E is a ﬁnite-dimensional Hilbert space. This enables to show that under mild conditions on Ω the Sobolev spaces in question are isomorphic as Banach spaces to classical Lp -spaces for 1 < p < ∞. The situation is different in the limit case p = 1 and p = ∞ for functions in two and more variables. These Sobolev spaces are not isomorphic to classical Banach spaces L1 and C, although they share some properties of the corresponding classical spaces. The proofs of these facts require various analytic tools like theory of Fourier multipliers, Sobolev embedding theorem, Marcinkiewicz interpolation theorem, etc. This is the obstacle which we had to overcome while writing the survey to make it accessible for a reader less familiar with hard analysis. The survey consists of 8 sections. Section 1 contains basic deﬁnitions. In Section 2 we study the embedding of a given scale of Sobolev spaces into the scale of ﬁnite-dimensional vector-valued Lp -spaces. We analyze analytic properties of the orthogonal projection onto the image of the embedding (for p = 2). We establish an explicit isomorphism of Sobolev spaces on Rn with Lp -spaces for 1 < p < ∞. In Section 3 we discuss isomorphisms of Sobolev spaces deﬁned on open subsets of Rn . Under mild conditions of regularity of the boundary of the domain, for ﬁxed p and ﬁxed order of derivatives, the topological dimension of the domain (= the number of variables) sufﬁces for the isomorphism of Sobolev spaces in question. The pioneering result in this direction is Mityagin’s theorem on the isomorphisms of spaces of k-times continuously differentiable functions. The result of this section depends on the existence of linear extension operators. In Section 4 we present “negative” results that Sobolev spaces in L1 -norm and L∞ -norm are not isomorphic to corresponding classical spaces. The results are based on the idea of S.V. Kislyakov of employing the Sobolev embedding theorem. The Sobolev spaces in L∞ -norm and the spaces of

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k-times continuously differentiable functions although non-isomorphic to spaces of continuous functions share various important properties of the latter spaces. These results mostly due to J. Bourgain are discussed in Section 5. Section 6 is devoted to the Sobolev embedding theorem and its improvement (like embeddings into Lorentz and Besov spaces). Section 7 concerns with interpolation properties of scales of Sobolev spaces. Section 8 reﬂects some research interest of the authors. We deal with non-classical anisotropic Sobolev spaces. 1. Classical Sobolev spaces ∂ |α| denote the partial derivative corresponding to the multi-index α α ∂x1 1 ∂x2 2 ...∂xnαn n n α = (αj )j =1 ∈ Z+ where Z+ := N ∪ {0}; here |α| = nj=1 αj is the order of the derivative * α ∂ α ; we use the convention x α := nj=1 xj j . Deﬁne α 0 β ≡def αj βj for j = 1, 2, . . . , n. If Ω is an open set in Rn then Ω denotes the closure of Ω and bd Ω = Ω \ Ω denotes the

Let ∂ α =

boundary of Ω. The set of scalars is denoted by K, it is, either R – the real numbers, or C – the complex numbers. Let D(Ω) be the space of all inﬁnitely many times differentiable scalar-valued functions f : Ω → K with compact support, supp f = {x: f (x) = 0} ⊂ Ω. A function g : Ω → K is said to be the α-th distributional partial derivative of an f : Ω → K, in symbol g = D αf , provided |α| gφ dx = (−1) f ∂ α φ dx for φ ∈ D(Ω). Ω

Ω

Here and in the sequel . . . dx denotes the integration against the n-dimensional Lebesgue measure λn ; by Lp (Ω) we denote Lp (Ω, λn ). If a partial derivative of f is continuous on Ω then the corresponding distributional derivative of f coincides with the partial derivative. We admit that the (distributional) derivative of order 0 of a function f coincides with f . Let 1 p ∞ and k = 0, 1, . . . . Let us put p L(k) (Ω) = f : Ω → K: D αf exists and D αf ∈ Lp (Ω) for |α| k . p

We equip L(k) (Ω) with the norm f Ω,(k),p =

α

D f (x)p dx 1/p , max|α|k essupx∈Ω D αf (x), |α|k Ω

for 1 p < ∞, for p = ∞.

By C0(k) (Ω) we denote the closure of D(Ω) in the norm · Ω,(k),∞ , and by C (k) (Ω) a subspace of L∞ (k) (Ω) consisting of functions which together with their partial derivatives of orders k are uniformly continuous and vanishes at inﬁnity (for unbounded Ω). Clearly C0(k) (Ω) ⊆ C (k) (Ω) ⊂ L∞ (k) (Ω).

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WARNING . Usually spaces in sup norms are deﬁned on closed subsets of Rn . However the Sobolev spaces in Lp -norms are naturally deﬁned on open subsets of Rn . To unify domains of functions we work with, we have deﬁned C (k) (Ω) as spaces of uniformly continuous functions on open Ω; the functions uniquely extend on the closure of Ω. On the other hand we work primarily with separable spaces thus we add the condition of vanishing at inﬁnity. This condition is meaningful only for unbounded domains. Thus in our (k) notation C (k) (Rn ) = C0 (Rn ) and C (k) (I n ) = C (k) (I n ). Here and in the sequel I n = (1/2, n n 1/2) ⊂ R . p

(k)

The spaces L(k) (Ω) (1 p ∞), C0 (Ω), C (k) (Ω) for k = 1, 2, . . . are called classical Sobolev spaces. A routine argument gives: P ROPOSITION 1. (i) The classical Sobolev spaces are Banach spaces; p (k) (ii) C0 (Ω), C (k) (Ω) and L(k) for 1 p < ∞ are separable. For regular domains Ω ⊂ Rn Sobolev spaces in Lp norms (1 p < ∞) can be deﬁned as completion of C ∞ -functions in the corresponding norms. Precisely we have p (k) P ROPOSITION 2. C ∞ (Ω) := ∞ k=1 C (Ω) is dense in L(k) (Ω) in the following cases n (i) Ω = R (cf. [88], Chapter V, §2, Proposition 1); p p (ii) there is a linear extension operator from L(k) (Ω) to L(k) (Rn ) which takes C ∞ (Ω) ∞ n into C (R ); (iii) Ω is bounded and has segment property, i.e., every x ∈ bd Ω has an open neighbourhood Ux in Rn and there exists a non-zero vector yx such that for every z ∈ Ω ∩ Ux one has z + tyx ∈ Ω for 0 < t < 1 (cf. [1], Chapter III, Theorem 3.18). R EMARKS . (1) In Proposition 2 one can replace C ∞ (Ω) by its subspace consisting with functions whose unique continuous extension to Ω has compact support. (2) A theorem of Stein (cf. [88], Chapter VI, §3, Theorem 5) provides for a large class p p of domains in Rn a construction of linear extension operators from L(k) (Ω) to L(k) (Rn ) taking C ∞ (Ω) into C ∞ (Rn ). (3) Proposition 2 extends also on bounded domains with so called cone property (cf. [62], §1.1.9 for deﬁnition, and §1.1.6 and §1.1.9 for the proof). (4) Proposition 2 extends to Sobolev spaces on compact manifolds. We outline brieﬂy how to deﬁne Sobolev spaces on manifolds. For simplicity let M be an n-dimensional compact Euclidean C k -manifold. Let (φj , Oj )j ∈A be a ﬁnite atlas of M into

consisting of open sets Oj and homeomorphisms φj : Oj −→ Rn such that Ωj := φj (Oj ) are open subsets of Rn whose closures are compact. Assume furthermore that φj φi−1 ∈ C (k) (Ωj ∩ Ωi )

((j, i) ∈ A × A).

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For 1 p ∞ we put p p L(k) (M) = f : M → K: f φj−1 ∈ L(k) (Ωj ) for j ∈ A , ⎧ −1 p

1/p ⎨ , for 1 p < ∞, j ∈A f φj Ωj ,(k),p f (φj ,Oj )j∈A ,(k),p = ⎩ max for p = ∞. j ∈A f Ωj ,(k),∞ , The deﬁnition of C (k) (M) is analogous. p Note that the topology of L(k) (M) does not depend of a particular choice of an atlas; for different atlases we get equivalent norms. In the case where M is a Lie group or a homogeneous subspace of a Lie group (we assume that the group operation is compatible with the differentiable structure of M, i.e., translation by element of the group are diffeomorp phisms) then the norm of L(k) can be naturally deﬁned in terms of the Haar measure of the group and partial derivatives deﬁned by elements of the Lie algebra of the group. This remark also applied to unimodular locally compact Lie groups. The most useful special models of Sobolev spaces on manifolds are the spaces on the groups Rn and the tori Tn , and the spaces on the Euclidean spheres Sn which are homogeneous spaces of the orthogonal groups. Since the spaces under consideration are translation invariant with respect to the group action, we can use powerful tools of Harmonic Analysis to study their structure. The torus Tn is usually identiﬁed with spaces Rn of 1-periodic functions with 1-periodic derivatives with respect to each variable. We end this section by introducing Sobolev spaces of measures – BV(k) (Ω). For a measure μ denote by v(μ) the positive measure being the total variation of μ (cf. [25], Vol. I, Chapter III, §1, Deﬁnition 1). M(Ω) stands for the Banach space of all scalar-valued Borel measures μ on Ω with bounded total variation with the norm μM(Ω) = v(μ)(Ω). A measure ν is said to be the distributional derivative of a measure μ corresponding to the multi-index α, in symbols D α μ = ν provided

∂ α ϕ dμ = (−1)α Ω

ϕ dν

for ϕ ∈ D(Ω).

Ω

We admit D 0 (μ) = μ. For an open Ω ⊂ Rn and for k = 1, 2, . . . by BV(k) (Ω) we denote the space of all μ ∈ BV (Ω) such that D α (μ) exists and belongs to M(Ω) for 0 |α| k, equipped with the norm D α μBV(k) (Ω) = . M(Ω) 0|α|k

The elements of BV(k) (Ω) can be regarded as functions on Ω. Precisely, it is not hard to show (cf., e.g., [81]) that if μ ∈ BVk (Ω) then μ and all the distributional derivatives D α (μ) for |α| < k are absolutely continuous with respect to the Lebesgue measure λn ; thus it is natural to identify them with functions in L1 (Ω). The space L1(k) (Ω) can be identiﬁed with the subspace of BV(k) (Ω) being the image of the isometric embedding f → f · λn

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1367

of L1(k) (Ω) into BV(k) (Ω). The space BV(1) (Ω) is often called the space of functions of bounded variation on Ω. For further information on spaces of functions with bounded variation the reader is referred to the books [27,62,104].

2. The canonical embedding and the Sobolev projection p

Given f ∈ L(k) (Ω) the tuple (D αf )|α|k is called a jet of f . A jet can be regarded as a vector-valued function from Ω into KK(n,k) . where K(n, k) is the number of partial derivatives of order k in n variables. Let |α|k Lp (Ω) be the product of K(n, k) copies of the space Lp (Ω) equipped with the norm (fα )p =

1/p p , |α|k fα Lp (Ω)

max|α|k fα L∞ (Ω) ,

for 1 p < ∞, for p = ∞.

. . The spaces |α|k C(Ω) and |α|k C0 (Ω) are deﬁned similarly. Clearly the space . p (Ω) is naturally isomorphic to the vector-valued Lp -space Lp (Ω; l 2 L |α|k K(n,k)). Sometimes it is more convenient to work with the second model. The canonical embedding is the map p

J = JΩ,(k),p : L(k) (Ω) →

/

Lp (Ω)

|α|k

deﬁned by

J (f ) = D αf |α|k

p

for f ∈ L(k) (Ω).

. Similarly one deﬁnes the canonical embedding J : BV(k) (Ω) → |α|k M(Ω). Clearly . p p |α|k L (Ω) is an L space on a measure space which is independent of p; the canonical embedding is an isometrically isomorphic embedding. Thus p

C OROLLARY 3. (a) If 1 < p < ∞ then the space L(k) (Ω) is superreﬂexive. (k) (k) (b) The spaces L1(k) (Ω), L∞ (k) (Ω), C (Ω), C0 (Ω), BV(k) (Ω) are not reﬂexive because they contain subspaces isomorphic either to l 1 or c0 spanned by functions with disjoint supports. (c) L∞ (k) (Ω) and BV(k) (Ω) are dual Banach spaces.

P ROOF. The statements (a) and (b) are obvious. We prove (c) for BV(k) (Ω) (cf. [80], p Proposition 6.1); the argument for L∞ (k) (Ω) is similar. Let lN for N = 1, 2, . . . denote the space of scalar-valued sequences x = (xj )N j =1 with the norm |x|p =

N 1

|xj |p

1/p

,

for 1 p < ∞,

max1j N |xj |, for p = ∞.

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1 and l ∞ are in duality, it follows from the Riesz representation theSince the spaces lN N 1 ) can be identiﬁed with the dual of the space orem that the vector-valued space M(Ω; lN ∞ ); the latter is the space of all continuous functions on Ω with values in l ∞ which C0 (Ω; lN N 1) vanish at inﬁnity and at the boundary of Ω. It is also convenient to identify M(Ω; lN ∞ (resp. C0 (Ω; lN )) with Cartesian products of N copies of M(Ω) (resp. C0 (Ω)) equipped with the suitable l 1 (resp. l ∞ ) norm. Now let N = K(k, n) be the number of multi-indices corresponding to all partial derivatives in n variables of order k. Let

J (μ) = D α μ 0|α|k

for μ ∈ BV(k) (Ω)

1 ). Then J (BV (Ω)) be an isometrically isomorphic embedding of BV(k) (Ω) into M(Ω; lN (k) 1 ∗ ∞ )-topology of is a subspace of M(Ω; lN ) which is closed in the w -topology (= C0 (Ω; lN 1 )). Indeed let F = (D αf ) M(Ω; lN m m 0|α|k for m = 1, 2, . . . and let F = (να )0|α|k . w∗

∞ ) we have Assume that Fm → F as m → +∞. Then for every N -tuple (f(α) ) ∈ C0 (Ω; lN

lim m

f(α) d D αfm =

0|α|k Ω

f(α) dνα .

(1)

0|α|k Ω

Fix a multi-index β with 0 |β| k and specify (f(α) )0|α|k by putting f(α) = 0 for α = β and f(β) = ϕ with ϕ ∈ D(Ω). The deﬁnition of distributional partial derivative combined with (1) gives

ϕ d D βfm

ϕ dνβ = lim m

Ω

Ω

= lim(−1)

|β|

m

∂ ϕ · fm dλn = (−1) β

Ω

|β|

∂ β ϕ dν(0,0,...,0) . Ω

Hence νβ = D β ν(0,0,...,0) for 0 |β| k. Thus (νβ )0|β|k ∈ J (BV(k) (Ω)). Therefore J (BV(k) (Ω)) is w∗ -closed. Thus J (BV(k) (Ω)) can be identiﬁed with the dual of the ∞ )/(BV (Ω)) , where quotient C0 (Ω; lN ⊥ (k)

∞ BV(k) (Ω) ⊥ = (fα )0|α|k ∈ C0 Ω, lN :

fα d D αf = 0 for f ∈ BV(k) (Ω) .

0|α|k Ω

The space !

.

2 |α|k L (Ω)

is a Hilbert space with the inner product deﬁned by

" (fα ), (gα ) := fα g¯α dx. |α|k Ω

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1369

Thus L2(k) (Ω) is a Hilbert space with the inner product f, g =

D αf D α g dx.

|α|k Ω

The orthogonal projection P = PΩ,(k) :

/

onto L2 (Ω) −→ J L2(k) (Ω)

|α|k

is called the Sobolev projection. For “nice” domains Ω the Sobolev projection has stronger analytic properties than . boundedness in |α|k L2 (Ω). In particular we have T HEOREM 4. Let n = 1, 2, . . . , k = 0, 1, . . . . Then (a) The Sobolev projection PRn ,(k) is of weak type (1, 1); (b) PRn ,(k) is of strong type (p, p) for 1 < p < ∞; (c) if either n = 1 or k = 0 then PRn ,(k) is of strong type (p, p) for 1 p ∞. Recall that a subadditive operator T whose domain contains a dense set Y ⊂ L1 (μ; E) ∩ (here E is a ﬁnite-dimensional Hilbert space) and whose range is contained in the space of μ measurable E-valued functions on some set is said to be of strong type (r, s) (resp. of weak type (r, s)) for some r, s with 0 r, s ∞ provided there is K ∈ (0, ∞) such that Tf s Kf r (resp. supc>0 μ({|Tf | c}) ( Kc f r )s ) for f ∈ Y . The l.u.b. of K satisfying the inequality in the parenthesis is called the weak type (r, s) constant of T . . Regarding |α|k L2 (Rn ) as vector-valued L2 (Rn ) we see that the shift operators induced by translations of Rn commute with PRn ,(k) . This allows to use the theory of Fourier multipliers in the proof of Theorem 4. Recall that a measurable m : Rn → C is a multiplier of weak type (r, s) (resp. of strong type (r, s)) provided that the operator Tm deﬁned by Tm (f ) = (mfˆ)∨ (for f in an appropriate subspace of functions on Rn ) is of weak type (r, s) (resp. strong type (r, s)). Here gˆ denotes the Fourier transform of g and g ∨ the inverse Fourier transform (cf. [42], Chapter 7). The operator Tm is called the multiplier transform of m. The Sobolev projection PRn ,(k) can be expressed by multiplier transforms of simple rational functions. Let S(Rn ) denote the Schwartz class of C ∞ -functions on Rn which together with all their derivatives are rapidly decreasing at inﬁnity (cf. [42], 7.1.2). Let . . n ) = {(f ) ∈ 2 (Rn ): f ∈ S(Rn ) for |α| k}. For α, β ∈ Zn with S(R L α α + |α|k |α|k |α| k, |β| k and for ξ ∈ Rn we put L∞ (μ; E)

Q(k) (ξ ) =

|α|k

ξ 2α ;

mα,β = i |α|−|β| ξ α ξ β Q−1 (k) (ξ );

Tα,β = Tmα,β .

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P ROPOSITION 5. One has

PRn ,(k) (F ) =

Tα,β (fβ )

|β|k

|α|k

for F = (fβ )|β|k ∈

/

S Rn .

(2)

|β|k

P ROOF. Denote by R(·) the right-hand side of (2). Taking into account that the Fourier transform and its inverse preserve S(Rn ) and the identities: ∨ fˆ = f ;

α f = i |α| ξ α fˆ ∂0

α ∈ Zn+ ; f ∈ S Rn ,

by simple algebraic calculation we verify for F ∈

.

|α|k S(R

n)

the identities:

!

" ! " R(F ), F = R(F ), R(F ) ; /

S Rn ∩ JRn ,(k) L2(k) Rn ; R(F ) ∈ |α|k

R(F ) = F

for F ∈ JRn ,(k) L2(k) Rn .

Thus R(F ) = PRn ,(k) (F ) for F ∈

.

|α|k S(R

n ).

P ROOF OF T HEOREM 4. (a) Since the Schwartz class is dense in L1 (Rn ), it follows from Proposition 5 that PRn ,(k) is of weak type (1, 1) iff all Tα,β have the same property, equivalently iff all mα,β are of weak type (1, 1) multipliers. One gets the latter property of mα,β routinely verifying that each mα,β satisﬁes the classical Hörmander–Mikhlin criterion (cf. [42], 7.9.5). (b) Since PRn ,(k) is of strong type (2, 2) and is selfadjoint, (b) follows by combining (a) with the Marcinkiewicz interpolation theorem (cf. [107], XII (4.6)). (c) Case k = 0 is trivial. If n = 1 then the functions mα,β are the Fourier transforms of linear combinations of the delta functions and functions from L1 (R) (cf. [75] for details). Thus the corresponding multiplier transforms are convolutions with these linear combinations. Hence they are of strong types (1, 1) and (∞, ∞). Combining Theorem 4 with an easy fact that a Sobolev space in Lp -norm contains a complemented subspace isomorphic to Lp (R) and using the decomposition method one can prove that whenever the Sobolev projection is of strong type (p, p) then the Sobolev space in question is isomorphic to Lp . However using Fourier multipliers we can construct an explicit isomorphism. T HEOREM 6. The multiplier transform T1/√Q(k) extends to an isomorphism from Lp (Rn ) p

onto L(k) (Rn ) for 1 < p < ∞ (k = 0, 1, . . . , n = 1, 2, . . .).

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1371

P ROOF. We use the following easy consequence of the Marcinkiewicz multidimensional multiplier theorem (cf. [90], Chapter IV, §6, Theorem 6) If |α| k then ξ α [Q(k) ]−1/2 is a strong type (p, p) multiplier on Rn for 1 0 does not depend on f . Since S(Rn ) is dense in L(k) (Rn ), Tm p has the unique extension to a bounded operator from Lp (Rn ) into L(k) (Rn ). 2α Conversely, the identity m−1 = implies Tm−1 (f ) = |α|k ξ m −|α| α αf ) . Therefore, by (3), α α i T (D f ). Thus T T (D −1 ξ m p ξ m p m |α|k |α|k T −1 (f ) Cα D αf p C1 f Lp , m p |α|k

(k)

where the constant C1 = C1 (p, k, n) does not depend on f .

Theorem 6 is false in the limit cases p = 1 and p = ∞ for n > 1 (cf. Section 4). However for n = 1 it extends on the limit cases (cf. [75]). In particular remembering that C(I ), where I = (−1/2; 1/2), is isomorphic to its Cartesian product with the ﬁeld of scalars we get (cf. Borsuk [9]) P ROPOSITION 7. The operator f → f is an isomorphism from the subspace {f ∈ 1/2 C (k) (I ): −1/2 f (x) dx = 0} of codimension one in C (k) (I ) onto C (k−1) (I ). Hence all the spaces C (k) (I ) are isomorphic to C(I ) for k = 1, 2, . . . . R EMARKS . (1) Theorems 4 and 6 have their counterparts for Sobolev spaces on Tn . To adopt the proofs we use the following T RANSFERENCE THEOREM . If m is a continuous multiplier on Rn of strong type (p, p) for some p with 1 p ∞ (resp. of weak type (1, 1)), then the “sequence” (m(2πa))a∈Zn is the multiplier on the group Zn – the dual of Tn ; the corresponding norm (resp. the weak type (1, 1) constant) of the multiplier transform on Lp (Tn ) is dominated by the norm of the multiplier transform on Lp (Rn ) (resp. the weak type (1, 1) constant). The strong type part is due to De Leeuw (cf. [92], Chapter 7, Theorem 3.8); for the weak type part cf. [103,2]. n n .(2) Letp apn(PR ,(k) ) denote the norm of PR ,(k) regarded as an operator on |α|k L (R ). One can show that if k = 1, 2, . . . , n = 2, 3, . . . then there are positive constants A(k, n) and B(k, n) such that A(k, n) max(p, p/(p − 1)) ap (PRn ,(k) )

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A. Pełczy´nski and M. Wojciechowski

B(k, n) max(p, p/(p − 1)) (cf., e.g., [72]). Thus in these cases PRn ,(k) is not of strong type (1, 1) and (∞, ∞). . (k) (3) JR,(k) (C0 (Rn )) is a complemented subspace of |α|k C0 (R). This is a consequence of the strong type (∞, ∞) of PR,(k) and the density of S(R) in C0 (R). (4) Theorems 4 and 6 and their counterparts for Tn are in fact results on the scales p p of Sobolev spaces (L(k) (Rn ))1
p/(p−1)

(Rn ), the duality is given P ROPOSITION 8. The dual of L(k) (Rn ) is isomorphic to L(k) by the bilinear form

p p/(p−1) n

R . (f, g) = D αf (x)D α g(x) dx f ∈ L(k) Rn ; g ∈ L(k) n |α|k R

An analogous result holds for Tn . 3. Isomorphisms of Sobolev spaces. Linear extension theorems Positive results on isomorphic classiﬁcation of Sobolev spaces heavily depend on linear extension theorems. Recall that if X and Y are (Banach) function spaces on topological spaces S and T respectively with S ⊂ T then a bounded linear operator E : X → Y is called a linear extension operator provided that E(f )(s) = f (s) for s ∈ S and f ∈ X. Assuming that E exists and the restriction operator R|S is continuous and takes Y onto X we infer that X and Y0 = {g ∈ Y : g(s) = 0 for s ∈ S} are isomorphic to complemented subspaces of Y ; the desired projection are E ◦ R|S and IdY − E ◦ R|S . We formulate (not in full generality) the most useful results on linear extension operators for Sobolev spaces: WET = W HITNEY E XTENSION T HEOREM . Let Ω ⊂ Rn be the quasi-Euclidean domain Ω : C (k) (Ω) → C (k) (Rn ). and let k = 0, 1, . . . . Then there is a linear extension operator E(k) Recall that Ω is a quasi-Euclidean domain if there exists C > 0 such that any two points x, y ∈ Ω could be joined by the rectiﬁable arc γ ⊂ Ω of length less than C|x − y|2 . JET = J ONES E XTENSION T HEOREM . Let Ω ⊂ Rn be the (ε, δ) domain. Then there is a linear extension operator p p n

ΛΩ (k),p : L(k) (Ω) → L(k) R ,

1 p ∞.

Recall that Ω is an (ε, δ) domain (cf. [35]) if there are ε, δ > 0 such that whenever y, x ∈ Ω with |x − y|2 < δ, there is a rectiﬁable arc γ ⊂ Ω joining x with y such that arclength(γ ) ε−1 |x − y|2 and infw∈Rn \Ω |z − w|2 ε|x − z|2 |z − y|2 /|x − y|2 for all z∈γ.

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1373

C OMMENT. For the proof of WET cf. [96], [42], Theorem 2.3.6, [90], Chapter VI, Theorem 4. For the proof of JET cf. [45]. For other linear extension operators with some additional properties the reader is referred to [90,62,15,16,105,106]. The (ε, δ) domains contain so called Lipschitz domains (cf. [90], Chapter VI, §3), in particular open convex sets in Rn as well as every bounded open set with C 1 -boundary are Lipschitz domains. The operator constructed by Stein for Lipschitz domains ([88]) does not depend on p, i.e., it is one operator which works for all p. For special sets, like cubes, there are simple constructions of linear extension operators (cf. [41], Addendum in [31], and this survey, Section 8, Theorem 57). The “positive” results on classiﬁcation of Sobolev spaces are still not satisfactory. T HEOREM 9 (Mityagin [65]). Let Ω = ∅ (resp. Rn \ Ω = ∅) be quasi-Euclidean subset of Rn . Then C (k) (Ω) (resp. C0(k) (Ω)) is isomorphic to C (k) (Rn ) (k, n = 1, 2, . . .). For p = 1 we have: T HEOREM 10. Let Ω ⊂ Rn be open non-empty. Assume that for some k = 0, 1, . . . there is a linear extension operator from L1(k) (Ω) into L1(k) (Rn ). Then L1(k) (Ω) is isomorphic to L1(k) (Rn ). For 1 < p < ∞ we have: T HEOREM 11. Let k, n with k = 0, 1, . . . , n = 1, 2, . . . be ﬁxed. Let Ω ∈ Rn be an open non-empty set such that for some p ∈ (1; ∞) either there is a linear extension operator p p from L(k) (Ω) into L(k) (Rn ) or the Sobolev projection PΩ,(k) is of strong type (p, p). Then p L(k) (Ω) is isomorphic to Lp = Lp (0; 1). S OME NOTATION . Let “∼” stand for “is isomorphic to”. Let I n = x = (xj ) ∈ Rn : max |xj | < 1/2 . 1j n

Let ej -stand for the j -th coordinate versor (j = 1, 2, . . . , n). P ROOF OF T HEOREM 9. The proof requires several steps. (I) Let B ⊂ Rn be a bounded open set. Deﬁne the new norm on C0(k) (B) by |||f ||| = max ∂ α f C(B) . |α|=k

Then the norm ||| · ||| is equivalent to the original norm · C (k) (B) , precisely there is a 0 non-decreasing function a : R+ → R+ with limt =0 a(t) = 1 such that |||f ||| f a(diam B)|||f |||

(k)

for f ∈ C0 (B).

(4)

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Indeed for z = (zj ) ∈ B pick z = (zj ) ∈ bd B so that z1 = max{x1 : (x1 , z2 , . . . , zn ) ∈ bd B and x1 < z1 }. Then, remembering that f ∈ C0(k) (B), we have f (z) =

z1 z1

∂ f (x1 , z2 , . . . , zn ) dx1. ∂x1

Hence (replacing the ﬁrst coordinate by the appropriate one) ∂ f C(B) diam B · max f j ∂x j

. C(B)

By a simple induction we get β ∂ f (diam B)k−|β| max ∂ α f C(B) C(B) |α|=k

for 0 |β| k.

(5)

Clearly (5) yields (4). (II)

(k)

(k) −1 n

(k) (k)

C0 10 I × C0 10−2 I n × · · · c ∼ C0 I n × C0 I n × · · · c . 0

0

√ Since supm diam(10−m I n ) n, it follows from (I) that for m = 0, 1, . . . the Banach– Mazur distance (cf. [44], p. 3) satisﬁes

√

d C0(k) 10−m I n , ||| · ||| , C0(k) 10−m I n , · C (k) (10−m I n ) a n . 0

On the other hand the space (C0 (10−m I n ), ||| · |||) is isometrically isomorphic to the space (C0(k) (I n ), ||| · |||) via the map Um : C0(k) (10−m I n ) → C0(k) (I n ) given by (k)

Um f (x) → 10mk f 10−m x x ∈ I n ; m = 1, 2, . . . . This implies (II), moreover the Banach–Mazur distance between the product spaces does not exceed n. (III) C0(k) (I n ) ∼ (C0(k) (I n ) × C0(k) (I n ) × · · ·)c0 . Let A = Km , where Km = 2e1 − (m + 1)−1 e1 + 10−m I n for m = 1, 2, . . . . Put B = Rn \ A. Then B is quasi-Euclidean. (For, note that two points, say x and y, of the boundary of a cube in Rn , can be join by a broken line living on the boundary of the cube whose length does not exceed the double distance between x and y. Thus given points z and w in B, we replace the interval

is J joining z and w by a broken line in which each of the non-empty intervals J ∩ Km

replaced by a suitable broken line living on the boundary of Km where Km is a homothetic image of Km with the same center and slightly bigger size (depending on z and w).) It B : C (k) (B) → C (k) (Rn ). follows from WET that there exists a linear extension operator E(k) (k)

B ◦ R . Let Q be the restriction of P to C (6I n ). The latter space Let P = IdC (k) (Rn ) − E(k) |B 0

Sobolev spaces

1375

is identiﬁed with the subspace of C (k) (Rn ) consisting of functions vanishing on the complement of 6I n . Identifying C0(k) (A) in the same way with a subspace of C (k) (Rn ) we infer (k) (k) that Q is a projection of C0 (6I n ) onto C0 (A). Clearly

(k) (k) (k) C0 (A) ∼ C0 10−1 I n × C0 10−2 I n × · · · c . 0

Now it follows from (II) that C0(k) (6I n ) ∼ C0(k) (I n ) contains a complemented subspace (k) (k) isomorphic to (C0 (I n ) × C0 × · · ·)c0 . Thus (III) follows from the decomposition method (cf. [44], p. 14). (IV) If Ω ⊂ Rn is a bounded non-empty quasi-Euclidean set then C (k) (Ω) ∼ C0(k) (I n ); (k) in particular C (k) (I n ) ∼ C0 (I n ). Clearly Ω ⊂ aI n for some a > 0. Since Ω is quasiΩ : C (k) (Ω) → Euclidean, it follows from WET that there is a linear extension operator E(k) Ω (C (k) (Ω)) ⊂ C (k) (2aI n ). (Otherwise we multiply C (k) (Rn ). We may assume that E(k) 0 Ω f by a ﬁxed function from D(Rn ) which equals 1 on aI n and vanishes outside 2aI n .) E(k) (k)

(k)

Thus C (k) (Ω) is isomorphic to a complemented subspace of C0 (2aI n ) ∼ C0 (I n ). On ¯ n is the other hand Ω contains a translate of the cube bI n for some b > 0. Since Rn \ bI (k) (k) quasi-Euclidean a similar argument as in (III) shows that C0 (bI n ) ∼ C0 (I n ) is isomorphic to a complemented subspace of C (k) (I n ). Now in view of (III) we get (IV) by the decomposition method. . It is convenient to denote by ( a∈Z Xa )c0 the c0 Cartesian product of a family (Xa )a∈Z . (V) C0(k) (I n ) ∼ C0(k) (Rn ). For m = 0, 1, 2, . . . , n put F0 = Rn and bd+ F0 = ∅, Fm = x ∈ Rn : 0 < xj < 1 for j = 1, 2, . . . , m , bd+ Fm = x ∈ F m : xj = 0 for j = 1, 2, . . . , m . For a ∈ Z we put Fma = Fm + a · em and bd+ Fma = bd+ Fm + a · em . Let

(k) a

C+ Fm = f ∈ C (k) Fma :

lim

α D f (x) = 0 for |α| k . a

x→bd+ Fm

Then (k)

C+ (Fm ) ∼

/ a∈Z

(k) a

C+ Fm+1

for m = 0, 1, 2, . . ., n − 1.

(6)

c0

Let φ ∈ D(3I ) satisfy φ(t) = 0 for t ∈ [2/3, 1] and φ(t) = 1 for t ∈ [0, 1/3]. For m = . (k) (k) a 0, 1, . . . , n − 1 deﬁne T : ( a∈Z C+ (Fm+1 ))c0 → C+ (Fm ) setting Tf = g where g = . (k) (k) a ))c0 and f ∈ C+ (Fm ) is given by (ga ) ∈ ( a∈Z C+ (Fm+1 k ∂i 1 · i fb−1 (πm+1 x) · (xm+1 − b)i , f (x) = gb (x) + φ(xm+1 − b) · i! ∂xm+1 i=0

1376

A. Pełczy´nski and M. Wojciechowski

where b ∈ Z and π(·) satisfy: b xm+1 < b + 1,

πm+1 (x) = (x1 , . . . , xm , b, xm+2 , . . . , xn ).

. (k) (k) (k) a Let S : C+ (Fm ) → ( a∈Z C+ (Fm+1 ))c0 be deﬁned by Sf = g where f ∈ C+ (Fm ) and . (k) a )) is given by g = (ga ) ∈ ( a∈Z C+ (Fm+1 c0 k ∂i 1 · i f (πm x) · (xm+1 − b)i gb (x) = (R|F b f )(x) − φ(xm+1 − b) · m+1 i! ∂xm+1 i=0 b for x ∈ Fm+1 .

It is not hard to check that T and S are bounded linear operators which satisfy and T ◦ S = IdC (k)(F ) .

S ◦ T = Id(.

(k) a a∈Z C+ (Fm+1 ))c0

+

m

Thus T is an isomorphism and we get (6). Since F0 = Rn we get by induction that / (k) C+ (Fn ) . C (k) (Rn ) ∼ c0

(7)

Similarly as in step (III), we get that C (k) (I n ) is isomorphic to a complemented subspace (k) (k) of C+ (Fn ). On the other hand C+ (Fn ) is isomorphic to a complemented subspace of (k) n C (2I ): the embedding is the formal identity (we extend the function on 2I n \ Fn by 0) and the projection is given by 2I n \F n

Pf = IdC (k) (2I n ) − E(k)

◦ R|2I n \F n .

(k) (Fn ). Combining Thus, by the decomposition method and (IV), we obtain C (k) (I n ) ∼ C+ with (7) we get (V). (VI) C (k) (Ω) ∼ C (k) (Rn ). If Ω ⊂ Rn is an arbitrary (not necessarily bounded) quasiEuclidean open set, then similarly as in step (IV) (replacing I n by Rn ) we show that C (k) (Ω) embeds as a complemented subspace into C (k) (Rn ). Then, combining steps (IV) and (V) and using the decomposition method we get (VI). (VII) C0(k) (Ω) ∼ C (k) (Rn ). If Ω ∈ Rn is such that Rn \ Ω is quasi-Euclidean then (k) C0 (Ω) is naturally identiﬁed with a subspace of C (k) (Rn ). This subspace is complemented via the projection Rn \Ω

P = IdC (k)(Rn ) − E(k)

◦ R|Rn \Ω . (k)

On the other hand C (k) (I n ) is isomorphic to a complemented subspace of C0 (Ω) exactly as in step (V). Thus combining steps (IV) and (V) and using the decomposition method we get (VII).

Sobolev spaces

1377

R EMARK . In the same way we prove that C (k) (Rn ) ∼ C (k) (M) for some compact Euclidean C k -manifold M. In particular we have C OROLLARY 12. C (k) (Tn ) ∼ C (k) (Sn ) ∼ C (k) (Rn ) for ﬁxed n and k with n = 1, 2, . . . , k = 0, 1, . . . . Theorem 10 is proved similarly. As in the proof of Theorem 8 the crucial role plays the counterpart of step (I) – the inﬁnite divisibility of L1(k) (I n ). For details cf. [79]. The argument in [79] does not use JET; instead it uses an ‘elementary’ explicit construction of an extension operator from a special domain. p

P ROOF OF T HEOREM 11. The assumptions on Ω imply that L(k) (Ω) is isomorphic to a . complemented subspace of Lp . Either we use Theorem 6 or we use that |α|k Lp (Rn ) p is isomorphic to Lp (cf. [44], Vol. I, pp. 14–15). Next we show that L(k) (Ω) contains a p complemented subspace isomorphic to L for arbitrary non-empty Ω ⊂ Rn . Clearly Ω contains a cube aI n + x for some a > 0 and some x ∈ Ω. Since there is a linear extension p p p operator from L(k) (aI n + x) into L(k) (Ω), the space L(k) (Ω) contains a complemented p subspace isomorphic to L(k) (I n ). Let p E = f ∈ L(k) (I n ): f depends on the ﬁrst coordinate only . p

p

Clearly E ∼ L(k) (I ). Moreover E is complemented in L(k) (I n ) via the projection f → I n−1 f (·, x2 , x3 , . . . , xn ) dx2 dx3 · · · dxn . An application of the decomposition method completes the proof.

4. Non-isomorphism of non-reﬂexive Sobolev spaces of several variables with classical Banach spaces To the contrary with the spaces of one variable (cf. Proposition 7 and the preceding comment) the non-reﬂexive Sobolev spaces of more than one variable are not isomorphic to corresponding L1 and C(K) spaces. The main analytic tool used in the proofs of next two theorems is a special case of the Sobolev embedding theorem (cf. Section 6 for more detailed discussion). We use the following notation: the characters of the group Tn are identiﬁed with exponents ea : I n → C deﬁned for a = (aj ) ∈ Zn by ea (x) = exp 2πi nj=1 aj xj for x = (xj ) ∈ I n . We put fˆ(a) = n f (x)ea (−x) dx (a ∈ Zn ; f ∈ L1 (I n )). I

T HEOREM 13. Let k, n (k = 1, 2, . . . , n = 2, 3, . . .) be given. Then L1(k) (Ω) is not a L1 space for every non-empty open Ω ⊂ Rn . P ROOF. Since every non-empty open Ω contains a cube tI n + x for some t > 0 and x ∈ Ω, it follows from JET that L1(k) (Ω) contains a complemented subspace isomorphic to L1(k) (I n ). Thus it is enough to restrict ourselves to Ω = I n .

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First we consider the case k = 1, n = 2 to which the general case reduces. Deﬁne V1 ﬁrst for trigonometric polynomials on I 2 (= ﬁnite linear combinations of the ea ’s for a = (a1 , a2 ) ∈ Z2 ) by

V1 (f ) =

fˆ(a)ea = f.

a∈Z2

C LAIM . V1 extends to a bounded linear operator from L1(1) (I 2 ) into L2 (I 2 ). We shall denote the extension also by V1 . Assume the claim. Deﬁne T1 : L2 (I 2 ) → by

L1(1) (I 2 )

T1 (f ) =

fˆ(a)ea(1),

where ea(1) = #

a∈Z2

ea 1 + (2πa1 )2 + (2πa2)2

.

T1 is bounded because (ea(1) )a∈Z2 is an orthonormal system in L2(1) (I 2 ), hence f 2L2 (I 2 )

2 (1) ˆ = f (a)ea 2

L(1) (I 2 )

a∈Z2

2 T1 (f )L1

(1) (I

2)

.

Thus V1 T1 : L2 (I 2 ) → L2 (I 2 ) is not a Hilbert–Schmidt operator because

−1 V1 T1 (ea )2 2 2 = = +∞. 1 + (2πa1 )2 + (2πa2 )2 L (I ) a∈Z2

a∈Z2

Thus L1(1) (I 2 ) is not an L1 -space because by a result of Grothendieck (cf. [44], Section 10, [38], [22], 4.12) every operator on a Hilbert space which factors through a L1 -space is Hilbert–Schmidt. P ROOF OF CLAIM . We start with the identity 1

2 ∂ ∂ h, h = h22 1R− ×R+ $ ∂x1 ∂x2

for h ∈ S R2 ,

(∗)

where 1R− ×R+ is the indicator function of the set {x ∈ R2 : x1 0, x2 0}, “$” denotes the operation of convolution, and ·, · stands for the usual inner product in L2 (Rn ); by 1R− and 1R+ we denote the indicator functions of the negative and the positive halﬂine respectively. To verify (∗) note that for g ∈ S(R) one has

1R− $ g (t) =

+∞

g (s) ds; t

(1R+ $ g)(t) =

t −∞

g(s) ds.

Sobolev spaces

Thus (1R− ×R+ $

x2 ∂ ∂x1 h)(x1 , x2 ) = − −∞ h(x1 , s2 ) ds2 .

1379

Hence integrating by parts we get

2 1 x2 ∂ ¯ ∂ ∂ h(x1 , x2 ) dx1 dx2 h, h =− h(x1 , s2 ) ds2 1R− ×R+ $ 2 ∂x1 ∂x2 ∂x 2 R −∞ h(x1 , x2 )2 dx1 dx2 = h2 . = 2 R2

It follows from (∗) and the Hausdorff–Young inequality that h22

∂ 1R− ×R+ ∞ h ∂x 1

∂ ∂ 2 −1 ∂ h 2 . ∂x h + ∂x h 1 2 1 ∂x2 1 1 1

(∗∗)

Thus the same inequality holds for h ∈ L1(1) (R2 ) with partial derivatives replaced by distributional derivatives. Now if Λ : L1(1)(I 2 ) → L1(1) (R2 ) is a linear extension operator then for f ∈ L1(1) (I 2 ) and h = Λ(f ) we get f L2 (I 2 ) h2

#

# 1/2 D (1,0) h1 + D (0,1) h1 1/2Λ f L1

(1) (I

2)

.

Clearly f 22 = a∈Z2 |fˆ(a)|2 . This completes the proof of claim and of Theorem 13 in the case k = 1, n = 2. Next consider the case k 2, n = 2. We deﬁne Vk : L1(k) (I 2 ) → L2 (I 2 ) and Tk : 2 L (I 2 ) → L1(k) (I 2 ) by Vk (f ) =

(2πa1)k−1 fˆ(a)ea ;

Tk (f ) =

a∈Z2

fˆ(a)ea(k),

a∈Z2

where ea ea(k) = ea L2

(k) (I

2)

and ea L2

(k)

3 4 l k 4 5 (2πa1 )2r (2πa2 )2(l−r). (I 2 ) = r=0 l=0

The boundedness of Tk is proved similarly as the boundedness of T1 using the orthonor(k) mality in L2(k) (I 2 ) of the system (ea )a∈Z2 . The boundedness of Vk uses inequality (∗∗) for the function

∂k h ∂x1k−1

instead of h. Since

(2πa1 )2(k−1) Vk Tk (ea )2 2 2 = = +∞, L (I ) ea 2 2 2 2 2

a∈Z

a∈Z

L(k) (I )

the operator Vk Tk is not Hilbert–Schmidt, hence L1(k) (I 2 ) is not a L1 -space.

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A. Pełczy´nski and M. Wojciechowski

The case k 1, n 3 follows from previous cases because L1(k) (I n ) contains the complemented subspace isomorphic to L1(k) (I 2 ) consisting of functions depending on the ﬁrst two variables; the averaging operator f→

I n−2

f (·, ·, x3 , . . . , xn ) dx3 · · · dxn

is the desired projection.

The operators Vk constructed above are not 1-summing, because 1-summing operators on a Hilbert space are Hilbert–Schmidt (cf. [44], Section 10). Thus C OROLLARY 14. There are bounded non-1-summing operators from L1(k) (Ω) into a Hilbert space (Ω ⊂ Rn , n = 2, 3, . . . , k = 1, 2, . . .). R EMARKS . (1) The operators Vk are special cases of the Sobolev embeddings (see Section 6). (2) Identify L1(k) (Tn ) with the subspace of L1(k) (I n ) being the closure of 1-periodic trigonometric polynomials. Clearly L2 (I n ) can be identiﬁed to L2 (Tn ) and L2 (Tn ) is translation invariant isometrically isomorphic with L2(k) (Tn ) for k = 1, 2, . . . . Note that (k)

Tk (L2 (T2 )) ⊂ L1(k) (T2 ). Let I2,1 : L2(k) (T2 ) → L1(k) (T2 ) be the natural embedding and let k denote the restriction of Vk to L1 (T2 ). Thus (V k , I (k) ) is the invariant factorization V 2,1 (k) (with respect to the action of the group T2 ) of an invariant operator on L2(k) (T2 ) which is not Hilbert–Schmidt. Next we discuss another property of Sobolev spaces in L1 -norms which differenties them from L1 (μ)-spaces. The latter spaces are by the Lebesgue decomposition theorem complemented in their second duals. We identify a Banach space with its canonical image in its second dual. T HEOREM 15 (cf. [80]). If n = 2, 3, . . . ; k = 1, 2, . . . then for every non-empty open Ω ⊃ Rn the space L1(k) (Ω) is uncomplemented in its second dual. Note that L1(k) (Ω) does not contain an isomorphic copy of c0 because L1(k) (Ω) is iso. metric to a subspace of |α|k L1 (Ω) (cf. Section 2) which is obviously isomorphic to an L1 (μ) space for some measure μ. Thus combining several facts on Banach lattices (cf. [60], Vol. II, Propositions 1.c.6, 1.a.11, Theorem 1.b.16) with Theorem 15 we get C OROLLARY 16. If Ω, k and n satisfy the assumption of Theorem 15 then L1(k) (Ω) is not isomorphic to any complemented subspace of a Banach lattice. To avoid technical complications we present the proof of Theorem 15 for k = 1. The argument requires some preparation. First observe that the general case reduces to the case

Sobolev spaces

1381

Ω = Rn because L1(1) (Ω) contains a complemented subspace isomorphic to L1(1) (I n ) ∼ L1 (Rn ) (by Theorem 10 and the argument in the beginning of the proof of Theorem 13) and the fact that the property “X is complemented in X∗∗ ” is inherited by complemented subspaces of X (cf. [59]). Next we introduce some notation. We represent Rn = Rn−1 × R and we write x = (y, xn ) with y ∈ Rn−1 and xn ∈ R. We identify Rn−1 with the hyperplane {x = (y, xn ) ∈ Rn : xn = 0}. We put Rn− = {x = (y, xn ) ∈ Rn : xn < 0} and Rn+ = {x = (y, xn ) ∈ Rn : xn > 0}. By D(Rn− ) we denote the space of scalar-valued inﬁnitely many times differentiable functions on Rn− which together with all their partial derivatives are uniformly continuous on Rn− and whose unique continuous extensions to Rn− have compact supports; we use the same symbol to denote the functions on Rn− and their extensions to Rn− . It is not hard to verify (cf. Section 1, Proposition 2) that D(Rn− ) is dense in L1(1) (Rn− ) in the norm · L1 (Rn ) . The next result is due to Gagliardo (cf. [33]); it belongs to so (1) − called “trace theorems” (cf. Section 6, Theorem 42). P ROPOSITION 17. There exists the unique bounded linear surjection (called the trace)

Tr : L1(1) Rn− → L1 Rn−1 such that Tr(φ) = φ|Rn−1 for φ ∈ D(Rn− ). Proposition 17 is an immediate consequence of the next two lemmas and the density of D(Rn− ) in L1(1) (Rn− ) (cf. [80] for details). L EMMA 18. One has φ|Rn−1 L1 (Rn−1 ) φL(1) (Rn− ) P ROOF. Fix φ ∈ D(Rn− ). Then φ(y, 0) = the absolute value against dy we get

φ ∈ D Rn− . 0

∂ −∞ ∂xn φ(y, xn ) dxn

∂ φ|Rn−1 L1 (Rn−1 ) φ ∂x 1 n φL1(1) (Rn− ) . n L (R− )

for y ∈ Rn−1 . Integrating

L EMMA 19. There exists C > 0 such that given ψ ∈ D(Rn−1 ) there exists φ ∈ D(Rn− ) such that φ|Rn−1 = ψ;

φL1

n (1) (R− )

CψL1 (Rn−1 ) .

P ROOF. Pick a non-negative h ∈ D(R) with h(0) = 1. It is not hard to verify that for t = t (ψ) > 0 small enough the function φ deﬁned by φ(y, xn ) = h( xtn )ψ(y) has the desired 0 property with C = ∞ h(xn ) dxn .

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Recall that a right inverse of a bounded linear operator T : X → Y (X, Y Banach spaces) is a bounded linear operator S : Y → X such that T S = IdY . The crucial analytic ingredient used in the proof of Theorem 15 is T HEOREM 20 (Peetre [71]). The trace Tr : L1(1) (Rn− ) → L1 (Rn−1 ) admits no right inverse. We postpone the discussion of Peetre’s Theorem until completing the proof of theorem 15. We also need the following result from the theory of Banach spaces L INDENSTRAUSS L IFTING P RINCIPLE = LLP. If a bounded linear surjection Q : X → Y (X, Y Banach spaces) has the property that ker Q is complemented in (ker Q)∗∗ then for every L1 -space E every linear operator T : E → Y admits a lifting T : E → X, i.e., T = QT. In particular if Y is isomorphic to L1 (0, 1), E = Y and T = IdE then T is a right inverse for Q. For a proof of LLP see [59,47]. P ROOF OF T HEOREM 15. It is enough to show that some complemented subspace of L1(1) (Rn ) is uncomplemented in its second dual. By Lemma 19, Theorem 20 and LLP, ker Tr is uncomplemented in its second dual. We show that ker Tr is isomorphic to a complemented subspace of L1(1) (Rn ). Let o L1(1) (Rn ) denote the subspaces of L1(1) (Rn ) consisting of the functions which are odd with respect to the variable xn . This subspace is complemented in L1(1) (Rn ) via the projection f → o f where o f (y, xn ) = (f (y, xn ) − f (y, −xn ))/2 for (y, xn ) ∈ Rn−1 × R a.e. For f ∈ ker Tr we deﬁne f˜ : Rn → R by f˜(y, xn ) =

f (y, xn ), −f (y, −xn ),

for xn 0, for xn > 0.

(8)

To prove that o L1(1) (Rn− ) is isomorphic to ker Tr we show that (i) the formula (8) deﬁnes a function in o L1(1) (Rn ); (ii) the operator f → f˜ is a surjection onto o L1(1) (Rn ). Note that (i) holds for f ∈ D(Rn− ) ∩ ker Tr. Thus it is enough to show that D(Rn− ) ∩ ker Tr is dense in ker Tr. Fix ε > 0 and f ∈ ker Tr. Since D(Rn− ) is dense in L1(1) (Rn− ), there is fε ∈ D(Rn− ) such that f − fε L1 (Rn ) < ε. Since Tr(fε ) = Tr(f − fε ), Lemma 18 (1) − yields fε |Rn−1 L1 (Rn−1 ) f − fε L1 (Rn ) < ε. Therefore, by Lemma 19, there exists (1)

−

gε ∈ D(Rn− ) such that gε |Rn−1 = fε |Rn−1 and gε L1 (Rn ) < Cε. Hence (fε − gε )|Rn−1 = 0 (1) − and f − (fε − gε )L1 (Rn ) < f − fε L1 (Rn ) + gε L1 (Rn ) < (C + 1)ε. (1)

−

(1)

−

(1)

−

For (ii) note that the map f → f˜ is an isomorphism because f L1

n (1) (R− )

2f L1

n (1) (R− )

f˜L1

(1) (R

n)

. For Φ ∈ o L1(1) (Rn ) ∩ D(Rn ) one has Φ| Rn− = Φ. Thus it sufﬁces to

show that o L1(1) (Rn ) ∩ D(Rn ) is dense in o L1(1) (Rn ). Fix ε > 0. Since D(Rn ) is dense in L1(1) (Rn ), given F ∈ o L1(1) (Rn ) (hence satisfying o F = F ) there is a Φ ∈

Sobolev spaces

1383

D(Rn ) such that F − ΦL1 (Rn ) < ε. Thus F − o ΦL1 (Rn ) = o (F − Φ)L1 (Rn ) (1) (1) (1) F − ΦL1 (Rn ) < ε. (1)

Next we discuss Peetre’s theorem. It can be reformated as follows: there is no linear extension operator from L1 (Rn−1 ) → L1(1) (Rn− ). In that form an elegant proof was given in [16]. Another proof is contained in [80]. The proofs in [16] and [80] use harmonic analysis. Here we present a simple proof, based on an idea from [81], which uses the following purely Banach space property of L1 (μ). L EMMA 21. Let μ be a non-purely atomic measure. Assume that weakly compact operators Tm : L1 (μ) → L1 (μ) satisfy ∞ Tm (g)

L1 (μ)

< +∞

for every g ∈ L1 (μ).

m=1

Then

∞

m=1 Tm

= IdL1 (μ) .

A proof of Lemma 21 can be obtain modifying the argument of Proposition 1.d.1 in [60], Vol. I, see also [81], Lemma 5.3. The next lemma is an improvement of Lemma 18. Let us put Dn = D (0,...,0,1) and U(a, b) = {x ∈ Rn : a < xn < b}. L EMMA 22. (j) Let f ∈ L1(1) (Rn− ). Let −∞ < c < d c∗ < d ∗ < 0. Let us deﬁne h : Rn−1 → C by h(y) = (d − c)−1

d

f (y, xn ) dxn c

−1 − d ∗ − c∗

d∗ c∗

f (y, xn ) dxn

(y a.e.-λn−1 ).

Then hL1 (Rn−1 )

U (c,d ∗ )

|Dn f | dλn .

(jj) If c < d < 0 then for f ∈ L1(1) (Rn− ) Tr(f ) − (d − c)−1

d c

f (·, xn ) dxn

L1 (Rn−1 )

U (c,0)

|Dn f | dλn .

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A. Pełczy´nski and M. Wojciechowski

P ROOF. Deﬁne e : R → R by ⎧ ⎪ (d − c)−1 (xn − c), ⎪ ⎨ 1,

e(xn ) = ∗ ∗ −1 c ∗ − x , ⎪ n ⎪ ⎩ d −c 0,

for c < xn < d, for d xn c∗ , for c∗ < xn < d ∗ , otherwise.

The derivative e exits at every point xn ∈ R \ {c, d, c∗ , d ∗ } and e L∞ (R) < +∞. Moreover h(y) = e (xn )f (y, xn ) dxn (y a.e.-λn−1 ). R−

Fix ε > 0. Pick ϕ ∈ D(Rn−1 ) so that ϕL∞ (Rn−1 ) = 1 and hL1 (Rn−1 ) − ε <

Rn−1

h(y)ϕ(y) dy =

R−

Rn−1

ϕ(y)e (xn )f (y, xn ) dy dxn .

For η > 0 deﬁne eη : R → R – the regularization of e by

η exp −(t/η)2 e(xn − t) dt, eη = ρ(xn ) · √ (2π) R where ρ ∈ D(R− ) does not depend on η and satisﬁes 0 ρ 1 and ρ(xn ) ≡ 1 for xn in some open interval containing [c, d ∗ ]. The regularization eη ∈ D(R− ) and it satisﬁes for sufﬁciently small η > 0,

ϕ(y)eη (xn )f (y, xn ) dy dxn > hL1 (Rn−1 ) − ε, (9) R−

Rn−1

/ (c, d ∗ )) and (because e is continuous and e(xn ) = 0 for xn ∈ eη (xn ) < ε Dn f

L1 (Rn− )

+1

−1

for xn ∈ / c, d ∗ .

(10)

Put φ(x) = ϕ(y)eη (xn ) for x = (y, xn ) ∈ Rn− . Combining the deﬁnition of distributional derivative with (9) and with the Fubini theorem we get ∂ = φ(x)D f (x) dx φ(x)f (x) dx n n ∂xn n R− R− ϕ(y)eη (xn )f (y, xn ) dx = =

Rn−

R−

Rn−1

ϕ(y)eη (xn )f (y, xn ) dy dxn

> hL1 (Rn−1 ) − ε.

Sobolev spaces

1385

On the other hand taking into account that |ϕ| 1 and (10) we infer that |φ(x)| = |ϕ(y)η(xn )| < ε(Dn f L1 (Rn− ) + 1)−1 whenever x ∈ Rn− \ U(c, d ∗ ). Thus

Rn−

φ(x)Dn f (x) dx

U (c,d ∗ )

φ(x)Dn f (x) dx

+

Rn \U (c,d ∗ )

U (c,d ∗ )

φ(x) · Dn f (x) dx

|Dn f | dλn + ε.

Hence U (c,d ∗ ) |Dn f | dλn hL1 (Rn ) − 2ε. Passing ε → 0 we get (j). To prove (jj) specify c, d, c∗ , d ∗ so that lim c∗ = lim d ∗ = 0. Note that if f ∈ D(Rn− ) then d∗ obviously if lim c∗ = lim d ∗ = 0 then lim (d ∗ − c∗ )−1 c∗ g(·, xn ) − Tr(g)L1 (Rn−1 ) = 0; thus by density of D(Rn− ) in L1(1) (Rn− ) the same formula holds for all functions in L1(1) (Rn− ). P ROOF OF T HEOREM 20. Assume to the contrary that there exists a right inverse of Tr : L1(1)(Rn− ) → L1 (Rn−1 ), say S. Let B = y ∈ Rn−1 : |y|2 1 . / 2B. Let Pick Φ ∈ D(Rn−1 ) so that Φ(y) = 1 for y ∈ B and Φ(y) = 0 for y ∈

X = g ∈ L1 Rn−1 : g(y) = 0 for y ∈ /B . Obviously X can be identiﬁed with L1 (B, λn−1 |B) = L1 (B) and gX = gL1 (Rn−1 ) for g ∈ X. Deﬁne S % : X → L1(1) (Rn− ) for g ∈ X by S % (g)(x) = Φ(y)S(g)(y, xn )

for x = (y, xn ) ∈ Rn λn -a.e.

Clearly Tr ◦S % = IdX . Deﬁne the operators Um : X → X by U0 = 0, and for m = 1, 2, . . . by Um g(y) = λ1 (Im )−1 S % (g)(y, xn ) dxn · 1B (y) for y ∈ Rn−1 λn−1 -a.e., Im

where Im = (−2−m , −2−m−1 ) and 1B denotes the indicator function of B. By Lemma 22(jj), for every g ∈ X, if m 1 then −1 % % (I ) S (g)(·, x ) dx − Tr ◦S (g) Um g − gL1 (Rn−1 ) λ n n 1 m Im

U (−2−m ,0)

Dn S % (g) dλn .

L1 (Rn−1 )

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A. Pełczy´nski and M. Wojciechowski

Thus lim Um g − gL1 (Rn−1 ) = 0 for g ∈ X.

(11)

m→∞

By Lemma 22(j) for m = 1, 2, . . . , we get −1 λ Um+1 g − Um gL1 (Rn−1 ) (I ) 1 m+1 − λ1 (Im )−1

Im+1

Im

S % (g)(·, xn ) dxn

U (−2−m ,−2−(m+2) )

S % (g)(·, xn ) dxn

L1 (Rn−1 )

Dn S % (g) dλn .

Taking into account that each x ∈ Rn− belongs to at most two of the sets U(−2−m , −2−(m+2) ) we get ∞

Um+1 g − Um gL1 (Rn−1 ) U1 gL1 (Rn−1 ) + 2S % (g)L1

n (1) (R− )

m=0

3S % · gL1 (Rn−1 ) .

(12)

Put q = n/(n − 1). By the Sobolev embedding theorem for L1(1) (Rn− ) (cf. Theorem 33 and remark (1) following its proof), % S (g) q n AS % (g) 1 L (R ) L

n (1) (R− )

−

for g ∈ X,

where A = A(k, n) is an absolute constant. Using the later inequality, the inequality between the ﬁrst and the q-th norms on a ﬁnite interval, and the Fubini theorem, for g ∈ X, we get q Um gLq (Rn−1 )

q −1 % S (g)(y, xn ) dxn dy λ1 (Im ) Rn−1 Im % S (g)(y, xn )q dxn dy λ1 (Im )−q

λ1 (Im )−q

Rn−1 Im

Rn−

q q Cm S % (g)L1

% S (g)(x)q dx

n (1) (R− )

q q q Cm S % gL1 (Rn−1 ) ,

m g = Um g regarded as an element of Lq (B) and let Jq,1 where Cm = A · λ1 (Im )−1 . Put U denote the natural embedding of Lq (B) into L1 (B). It follows from the previous inequality

Sobolev spaces

1387

m : L1 (B) → Lq (B) is bounded; obviously Jq,1 is bounded because the measure that U m admits a factorization through the reﬂexive space λn−1 |B is ﬁnite. Hence Um = Jq,1 ◦ U Lq (B). Thus Um and Um − Um−1 are weakly compact for m = 1, 2, . . . . In view of (11) and (12) the desired contradiction follows from Lemma 21. Recall that a Banach space X has local unconditional structure (cf. [36], [22], p. 345, [44], p. 59) if there is a constant C > 0 such that for every ﬁnite-dimensional subspace F of X there are a ﬁnite-dimensional Banach space E with a basis with unconditional constant one (cf. [44], p. 14) and operators u : F → E and v : E → X such that u · v C and v ◦ u : F → X is the natural (set theoretical) embedding. One has (cf. [32], [22], Theorem 17.5) (FJT). A Banach space has local unconditional structure iff its second dual is isomorphic to a complemented subspace of a Banach lattice. Thus the following result [81] is an improvement of Corollary 16. T HEOREM 23. If Ω ⊂ Rn is an open non-empty set then for n = 2, 3, . . . and k = 1, 2, . . . the spaces L1(k) (Ω) and BV(k) (Ω) do not have local unconditional structure. The proof of Theorem 23 is lengthy (cf. [81] for details). It starts with (FJT) and uses the method of the proof of Theorem 15 to show that members of a certain net of separable subspaces of BV(k) (Rn ) containing L1(k) (Rn ) are not isomorphic to complemented subspaces of Banach lattices. This allows us to show that BV(k) (Rn ) does not have local unconditional structure. This implies that L1(k) (Rn ) does not have local unconditional structure by the following result: T HEOREM 24 (cf. [80], Proposition 6.2, [81], Proposition 7.3). There exists an isomorphic embedding ðk : BV(k) (Rn ) → [L1(k)(Rn )]∗∗ such that ðk (BV(k) (Rn )) is a complemented subspace of [L1(k) (Rn )]∗∗ and ðk ◦ ιk = κk , where ιk : L1(k) (Rn ) → BV(k) (Rn ) is the isometric embedding deﬁned by ιk (f ) = f ◦ λn and κk is the natural embedding of L1(k) (Rn ) into its second dual. 24. For simplicity we identify BV(k) (Rn ) with O UTLINE OF THE PROOF OF T HEOREM. n n J (BV(k) (R )), where J : BV(k) (R ) → |α|k M(Rn ) is the canonical embedding, and we identify L1(k) (Rn ) with its image via ιk . Thus L1(k) (Rn ) can be regarded as the subspace 1 ) deﬁned by of L1 (Rn ; lN

(fα )0|α|k ∈ L1 Rn ; l 1 : fα = D αf for 0 |α| k and for f ∈ L1(k) Rn , where N = K(k, n) is the number of partial derivatives in n variables of order k. 1 ) can be identiﬁed with L∞ (Rn ; l ∞ ). Thus, by the Hahn–Banach The dual of L1 (Rn ; lN N extension principle every z∗ ∈ (L1(k) (Rn ))∗ has a norm preserving extension to some ∗

∞ ). Now let (G ) ∞ 1 n (φα[z ] ) ∈ L∞ (Rn ; lN ε ε>0 be a C -approximate identity of L (R ), for

1388

A. Pełczy´nski and M. Wojciechowski

instance Gε (x) = (ε)−n G(x/ε) for x ∈ Rn where G(x) = (2π)−n/2 exp(−|x|22/2). Let Φε be the operator of convolution with Gε , i.e., Φε (ν)(x) = Rn Gε (x − y)ν(dy) for x λn -a.e. Then Φε (BV(k) (Rn )) ⊂ L1(k) (Rn ) for k = 0, 1, . . . and lim ε

Rn

Φε (ν)(x)f (x) dx =

Rn

f (x) dν

for f ∈ D Rn and ν ∈ M Rn .

Given ν ∈ BV(k) (Rn ) we deﬁne ð(k) (ν) by

ð(k) (ν) z∗ = LIM z∗ Φε (ν) ε→0

= LIM ε→0

0|α|k

Rn

∗ Φε D α ν φα[z ] dx

∗ for z∗ ∈ L1(k) ,

where LIMε→0 denotes a generalized (Banach) limit (cf. [25], Chapter II.3 (23)). The desired projection from (L1(k) (Rn ))∗∗ onto ðk (BV(k) (Rn )) is the operator ðk ◦ U ∗ where U ∗ : (L1(k) (Rn ))∗∗ → BV(k) (Rn ) is the adjoint operator to the isometric embedding ∞ )/(BV (Rn )) → (L1 (Rn ))∗ deﬁned as follows. Let g = (g ) U : C0 (Rn ; lN ⊥ α 0|α|k be (k) (k) a representative of a coset [g]. Then U ([g]) ∈ L1(k) (Rn ))∗ is deﬁned by

U [g] (f ) =

n 0|α|k R

gα · D α f dλn

f ∈ L1(k) Rn .

R EMARKS . (1) By Corollary 3(c) BV(k) (Ω) is a dual Banach space hence it is always complemented in its second dual (cf. [24]). Thus Theorem 15 does not extend on BV(k) (Ω), moreover ιk (L1(k) (Ω)) is not complemented in BV(k) (Ω). (2) The isometric embedding ðk is not unique and depends of the choice of a Banach limit. (3) One can extend the operator Vk deﬁned in the proof of Theorem 13 on BVk (I n ); one can show in that way that there are bounded non-absolutely summing operators from BVk (Ω) into a Hilbert space. Next we pass to Sobolev spaces in sup norm. We exhibit a pathological property of these spaces much stronger than the non-isomorphism with L∞ -spaces. Recall that a Banach space has GL (= Gordon–Lewis property) provided that every 1-summing operator from X into a Hilbert space factors through L1 (μ) (cf. [36], [22], p. 350, [44], Section 9). T HEOREM 25. Let k, n (k = 1, 2, . . . , n = 2, 3, . . .) be given. Then for every open nonempty Ω ⊂ Rn the spaces C (k) (Ω) and L∞ (k) (Ω) fail GL. n C OROLLARY 26. C (k) (Ω) and L∞ (k) (Ω) (∅ = Ω ⊂ R , n = 2, 3, . . . , k = 1, 2, . . .) are not isomorphic either to quotients of L∞ -spaces or to Banach spaces with local unconditional structure or all the more to complemented subspaces of Banach lattices.

Sobolev spaces

1389

P ROOF. Each of the above properties implies GL (cf. [36], [22], Chapter 17, [44], Section 9). The proof of Theorem 25 uses the theory of invariant r-summing operators for 0 < r < 1 and the weak type (1, 1) of the Sobolev projections PTn ,(k) (cf. Section 2). Recall that a linear operator T : X → Y is r-summing (0 < r < ∞, X, Y normed spaces) provided T (xj )r C r sup x ∗ (xj )r πr (T ) := inf C: x ∗ 1 j

j

(xj )nj=1 ⊂ X, n = 1, 2, . . .

.

We need (cf. [61], [101], III.F.35) G ROTHENDIECK –M AUREY T HEOREM . Every bounded linear operator from an L1 space to a Hilbert space is r-summing for 0 < r 1. Hence every bounded linear operator from a Banach space to a Hilbert space which factors through an L1 -space is r-summing for 0 < r 1. The next proposition on invariant r-summing operators for 0 < r < 1 is crucial for our proof of Theorem 25. P ROPOSITION 27. Let U : C (k) (Tn ) → L2(k) (Tn ) be an invariant r-summing operator for some 0 < r < 1. Then a∈Zn U (ea(k) )2 2 n < ∞. L(k) (T )

Throughout the proof we write P instead of PTn ,(k) . It is.convenient to idenp n tify canonical images in |α|k L (T ) (resp. . Sobolevn spaces in question.with their . p n n |α|k C(T )) and to regard |α|k L (T ) (resp. |α|k C(T )) as vector-valued spaces Lp (Tn , E) (resp. C(Tn , E)) where E = (E, | · |E ) is a ﬁnite-dimensional Hilbert space. By Trig(Tn , E) we denote the linear span of the functions ea · ξ (a ∈ Zn , ξ ∈ E). The orthonormal basis (ea(k))a∈Zn of L2(k) (Tn ) is identiﬁed via the canonical embedding (k)

with (ea )a∈Zn where e(k) a

= ea · ξa ,

where ξa =

%

(2πi)|α| a α

|β|k (2π)

2|β| a 2β

∈ E. |α|k

The bold letters are used to denote vector-valued functions. We apply the standard notation, * α a α = nj=1 aj j for a = (aj ) ∈ Zn and α = (αj ) ∈ Zn+ . Note that P(ea · ξ ) = ξ, ξa E · e(k) a for ξ ∈ E and a ∈ Zn . Hence P(Trig(Tn , E)) ⊂ Trig(Tn , E). Moreover for each r with 0 < r < 1 there is Ar ∈ (0, ∞) such that

P(f )(x)r dx Ar f r for f ∈ Trig Tn , E . (13) r 1 E Tn

1390

A. Pełczy´nski and M. Wojciechowski

The latter inequality is a consequence of a general easy fact that similarly as in the scalar case for E-valued functions a weak type (1, 1) operator is of strong type (1, r) for 0 < r < 1. P ROOF OF P ROPOSITION 27. We shall show that there is A ∈ (0, ∞) such that U Pf 2 Af 1

f ∈ Trig Tn , E .

(14)

Clearly (14) implies the existence of a bounded operator U1 : L1 (Tn , E) → L2(k) (Tn , E) such that U P = U1 I∞,1 . Here I∞,1 : C(Tn , E) → L1 (Tn , E) denotes the natural embedding. Similarly as in the scalar case every invariant operator, say U1 , from L1 (Tn , E) into L2 (Tn , E) is given by a convolution with an L2 function; precisely there is a U1 ∈ L2 (Tn , L(E, E)) such that

U1 (f ) (s) =

Tn

U1 (s − t) f (t) dt

for s λn -a.e.

Here L(E, E) denotes the (dim E)2 -dimensional space of linear operators on E. In our case where U1 (L1 (Tn , E)) ⊆ L2(k) (Tn ) there is a scalar sequence (ua ) such that the Fourier coefﬁcients of U1 satisfy 1 (a)(ξ ) = U

Tn

ea (−t) · U1 (t) dt (ξ ) = ua ξ, ξa E · ξa

2 a∈Zn U1 (a)L(E,E) = (k) (k) = U1 I∞,1 implies U (ea ) = ua ea

Clearly U1 ∈ L2 (Tn , L(E, E)) yields other hand the identity U P To prove (14) deﬁne f y by f y (·) = f (· + y)

ξ ∈ E, a ∈ Zn .

a∈Zn |ua | for a ∈ Zn .

2

< ∞. On the

f ∈ Trig Tn , E , y ∈ Tn .

The deﬁnition of r-summing operator yields Tn

U Pf y r2 dy

r π(U ) sup

x ∗ ∈Z

Tn

∗ x (Pf y )r dy,

(15)

where the supremum extends on an arbitrary subset Z of the unit ball of [C (k) (Tn )]∗ which is weak-star dense in the ball. It is convenient to take as Z the set of those x ∗ ∈ [C (k)(Tn )]∗ that there exists h = (hα )|α|k ∈ Trig(Tn , E) such that h1 1 and ∗

x (g) =

Tn

! " g(x), h(x) E dx

g ∈ C (k) Tn ∩ Trig Tn , E .

(16)

Sobolev spaces

1391

Note that the integrand on the left-hand side of (15) is a constant function. Hence U Pf y r2 dy = U Pf r2 . Next we estimate the right-hand side of (15). For a scalarTn valued trigonometric polynomial h deﬁne ho by ho (x) = h(−x). We need the identity: Tn

(Pf y )(x)h(x) dx = P ho ∗ f (y) f ∈ Tn ,

(17)

where the convolution ho ∗ f is taken coordinatewise. (It is enough to verify (17) for h = ea n and f = ξ · eb (a, ∈ E).) Let {ξ (α) : |α| k} be an orthonormal basis for E. Fix b ∈ Z ; ξ (α) ∗ x ∈ Z and h = |α|k hα ξ ∈ Trig(Tn , E). Then (16) and (17) yield 1 r 2 ∗ (α) x (Pf y )r = (x), h (x)ξ dx Pf α y n T

E

|α|k

1 2 = (Pf y )(x)hα (x) dx, ξ (α) n |α|k

T

E

r

r n (Pf y )(x)hα (x) dx T

|α|k

=

|α|k

E

o

P h ∗ f (y) α E

r [Br ]r

P ho ∗ f (y)r , α E

|α|k

where Br is an absolute constant which depends only on r, k and n. Integrating dy, against r o using (13), the Hausdorff–Young inequality, and taking into account that |α|k hα 1 [Cr ]r hr1 for some absolute constant Cr we get Tn

∗ x (Pf y )r dy [Br ]r

n |α|k T

[Br Ar ]r

o

P h ∗ f (y)r dy α E

ho ∗ f r α 1

|α|k

[Ar Br ]r

r ho f r α 1 1

|α|k

[Ar Br Cr ]r hr1 f r1 = [Ar Br Cr ]r f r1 . Therefore (U P)(f )2 πr (U )Ar Br Cr f 1 which proves (14).

k I (k) : P ROOF OF T HEOREM 25. First we show that C (k) (T2 ) fails GL. Let U = V ∞,1 (k) k : C (k) (T2 ) → L1 (T2 ) is the natural embedding and V C (k) (T2 ) → L2 (T2 ) where I (k)

∞,1

(k)

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A. Pełczy´nski and M. Wojciechowski

is the operator constructed in remark (2) after Corollary 14. It follows from the construction k that of V 2 U e(k) 2 a L a∈Z2

(k)

(T2 )

=

(2πa1 )2k−2 = +∞. ea 2 2 2 2

a∈Z

L(k) (I )

Thus, by Proposition 27 and the Grothendieck–Maurey theorem, U does not factor through (k) any L1 -space. On the other hand U is 1-summing because I∞,1 is 1-summing and therefor r-summing for 0 < r < 1. Thus C (k) (T2 ) and C (k) (I 2 ) isomorphic with the previous one (by Theorem 9) fail GL. The GL property is inherited by complemented subspaces of a space with GL. Since C (k) (Ω) and C0(k) (Ω) contain a complemented subspace isomorphic to C (k) (T2 ) (same ar(k) gument as in the proof of Theorem 13 with JET replaced by WET ), C (k) (Ω) and C0 (Ω) n fail GL for ∅ = Ω ⊂ R (k = 1, 2, . . . , n = 2, 3, . . .). The proof for L∞ (k) (Ω) is similar. R EMARKS . (1) Theorem 13 and the idea to apply Sobolev embeddings is due to Kislyakov [49]. One can prove directly that L1(k) (I n ) is not an L1 -space using the fact that the n-dimensional Sobolev embedding of L1(1) (I n ) into Lp (I n ) for p = n/(n − 1) (cf. Section 6) is a bounded operator but is not (n, 2)-summing while by Kwapie´n’s result (cf. [57], [93], Corollary 11.12) every operator from an L1 -space to Lp is (p/(p − 1), 2)-summing for 1 < p 2 (cf. [99] for details). Similarly one can show that C (1) (Tn ) is not isomorphic I∞,n/(n−1)

n/(n−1)

S

to an L∞ -space considering the operator C (1) (Tn ) −→ L(1) (Tn ) −→ Ln (Tn ), where I∞,n/(n−1) – the natural embedding, S – the appropriate Sobolev embedding. Now use [56], Theorem 2.b.8, which says that every operator from an L∞ -space to Lp -space is (p, 2)-summing for p 2. (2) The non-isomorphism of C (1) (Sn ) with C(K)-spaces for n 2 is due to Grothendieck who gave in [37] some indication for the proof. Henkin [40] published the complete proof that C (k) (Sn ) for n 2 is not isomorphic to any C(K)-space; he even showed that C (k) (Sn ) is not uniformly homeomorphic to any C(K)-space (cf. also [3], Theorem 10.8). The Grothendieck–Henkin argument is simpler if one uses the spaces on Tn instead of on Sn . The idea bases upon the following FACT (folklore). Let G be a compact Abelian group, E a ﬁnite-dimensional Hilbert space, X a translation invariant L∞ -subspace of C(G, E). Then X is complemented in C(G, E) via the orthogonal projection from L2 (G, E) onto X, i.e., this orthogonal projection is of strong type (∞, ∞). O UTLINE OF THE PROOF. The injectivity of ln∞ and the deﬁnition of an L∞ space imply the existence of a net of ﬁnite-dimensional operators {Tu : C(G, E) → X: u ∈ Σ} whose restrictions Tu |X tend pointwise to the identity on X. Averaging the Tu ’s with respect to the Haar measure of G we get the net {Tu : u ∈ Σ} of invariant operators which tends to the desired projection.

Sobolev spaces

1393

Now specifying G = Tn and X = C (k) (Tn ) we conclude that C (k) (Tn ) is not an L∞ space for n 2 because the corresponding orthogonal projection, which is PTn ,(k) , is not of strong type (∞, ∞) (cf. remark (2) after Proposition 7). (3) Kislyakov [49] ﬁrst established that C (k) (Tn ) is not isomorphic to a quotient of a C(K)-space. Theorem 25 was proved independently in [50] and [58]. The proofs of Proposition 27 and Theorem 25 presented here are taken from [58]. Actually the inverse to Proposition 27 also holds and it also characterizes invariant nuclear operators (cf. [58], Proposition 1.3 and Theorem 4.1). (4) The space C (k) (Tn ) is not isomorphic to any complemented subspace of the Disc Algebra for k = 1, 2, . . . , n = 2, 3, . . . (cf. [74]). (5) The main classiﬁcation problem which remains open is: are the integers k and n linear topological invariants for C (k) (I n ) and L1(k) (I n )? Precisely, does the isomorphism

of the spaces C (k) (I n ) and C (k ) (I n ) imply k = k and n = n (n, n = 2, 3, . . . , k, k = 1, 2, . . .)? The same question remains open for Sobolev spaces in L1 -norms.

5. Properties of C(Q) spaces shared by C (k) (Ω) Denote by M = M(I ) the dual of C(I ). If Q is an uncountable compact metric space then [C(Q)]∗ is isometrically isomorphic to M (cf. [44], Section 4). A dual X∗ of a Banach space X is said to be a separable perturbation of M provided that X∗ is isomorphic to M ⊕ F for some separable space F . We have T HEOREM 28. If open ∅ = Ω ⊂ Rn (resp. Rn \ Ω) is quasi-Euclidean then the ﬁrst dual of C (k) (Ω) (resp. C0(k) (Ω)) is a separable perturbation of M for k = 1, 2, . . . , n = 1, 2, . . . . The proof of Theorem 28 requires some preparation. Note that for a Banach space being a separable perturbation of M is an isomorphic invariant. Hence, by Theorem 9, it is enough to prove that [C (k) (Tn )]∗ ∼ M ⊕ F . By the Riesz representation theorem M . we identify n . The space [ n )]∗ is with the space of all Borel complex-valued measures on T C(T |α|k . therefore identiﬁed with the space |α|k M of all tuples (μα )|α|k of measures in M with the norm (μα ). = |α|k μα ; the duality.is given by f , Υ = |α|k Tn fα dμα for Υ = (μα ) . ∈ |α|k M and f = (fα ) ∈ |α|k C(Tn ). Recall that a closed linear subspace G ⊂ |α|k M is a C(Tn )-module provided that

(μα ) ∈ G and f ∈ C Tn implies (f μα ) ∈ G. Here f μ is deﬁned by (f μ)(A) = A f dμ for Borel A ⊂ Tn . Note that since the exponents are linearly dense in C(Tn ) in the sup norm, in the deﬁnition of C(Tn )-module one can replace C(Tn ) by the set {ea : a ∈ Zn }. FACT. If G ⊂

.

|α|k M

is a C(Tn )-module then G is complemented in

.

Fact is a particular case of much more general result (cf. [73], Section 1).

|α|k M.

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A. Pełczy´nski and M. Wojciechowski

By the standard duality argument the quotient space ﬁed with [C (k) (Tn )]∗ , where

C

(k)

T

n

⊥

= Υ∈

/

.

|α|k M/[C

(k) (Tn )]⊥

is identi-

(k) n

M: f , Υ = 0 for f ∈ JTn ,(k) C T .

|α|k

The crucial role in the proof of Theorem 28 is played by the concept of Henkin measure. Given a sequence (fm ) ⊂ C (k) (Tn ) we write fm ⇒ 0 provided that (i) supm fm C (k) (Tn ) < ∞; α n (ii) lim. m ∂ fm (x) = 0 (x ∈ T , |α| < k). Call Υ ∈ |α|k M a Henkin measure provided that (iii) limm JTn ,(k) (fm ), Υ = 0 whenever fm ⇒ 0. . Denote by MH = MH (n, k) the set of all Henkin measures in |α|k M. L EMMA 29. ⊥

MH ⊃ C (k) Tn ;

(18)

MH is a normed closed subspace of

/

M;

(19)

|α|k

MH is a C Tn -module.

(20)

P ROOF. (18) is obvious. (19) is routine and easy. To obtain (20) it sufﬁces to show that ea Υ ∈ MH for every exponent ea with a ∈ Zn and for every Υ ∈ MH ; equivalently lim m

n |α|k T

α

∂ fm ea dμα = 0 whenever fm ⇒ 0 Υ = (μα ) ∈ MH , a ∈ Zn .

To verify the latter statement we note that fm ⇒ 0 implies fm ea ⇒ 0 for a ∈ Zn . Now invoking (i)–(iii) we use the Domination Convergence Theorem, combined with the identity

∂ αfm ea = ∂ α (fm ea ) −

n $ β≺α j =1

αj ! (2πi)|α−β| a α−β ∂ βfm ea , βj !(αj − βj )!

(21)

where α ≺ β ≡def α = β and αj βj for j = 1, 2, . . . , n. The identity (21) follows from the Leibniz formula for the derivatives of the product.

Sobolev spaces

1395

P ROOF OF T HEOREM 28. Let δx,α (f ) = fα (x) for |α| k, x ∈ Tn , and f = (fα )|α|k ∈ . n ∗ (k) n ∗ |α|k C(T ). Let δx,α be the corresponding functional on C (T ), i.e., δx,α (f ) = δx,α (JTn ,(k) (f )) = ∂ α f (x) for f ∈ C (k) (Tn ). Let

∗ ∗ F = the closed subspace of C (k) Tn generated by δx,α for x ∈ Tn , |α| < k. Then F is separable;

(22)

⊥

F = MH / C (k) Tn .

(23)

To verify (22) recall that if k 1 then the natural embedding C (k) (Tn ) → C (k−1) (Tn ) is compact (the Ascoli theorem). Therefore operator [C (k−1)(Tn )]∗ → the adjoint (k) n ∗ ∗ [C (T )] is also compact. Thus Z = x∈Tn ;|α|
and

! " x ∗ (f ) = JTn ,(k) (f ), Υ

for f ∈ C (k) Tn .

(24)

Since x ∗ = 0, there would exist an x ∗∗ ∈ [C (k) (Tn )]∗∗ such that ∗∗ x = 1;

x ∗∗ x ∗ = 0;

x ∗∗ f ∗ = 0 for f ∗ ∈ F .

∗ ) = 0 for x ∈ Tn and |α| < k. Pick a sequence (x ) ⊂ Tn dense in Tn . In particular x ∗∗ (δx,α l By the Goldstine theorem ([25], Chapter V, §4, Theorem 5) there would exist a sequence (fm ) ⊂ C (k) (Tn ) such that for m = 1, 2, . . .

x ∗ (fm ) = x ∗∗ x ∗ = 0;

δx∗l ,α (fm ) = x ∗∗ δx∗l ,α = 0 (|α| < k; l = 1, 2, . . . , m). fm 2;

(25)

Since (xl ) is dense in Tn and the natural embedding C (k) (Tn ) → C (k−1) (Tn ) is compact, (25) would imply the existence of a subsequence (fm ) of (fm ) such that limm ∂ α fm (x) = 0 for all x ∈ Tn and for |α| < k. Hence fm ⇒ 0. Thus, by (24), ! "

0 = lim JTn ,(k) fm , Υ = lim x ∗ fm . m

m

On the other hand, by (25), lim x ∗ (fm ) = x ∗∗ (x ∗ ) = 0; a contradiction. This proves (23).

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A. Pełczy´nski and M. Wojciechowski

From (20), Fact, and (18) we get / ( (k) n ⊥

(k) n ∗ C T C T = M |α|k

⊥ = (MH ⊕ V )/ C (k) Tn = E ⊕ V,

(26)

. (k) n where V is a complementary subspace to MH in |α|k M. Since C (T ) contains (k) n ∗ C(T) ∼ C(0, 1) as a complemented subspace, [C (T )] has a complemented subspace isomorphic to M. Recall that M is an uncountable l 1 sum of L1 (cf. [19], [44], Section 4). Now the separability of F combined with (26) yields that V contains a complemented sub. space isomorphic to M. Also |α|k M ∼ M being the dual of a “large” separable C(Q). Thus the standard decomposition argument yields V ∼ M. H ISTORICAL COMMENT. Theorem 28 is a substitute of the F. and M. Riesz theorem for the Disc Algebra. A ﬁrst similar decomposition for Ball Algebras was discovered in 1968 by Henkin [40]. Around 1970 he communicated to the ﬁrst author of this survey the statement of Theorem 28 indicating that the proof requires hard analytic tools. Bourgain in [12] stated Theorem 28 without a proof. The ﬁrst proof seems to appear in [73]. The proof presented here is a modiﬁcation of this argument. Next we enlist several common properties of C(Q) spaces shared by C (k) (Ω) concerning weakly compact operators and related topics. Recall (cf. [23], [44], Section 10) that a Banach space X has DP (= the Dunford–Pettis property) provided that every weakly compact operator from X to arbitrary Banach space takes weakly convergent to zero sequences into norm convergent. Note that if X∗ posseses DP so does X (cf. [23]). Consider the following properties of a Banach space X: (a) X∗ has DP; (b) for every operator T from X to a Banach space, T is weakly compact iff T takes weakly unconditionally convergent series into unconditionally convergent series; (c) X∗ is weakly complete. It has been known for more than 40 years that the spaces C(Q) have the properties (a), (b), (c) (cf. [23]). The next concept due to Bourgain allows us to extend these results to a broad class of spaces including separable Sobolev spaces in sup norm. Let E be a ﬁnite-dimensional Banach space, Q – a compact Hausdorff space, X – a closed linear subspace of C(Q, E). We call X a rich subspace of C(Q, E) provided that there exists a positive linear functional F ∗ ∈ [C(Q)]∗ such that for every bounded sequence (f l ) ∈ X and every g ∈ C(Q) if liml F ∗ (|f l (·)|E ) = 0 then liml dist(gf l , X) = 0. T HEOREM 30 (Bourgain [12]). A rich subspace of C(Q, E) has (a), (b), (c). For the proof the reader is referred to [12] and to the book [101]. Theorem 30 is applied to Sobolev spaces via L EMMA 31. J (C (k) (Tn )) is a rich subspaces of 1, 2, . . .).

.

n |α|k C(T )

= C(Tn , E) for (k, n =

Sobolev spaces

1397

P ROOF. . The positive functional F ∗ is induced by the Haar measure of Tn . Note that regarding |α|k C(Tn ) as C(Tn , E) one has

F J f (·) E C ∗

|α|k

Tn

α ∂ f (x) dx

for some absolute constant C independent of f ∈ C (k) (Tn ). It follows from (21) that for every trigonometric polynomial g on Tn there is a constant Cg such that

gJ (f ) − J (fg) n Cg f (k−1) n for f ∈ C (k) Tn . (27) C (T ) C(T ,E) Since the natural embedding C (k) (Tn ) → C (k−1) (Tn ) is compact, the condition that (fm ) ⊂ C (k) (Tn ) is bounded and satisﬁes limm F ∗ (|J (fm (·))|E ) = 0 implies limm fm C (k−1) (Tn ) = 0. Thus (27) yields

lim dist gJ (fm ), J C (k) Tn m lim gJ (fm ) − J (gfm )C(Tn ,E) lim Cg fm C (k−1) (Tn ) = 0. m

m

This completes the proof because trigonometric polynomials are dense in C(Tn ).

Combining Theorem 30 with Lemma 31 and Theorem 9 we get (cf. [11,12]) C OROLLARY 32. Let open ∅ = Ω ⊂ Rn (resp. Rn \ Ω ) be quasi-Euclidean. Then C (k) (Ω) (k) (resp. C0 (Ω)) has (a), (b), (c). R EMARKS . (1) It seems to be interesting to determine whether the dual of C (k) (Ω) has always a ﬁnite cotype; in particular whether [C (1)(T2 )]∗ has cotype 2. (2) We refer the reader to the survey of Figiel and Wojtaszczyk (cf. [26], pp. 561–596) for constructions of Schauder bases in C (k) (Ω) and L1(k) (Ω) for “nice” Ω ⊂ Rn . Hence, by (k)

Theorem 9, if Ω ⊂ Rn (resp. Rn \ Ω) is quasi-Euclidean then C (k) (Ω) (resp. C0 (Ω)) has n a basis. Similarly as in the case of analytic functions it is not known whether L∞ (k) (T ) has the approximation property and whether C (k) (Tn ) has the uniform approximation property for k = 1, 2, . . . , n = 2, 3, . . . . The Hardy space H 1 has the uniform approximation property [46]; a counterpart for L1(k) (Tn ) is unknown (k = 1, 2, . . . , n = 2, 3, . . .). (3) It is not known whether L1(k) (Tn ) has DP for k = 1, 2, . . . , n = 2, 3, . . . . The Hardy space H 1 fails DP.

6. Embedding theorems of Sobolev type The assumption on integrability of derivatives of function yields “better” integrability of function itself. We begin with the classical Sobolev embedding theorem discovered by

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A. Pełczy´nski and M. Wojciechowski

Sobolev [88] for p > 1 and extended independently by Gagliardo [34] and Nirenberg [66] to p = 1. Here “→” stands for the set theoretical inclusion. T HEOREM 33 (Sobolev embedding theorem). p (i) If 1 p < n/k, then L(k) (Rn ) → Lq (Rn ) for 1/p 1/q 1/p − k/n. p (ii) If p = n/k > 1, then L(k) (Rn ) → 1r<∞ Lr (Rn ). (iii) If p = n/k = 1, then L1(k) (Rn ) → L∞ (Rn ). p (iv) If p > n/k and p 1 then L(k) (Rn ) → C0 (Rn ). The reader is referred to [1,7,90,62] for various proofs of Theorem 33. We present here brieﬂy the original Sobolev approach for p > 1 (cf. [90], Chapter V), combined for p = 1 with a “weak type” argument which we learned from P. Hajłasz. O UTLINE OF THE PROOF. We restrict ourselves to part (i). It is enough to prove (i) for the “maximal possible” exponent q = pn/(n − pk). The proof reduces, by simple induction, to the case k = 1 and to real-valued functions. We use the Sobolev identity (cf. [90], Chapter V, §2, (18)) f (x) =

n

1 ωn−1

i=1

Rn

∂f yi (x − y) · n dy, ∂xi |y|

where ωn−1 denotes the surface measure of (n − 1)-dimensional sphere. Consequently we have n f (x) C j =1

∂f (x − y) · |y|−n+1 dy. n ∂x R

j

The right-hand side of the above formula is the convolution of the sum of absolute values of derivatives of f with the Riesz potential y → |y|−n+1 . By the Hardy–Littlewood–Sobolev theorem on fractional integration (cf. [90], Chapter V, §1, Theorem 1), this convolution is an operator of weak type (1, 1 + 1/(n − 1)) and it is a bounded operator from Lp (Rn ) to Lq (Rn ) for 1 1. For p = 1 the previous argument gives only that f is in “weak Lq ”. To show that f ∈ q L (Rn ) observe that the weak Lq estimate depends only on the gradient of f . Let f be a smooth positive function with bounded support (differences of such functions are dense in L1(1) (Rn )). Put

fm = max 0, min f − 2m , 2m One shows that fm ∈ L1(1) (Rn ) and f =

(m ∈ Z).

∞

−∞ fm

with

∞

Am = x : f (x) 2m+1 = x : fm (x) = 2m ,

−∞ fm L1(1) (Rn )

< ∞. Setting

Sobolev spaces

1399

we infer that the support of the gradient satisﬁes supp ∇fm ⊂ Am−1 \ Am . Therefore, by the weak type (1, q) inequality, we get ∇fm 1 q λn (Am ) C 2m (here λn stands for the usual Lebesgue measure on Rn ). Thus Rn

|f |q =

∞ m=−∞ Am−1 \Am ∞

2

(m+1)q

|f |q

4q C

∞

2(m+1)q λn (Am−1 \ Am )

m=−∞

λn (Am−1 ) 4 C

m=−∞

∞

q

q

∇fm 1

m=−∞

q ∇fm 1

∞

q

= 4q C∇f 1 .

m=−∞

The last equality holds because the functions ∇fm have pairwise disjoint supports.

R EMARKS . (1) Theorem 33 remains valid if we replace Rn either by a domain with suitable regularity, e.g., any domain satisfying JET (cf. Section 3), or by a compact manifold without boundary, in particular by Tn . (2) The weak type argument is implicitly contained in [62], Chapter I. (3) For general domains the theory is more complicated; there are counterexamples (cf. [1,62]). (4) The reader is referred to books [62,7], the survey [55], and the memoir [39] for extensive literature and comprehensive discussion of the subject. Next we discuss embeddings of L1(1) (Rn ) into Lorentz and Besov spaces. Let h∗ : R → R denote the non-increasing rearrangement of the function h : Rn → R (cf. [92] for deﬁnition). Put hp,q =

p q

∞

t

0

q dt h (t) t

1/p ∗

1/q for 0 < p, q < ∞.

The Lorentz space Lp,q (Rn ) consists of all functions f for with f p,q < ∞. Since Ln/(n−1),1 (Rn ) → Ln/(n−1) (Rn ), the next result slightly improves Theorem 33. T HEOREM 34. One has L1(1) (Rn ) → Ln/(n−1),1 (Rn ) for n = 2, 3, . . . . Theorem 34 is a simple consequence of the more subtle embedding theorem into suitable Besov spaces which are in contained the Lorentz spaces in question. θ(p,n)

T HEOREM 35. Let n = 1, 2, . . . . Then L1(1) (Rn ) → Bp,1 1/n − 1) and 1 < p n/(n − 1).

(Rn ) where θ (p, n) = n(1/p +

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A. Pełczy´nski and M. Wojciechowski

Recall the deﬁnition of Besov spaces due to Peetre [70]. n Let Ψ (Rn ) be the family of all partitions of unity ψ = {ψj (x)}∞ j =0 ⊂ D(R ) such that: (i) supp ψ0 ⊂ {|ξ | 2}, supp ψj ⊂ {2j −1 |ξ | 2j +1 } for j = 1, 2, . . . , (ii) supj ψˆ j 1 < ∞, ∞ n (iii) j =0 ψj (ξ ) = 1 for ξ ∈ R . For ﬁxed ψ ∈ Ψ , 1 p < ∞, 1 q ∞, and 0 θ < ∞ we deﬁne for f ∈ S(Rn ) the norm · Bp,q θ (ψ) by θ f Bp,q θ (ψ) = f p + Bp,q (f ; ψ),

where θ Bp,q (f ; ψ) =

∞ q jθ 2 ψˆ j ∗ f p

1/q .

j =0

One can prove that the above norms are equivalent for different partitions of unity in Ψ (cf. [70,78]). θ (Rn ) is the completion of S(Rn ) in the norm · The Besov space Bp,q θ (ψ) for some Bp,q ψ ∈ Ψ . We will also need another norm deﬁned only for θ > 0 which involves the concept of p-th modulus of smoothness ωp (f ; ·). For 1 p < ∞ and for f ∈ Lp (Rn ) we put ωp (f, t) = sup fs − f p , |s|t

where fs (x) = f (x + s) for x, s ∈ Rn .

by Next for 1 q < ∞ and for 0 < θ < 1 we deﬁne the norm · Bp,q θ f Bp,q θ

θ = f p + Bp,q (f ),

where

θ Bp,q (f ) =

∞

t

0

−θ

q dt ωp (f, t) t

1/q .

L EMMA 36. For 1 p < ∞, 1 q < ∞, 0 < θ < 1 the norms · Bp,q and · Bp,q θ θ (ψ) are equivalent. For the proof see ([94], Sections 2.3.2 and 2.5.1, [78], Proposition 3.1). Now we are ready for O UTLINE OF THE PROOF OF T HEOREM 35. It is enough to restrict ourselves to a dense set in the cone of non-negative functions in S(Rn ) which are “non-ﬂat”, i.e.,

λn supp f \ {f > 0} = 0 and λn {f = c} = 0

for c > 0.

Our argument bases on a decomposition of non-ﬂat functions in a sum of slices,

fα,β = max α, min(f, β) − α

(α < β).

Sobolev spaces

1401

One shows (cf. [78], Theorem 2.1) that for non-ﬂat f there is an increasing sequence (αm )∞ m=1 such that if we put fm = fαm ,αm+1 then

fm (x) = f (x);

m

∇fm (x) = ∇f (x)

(for x λn -a.e.);

(28)

m

fm 1 = f 1 ;

m

∇fm 1 = ∇f 1 ;

(29)

m 1/n

(n−1)/n

fm ∞ fm 1

C · ∇fm 1

(m = 1, 2, . . .).

(30)

The inequality (30) is non-trivial; it follows from Federer–Kronrod coarea formula (cf. [30], Theorem 3.2.12, [62], 1.2.4, [78], pp. 76–77). Here C, C1 , C2 , . . . denote absolute constants. Using the above decomposition, to prove Theorem 35 for p < n/(n − 1) it is enough to establish for a single function g the inequality

∞

t −θ(p,n) ωp (g, t)

0

dt C1 · ∇g1 , t

(31)

assuming that g satisﬁes (n−1)/n

g1

1/n

g∞ C · ∇g1 .

(32)

Combining two elementary inequalities: ωp (g, t) ω1 (g, t)1/p (2g∞ )(p−1)/p and ω1 (g, t) t∇g1 , we get

(p−1)/p 1/p ∇g1 . ωp (g, t) t 1/p 2g∞ We split the left-hand side integral of (31),

∞ 0

=

(33) s 0

+

∞ s

where

1/n . s = g1 /g∞ Thus putting Kp,n =

s 0

21−1/p (n−1)(1−1/p) ,

by (33) and (30),

(p−1)/p 1/p t n−2−n/p ωp (g, t) dt 2g∞ ∇g1 =

(p−1)/p 1/p Kp,n g∞ ∇g1

s

t n−2−n/p+1/p dt

0

· s (n−1)(1−1/n) 1/p (n−1)/n 1/n 1−1/n = Kp,n ∇g1 g1 g∞ C2 · ∇g1 .

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A. Pełczy´nski and M. Wojciechowski

To estimate the second integral we need the trivial inequality ωp (g, t) 2gp and the inequality 1−n(1−1/p)

gp C3 · g1

n(1−1/p)

∇g1

which follows immediately from the Hölder inequality 1−n(1−1/p)

gp g1

n(1−1/p)

gn/(n−1)

and the Sobolev embedding (Theorem 33). Putting b = 1 + n/p − n and Qp,n = 2C3 b−1 we get ∞ ∞ b t n−2−n/p ωp (g, t) dt C3 · ∇g1−b g t n−2−n/p dt 1 1 s

s

−b/n b = Qp,n ∇g1−b 1 g1 · g1 /g∞ 1−1/n 1/n b g1 = Qp,n ∇g1−b g∞ 1 C4 · ∇g1 .

This completes the proof in the case p < n/(n − 1). The limit case p = n/(n − 1) follows from the reciprocal relations between the Besov spaces with different parameters. Precisely we need the following (cf. [78], Lemma 3.3) L EMMA 37. If 0 1/p − 1/r < 1/n, p 1, and n = 1, 2, . . . , then n(1/p−1/r) n

Bp,q

R

n

0 → Br,q R for 1 q ∞.

This ends the outline.

Using an improvement of Lemma 37 (cf. [78], Lemma 3.3), a norm of Besov space induced by an appropriate partition of unity and analysing the proof of Theorem 35 one gets C OROLLARY 38. For every partition of unity ψ ∈ Ψ there is a constant Cψ such that 0 Bn/(n−1),1 (f ; ψ) Cψ ∇f 1 for f ∈ L1(1) (Rn ). Specifying the partition of unity in Corollary 38 and applying the Hausdorff–Young and Hölder inequalities we obtain T HEOREM 39. There are positive constants C(n) for n = 2, 3, . . . such that

fˆ(ξ ) 1 + |ξ | 1−n dξ C(n)∇f 1 for f ∈ L1 Rn ; (1)

Rn

Rn

fˆ(ξ ) 1 + |ξ | k−n dξ < ∞ for f ∈ L1 Rn (k = 1, 2, . . .). (k)

Sobolev spaces

1403

The counterparts of all the results of this section for periodic functions are also valid. They either can be obtained by adopting the proofs or can be derived from the results for functions in Rn using the Poisson summation formula (cf. [92], Chapter VII, §2). In particular the periodic counterpart of the second inequality of Theorem 39 is C OROLLARY 40. If f ∈ L1(k) (Tn ) then 2, 3, . . .). Here |a|2 =

%

n 2 j =1 aj

ˆ

a∈Zn f (a)(1 + |a|2 )

k−n

< ∞ (k = 1, 2, . . . , n =

for a = (aj ) ∈ Zn .

R EMARKS . (1) Using the duality between Lorentz spaces one can immediately get Theorem 34 from a result of Faris [29]. In the present form it was proved by Poornima [83]. (2) The counterpart of Theorem 35 and Theorem 39, in particular Corollary 40, was proved by Bourgain in unpublished preprints [10,13]. The outline of the proof of Theorem 35 presented here is taken from [78]. Our proof was strongly inﬂuenced by Bourgain’s technique developed in the preprints [10,13]. Earlier by a different method Theorem 35 was obtained by Kolyada [55] (cf. also [53] and [54]). Kolyada’s method allows to strengthen Theorem 39 and Corollary 40, roughly speaking, by controlling weighted norms of Fourier transforms by Sobolev type norms involving only pure derivatives. In particular one gets (cf. [53]) C OROLLARY 41. If f and its pure distributional derivatives D (2,0) f and D (0,2) f belong to L1 (R2 ) then fˆ belongs to L1 (R2 ). In view of Theorem 49, Corollary 41 improves the ﬁrst part of Theorem 39. (3) Theorems 33, 34, 35, as well as equivalence of norms in Besov spaces (Lemma 36) extend to functions with values in an arbitrary Banach space (cf. [78]). It does not apply to results involving weighted norms of Fourier transforms. (4) The Banach ideals properties of Sobolev embeddings have been extensively investigated. For the classical Rellich–Kondrachov theorem on compactness of Sobolev embeddings and its generalizations the reader is referred to [1], Chapter VI, and [62], Chapter V, §5.5. The problematic of s-numbers of Sobolev embeddings is well presented in the book [56]. Recently in [99] it has been observed that the Sobolev embedding p L(k) (Rn ) → Ls (Rn ) where 1/s = 1/p − k/n is (v, 1)-summing for v > v0 but is not 2n , p}. (v0 , 1)-summing for v0 = max{ 2k+n (5) For other aspects of Sobolev embeddings the reader is referred to the survey of Schechtman [86], and the memoir [39]. Embeddings theorems admit generalizations to embeddings with respect to other measures than the Lebesgue measure on Rn . In particular if the measure in question is concentrated on a submanifold in Rn of the n-dimensional Lebesgue measure zero we get so called trace theorems. The simplest example is the case of the Lebesgue measure of a lower-dimensional linear manifold in Rn .

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T HEOREM 42. Let H be a m-dimensional linear manifold in Rn (1 k < n). Let 1 p p q mp/(n − kp). There is the unique bounded linear operator TrH : L(k) (Rn ) → Lq (H ) such that TrH (φ) = φ|H for φ ∈ H . The reader is referred to the books [1,62] and the survey [55].

7. Interpolation in Sobolev spaces For the real interpolation method (cf. [4], [44], Section 11), we know the complete dep q scription of the interpolation spaces between L(k) (Rn ) and L(k) (Rn ) as well as between p p L(k1 ) (Rn ) and L(k2 ) (Rn ). Let us recall some basic concepts of the real method. Let (X0 , X1 ) be a compatible couple of Banach spaces and Yi be a closed subspace of Xi , i = 0, 1. The couple (Y0 , Y1 ) is said to be K-closed in (X0 , X1 ) if there is C > 0 such that whenever y ∈ Y0 + Y1 is represented in the form y = x0 + x1 with xi ∈ Xi for i = 0, 1, then there exists another representation y = y0 + y1 , with yi ∈ Yi and yi Xi Cxi Xi for i = 0, 1. Recall that for the a compatible couple (X0 , X1 ) of Banach spaces the K-functional is deﬁned for each x ∈ X0 + X1 and t > 0 by K(x, t; X0 , X1 ) = inf x0 X0 + tx1 X1 : x = x0 + x1 , where the inﬁmum extends over all representations x = x0 + x1 with x0 ∈ X0 and x1 ∈ X1 . The real interpolation space (X0 , X1 )θ,q consists of all x in X0 + X1 for which the functional

1

xθ,q = 0

q dt t K(x, t; X0 , X1 ) t θ

1/q ,

0 < θ < 1, 1 q < ∞,

is ﬁnite. p p The real interpolation for couples (L(k1 ) (Rn ), L(k2 ) (Rn )) leads to Besov spaces α (Rn ) mentioned in Section 6 (cf. [4,5] for deﬁnition). Here Lp (Rn ) = Lp (Rn ). Bp,q (0) p

T HEOREM 43 ([4,5,94]). If k1 , k2 are integers with 0 k1 < k2 then (L(k1 ) (Rn ), p α (Rn ) where α = (1 − θ )k + θ k (1 p ∞). L(k2 ) (Rn ))θ,q = Bp,q 1 2 p

The main ingredient in the proof is to show that the K-functional K(f, t; L(0) (Rn ), is equivalent to min(1, t)f p + ωp (f, t). We consider next the case of ﬁxed smoothness and the exponent varying. Let E be a suitable ﬁnite-dimensional Hilbert space and L(E, E) denote the space of linear operators on E. Notice that the description of the interpolation space for the couple p q p (L(k) (Rn ), L(k) (Rn )) with 1 < p, q < ∞ is trivial because J (L(k) (Rn )) is a complemented subspace of Lp (Rn , E) via the Sobolev projection. The limit case is much more delicate. The result is due to DeVore and Scherer (cf. [20,21]). It is a consequence of the next result. p L(1) (Rn ))

Sobolev spaces

1405

n T HEOREM 44. An interpolation couple (J (L1(k) (Rn )), J (L∞ (k) (R ))) is K-closed in the couple (L1 (Rn , E), L∞ (Rn , E)). Hence

1 n ∞ n

p L(k) R , L(k) R θ,p = L(k) Rn for θ = 1 − 1/p. We present after [52] a sketch of the proof of Theorem 44 based on an idea of Bourgain [14]. It requires some preparation which involves the concept of Calderon–Zygmund integral operators. Recall that the translation invariant linear operator T : L2 (Rn , E) → L2 (Rn , E) is called a Calderon–Zygmund singular integral operator if there exist a measurable function K : Rn → L(E, E) and a constant C such that (i) for every x0 ∈ Rn , r > 0 and f ∈ L2 (Rn , E) such that supp f ⊂ B(x0 , r) (Tf )(x) = K(x − y)f (y) dy for x ∈ B(x0 , Cr), (ii) for every y ∈ Rn we have K(x + y) − K(x) L(E,E) dx C. |x|>C|y|

By the classical theory of Calderon–Zygmund operators (cf., e.g., [90], Chapter II) T , being a priori of strong type (2, 2), can be extended to the operator of strong type (p, p) for 1 < p < ∞ and of weak type (1, 1) operator. Hence for every f ∈ Lp (Rn , E) (1 p < ∞), Tf is a measurable E-valued function well deﬁned up to the set of measure zero. Let Q : L2 (Rn , E) → L2 (Rn , E) be a Calderon–Zygmund projection (i.e., projection which is a Calderon–Zygmund operator). One can prove that then Q is also a projection on Lp (Rn , E) for 1 < p < ∞; moreover QQf = Qf if f, Qf ∈ L1 (Rn , E) (cf. [52], Lemma 1). This allows us to introduce the following spaces:

HpQ = f ∈ Lp Rn , E : Qf = f a.e. , 1 p < ∞,

Q H∞ = f ∈ L∞ : f, gE dλn = 0 whenever g ∈ L1 Rn , E and Q∗ g = 0 (clearly the conjugate Q∗ is also a Calderon–Zygmund projection). It is well known that the multiplier transforms of multipliers satisfying Hörmander– Mikhlin criterion are translation invariant Calderon–Zygmund operators (cf. remark after this proof). Hence, by the proof of Theorem 4(a), the Sobolev projection PRn ,(k) is P

a Calderon–Zygmund projection for every n, k = 1, 2, . . . and Hp R p J (L(k) (Rn )).

n ,(k)

coincides with

Q

Q

O UTLINE OF THE PROOF OF T HEOREM 44. First we establish that the couple (H1 , H2 ) is K-closed in (L1 , L2 ). We have to show that there exists C > 0 such that for f ∈

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L1 (Rn , E) + L2 (Rn , E) such that Qf = f , and for every decomposition f = g + h with g ∈ L1 (Rn , E), h ∈ L2 (Rn , E) there exists a decomposition f = g˜ + h˜ with g˜ ∈ H1Q , Q ˜ 2 Ch2 . Let a = g1 , b = h2 . We apply now h˜ ∈ H2 such that g ˜ 1 Cg1 , h the Calderon–Zygmund decomposition of g on the level t = b2 a −1 (cf. [18]), i.e., the decomposition g = g0 + g1 for which there exists a measurable set Ω ⊂ Rn such that |g0 |E C1 · t,

g0 1 C1 a, g1 1 C1 a, |Qg1 |E C1 a λn (Ω) C1 at −1 , Rn \Ω

(here C1 > 0 is a constant independent of g). We put g˜ = Qg1 , h˜ = Q(g0 + h). We claim 1/2 1/2 that f = g˜ + h˜ is the required decomposition. Indeed, since g0 2 g0 ∞ g0 1 ˜ 2 C2 b where C2 = Q2→2 (C1 + 1). Since Qg1 = g1 + (I − Q)(g0 + C1 b, we have h h), (I − Q)(g0 + h) |Qg1 |E + |g1 |E + g ˜ 1= Rn \Ω

Ω

E

Ω

2C1 a + λn (Ω)1/2 · (I − Q)(g0 − h)2 # 2C1 a + C1 b−1 ag0 − h2 # 2C1 a + C1 b−1 a(C1 b + Cb) C3 a, where C3 > 0 does not depend on the functions involved. To prove that the couple Q Q (H2 , H∞ ) is K-closed in (L2 (Rn , E), L∞ (Rn , E)) we use the trick based on the following lemma due to Pisier (cf. [82,52]). P ISIER ’ S LEMMA . Let (X1 , X2 ) be a compatible couple with X1 ∩ X2 dense in X1 ∪ X2 , and Yi be the closed subspace of Xi (i = 1, 2). Then (Y1 , Y2 ) is K-closed in (X1 , X2 ) iff (Y1⊥ , Y2⊥ ) is K-closed in (X1∗ , X2∗ ). Clearly in view of the lemma it is sufﬁcient to show that ((H∞ )⊥ , (H2 )⊥ ) is K-closed Q I −Q∗ in (L1 (Rn , E), L2 (Rn , E)). But (Hp )⊥ = Hp and we can use what we already established, since I − Q∗ is also Calderon–Zygmund projection. To complete the proof we use Wolff type theorem for K-closedness. It says that one can derive K-closedness of the couple (Y1 , Y2 ) in (X1 , X2 ) from the K-closedness of couples (Y1 , F1 ) in (X1 , E1 ) and (F2 , Y2 ) in (E2 , X2 ), where E0 = (X0 , X1 )θ,p , E1 = (X0 , X1 )δ,q , F0 = (Y0 , Y1 )θ,p , F1 = (Y0 , Y1 )δ,q (0 < θ < δ < 1 and 0 < p < q ∞); for details see (cf. [52], Theorem 2). Q

Q

R EMARK . The fact that the multiplier transform of a function kˆ which satisﬁes Hörmander’s integral condition is a Calderon–Zygmund operator follows, for example, directly

Sobolev spaces

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from Hörmander’s proof (cf. [42], Theorem 7.9.5). If we replace w in formula (7.9.17) in [42] by the difference of point masses δy − δ0 and if I (preserving Hörmanders notation) is a cube centered at 0 and containing y, the same argument as in the original proof of Theorem 7.9.5 yields the condition (ii) of the deﬁnition of Calderon–Zygmund (scalar-valued) operators. The case of operators in a ﬁnite-dimensional space is similar, it is sufﬁcient to consider separately every entry of the multiplier matrix. Our knowledge of the complex interpolation method in the context of Sobolev spaces is still not satisfactory. For the deﬁnition of the complex interpolation method see the survey by Johnson and Lindenstrauss, Section 11 in [44], p. 76. Clearly, as in the case of real interpolation, any non-trivial result has to include at least one limit exponent (i.e., 1 or ∞). To state the next result which provides link between the real and the complex interpolation method we need the concept of Fourier type introduced by Peetre [70]. We say that a Banach space X has a Fourier type p for 1 p 2 if the vector-valued

Fourier transform maps Lp (X) into Lp (X) where 1/p + 1/p = 1. Clearly every Banach space has Fourier type 1, and every Hilbert space has Fourier type 2. The following result from the interpolation theory, due to Peetre (cf. [69,63]), provides the link between the real and the complex interpolation method. L EMMA 45. Let (X0 , X1 ) be a Banach couple such that Xj has Fourier type pj , j = 0, 1. Then (X0 , X1 )θ,pθ ⊂ (X0 , X1 )[θ] , where 0 < θ < 1, and 1/pθ = (1 − θ )/p0 + θ/p1 . The above result yields the following T HEOREM 46 (cf. [63]). For the complex interpolation scale one has 1 n p n

p L(k) R , L(k) R [θ] = L(k)θ Rn , 1/pθ = (1 − θ ) + θ/p, 1 < p < ∞. p

P ROOF. The embedding (L1(k) (Rn ), L2(k) (Rn ))[θ] ⊂ L(k)θ (Rn ) for 1/pθ = 1 − θ/2 is obvious. Since L2(k) (Rn ) is a Hilbert space, it has Fourier type 2. Thus by Theorem 44 and Lemma 45 we get

p

p L(k)θ Rn = L1(k) Rn , L2(k) Rn θ,p ⊂ L1(k) Rn , L(k) Rn [θ] . θ

Hence

p (L1(k) (Rn ), L(k) (Rn ))[θ]

p = L(k)θ (Rn ) for p 2. If p > 2, the boundedness of the p/(p−1) n p (L(k) (R ), L(k) (Rn ))[ 1 ] = L2(k) (Rn ). Thus we complete 2

Sobolev projection implies the proof applying the following result of Wolff (cf. [4,102]) which allows us to “glue” together two interpolation scales.

W OLF ’ S THEOREM . Let Xi (i = 0, 1, 2, 3) be the Banach spaces continuously embedded in a suitable topological vector space. Let 0 < θ < η < 1, θ = λμ and η = (1 − μ)θ + μ. If X1 = (X0 , X2 )[λ] and X2 = (X1 , X3 )[μ] then X1 = (X0 , X3 )[μ] and X2 = (X0 , X3 )[η] . It is an open problem, attributed to P. Jones, to describe the complex interpolation scale n (L1(k) (Rn ), L∞ (k) (R ))[θ] .

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8. Anisotropic Sobolev spaces An n-dimensional smoothness is a ﬁnite non-empty subset S of Zn+ such that if α ∈ S and p β 0 α ∈ Zn+ then β ∈ S. The anisotropic Sobolev space LS (Ω) (1 p ∞) is the space n of all scalar-valued functions f on open Ω ⊂ R having distributional partial derivatives D αf ∈ Lp (Ω) for α ∈ S, with the norm f S,p,Ω =

α

D f (x)p dx 1/p , for 1 p < ∞, maxα∈S essupx∈Rn D αf (x), for p = ∞. α∈S Ω

By C0S (Ω) we denote the closure of D(Ω) in the norm · S,∞ , and by C S (Ω) – the subspace of L∞ S (Ω) consisting of functions which together with their partial derivatives ∂ α with α ∈ S are uniformly continuous on Ω and vanish at inﬁnity. The smoothnesses {Zn+ : |α| k} are called isotropic smoothnesses (k = 0, 1, . . . , n = 1, 2, . . .). Note that every one-dimensional smoothness is isotropic. The classical Sobolev spaces are these which correspond to isotropic smoothnesses; they are also called isotropic Sobolev spaces. The Sobolev spaces on Tn can be deﬁned, for instance, as the closure of trigonometric polynomials with period 1 with respect to each coordinate in the norm · S,p,I n . In general rotations of coordinates do not preserve anisotropic smoothnesses. Thus only for a very few manifolds the anisotropic Sobolev spaces can be reasonably deﬁned. In particular we do not know how to deﬁne them on Sn for n 2. A smoothness S is uniquely deﬁned by the set S # of its maximal elements (in the partial order 0); α ∈ S is maximal provided that for every β ∈ S the condition α 0 β implies β = α. It is natural to ask the following question: given a subset S0 of a set S ⊂ Zn+ , not necessarily a smoothness, under what conditions the norms · S,p,Ω and · S ,p,Ω are equivalent? A satisfactory solution for Ω = Rn and p = 1 follows from the next two results. It is convenient to identify Zn+ , hence also S, with an appropriate subset of Rn . T HEOREM 47 ([43]). Let 1 0 such that γ D f C · D αf p p

(34)

α∈S

if and only if γ ∈ conv S. For p = ∞ the condition is more involved. T HEOREM 48 ([8]). Let S be as above and let γ ∈ Zn+ \ S. Then for p = ∞ (34) holds if and only if (∗) there exists an integer k, 0 k n, and a k-dimensional afﬁne subspace Lk ⊂ Rn parallel to some k-dimensional coordinate space, such that γ is an internal point (with respect to Lk ) of conv(Lk ∩ S).

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For p = 1 the condition (∗) is known to be necessary. It is not known whether (∗) is sufﬁcient. A partial result is due to Ornstein. T HEOREM 49 ([68]). If γ ∈ Zn+ \ S and all the maximal elements of S (with respect to the partial order “0”) are of the same order as γ then (34) does not hold for p = 1. In particular Theorem 49 yields that if S = {(0, 0), (1, 0), (0, 1), (2, 0), (0, 2)} then the embedding L1(1) (R2 ) → L1S (R2 ) is not a surjection; in other words the L1 -norm of the mixed derivative D (1,1) is not controlled by the L1 -norms of the pure derivatives of the second order. R EMARKS . (1) Using the Transference Theorem (cf. Section 2) we derive from Theorems 47 and 48 their counterparts for Tn . They also imply similar results for domains with the following individual extension property: for each multiindex α and every p ∈ [1, ∞] there is a positive constant Cα,p such that every C ∞ function f on Ω extends to a C ∞ function f˜ on Rn so that ∂ α f˜Lp (Rn ) Cα,p ∂ α f Lp (Ω) . It is easy to see that starlike domains have this property. (2) The reader is referred to [7] for far reaching generalizations of Theorems 47 and 48 involving fractional derivatives and mixed norms. (3) Mityagin [64] after Il’in [43] and prior to Boman [8] constructed elegant examples showing that condition γ ∈ conv S is not sufﬁcient for (34) in the case p = ∞. (4) The reader is referred to [97] for results partially complementing Theorem 49. They are based on Theorem 65 below. Similarly as in the case of isotropic Sobolev . spaces in Section 2 we deﬁne the canonp ical embedding J = JΩ,S,p : LS (Ω) → α∈S Lp (Ω) and the Sobolev projection P = . . p PΩ,S : α∈S Lp (Ω) → J (LS (Ω)) where α∈S Lp (Ω) is the space of tuples (fα )α∈S equipped with the norm (fα )p =

α∈S

p

fα Lp (Ω)

1/p

, for 1 p < ∞,

maxα∈S fα L∞ (Ω) ,

for p = ∞.

. . . The spaces α∈S C(Ω) and α∈S C0 (Ω) are deﬁned similarly. The space α∈S Lp (Ω) is naturally isomorphic to Lp (Ω, E), where E is an appropriate ﬁnite-dimensional Hilbert space. For a ﬁxed n-dimensional smoothness S we put for ξ ∈ Rn QS (ξ ) =

ξ 2α ;

mα,β (ξ ) = i |α|−|β| ξ α ξ β Q−1 S (ξ );

α∈S

Tmα,β = Tα,β

(α, β ∈ S).

QS is called the fundamental polynomial of S.

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The Sobolev projection is given by

PRn ,S (fα ) =

Tα,β (fβ )

α∈S

α∈S

/ (fα ) ∈ S Rn . α∈S

As in the isotropic case the Lp -boundedness and weak type (1, 1) of a Sobolev projection PRn ,S is equivalent to the same properties of multiplier transforms Tmα,β for all α, β ∈ S. This allows us to extend part (b) of Theorem 4 to arbitrary smoothnesses. T HEOREM 50 ([75]). PRn ,S is of strong type (p, p) for 1 < p < ∞. The proof differs from the proof of Theorem 4. Instead of the Hörmander–Mikhlin criterion one uses the Marcinkiewicz multidimensional multiplier theorem (cf. [90], Chapter IV, §6, Theorem 6) to show that each of the multipliers mα,β is of strong type (p, p) for 1 < p < ∞. Note that part (c) of Theorem 4 extends trivially to arbitrary smoothnesses because every one-dimensional smoothness is isotropic. Part (a) of Theorem 4 does not extend to arbitrary smoothnesses. Examples of smoothnesses whose Sobolev projections fail to be of weak type (1, 1) are shown later. There is a transparent characterization of smoothnesses whose Sobolev projections are of strong types (1, 1) and (∞, ∞). T HEOREM 51. For an n-dimensional smoothness S the following are equivalent (i) PRn ,S is of strong type (1, 1); (ii) PRn ,S is of strong type (∞, ∞); (iii) PTn ,S is of strong type (1, 1); (iv) PTn ,S is of strong type (∞, ∞); (v) S has exactly one maximal element. P ROOF. The equivalences (i) ⇔ (ii) and (iii) ⇔ (iv) follow by duality. The implication (i) ⇒ (iii) is a consequence of the Transference Theorem (cf. Section 2). It can be easily veriﬁed that (v) implies that S is a Cartesian product of one-dimensional smoothnesses, hence PS is a tensor product of Sobolev projections of these one-dimensional smoothnesses. Since every one-dimensional smoothness is of strong type (1, 1) (cf. Section 2, Theorem 4(c)), its tensor powers have the same property. Thus (v) ⇒ (i). It remains to show that “non(v)” ⇒ “non(iii)”. Recall that PTn ,S is of strong type (1, 1) for some smoothness S iff mα,β is a Fourier transform of a measure from M(Tn ) for every α ∈ S and β ∈ S, where M(Tn ) denotes the space of scalar-valued Borel measures on Tn with ﬁnite total variation. We need (for the proof see [48,72]) W IENER CRITERION . If f = μˆ then lim

d(I )→∞ I

f dσ = μ {0} f : Zn → C, μ ∈ M Tn . σ (I )

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Here I = I (b; r) := {a ∈ Zn : |aj − bj | rj for j = 1, 2, . . . , n} for r, b ∈ Zn , d(I ) = min1j n |rj |, and σ is the counting measure on Zn . We shall show that “non(v)” implies that mα,α violates Wiener criterion for some α ∈ S. Indeed “non(v)” yields that given α ∈ S there is a β ∈ S such that αj < βj for some (k) j ∈ {1, 2, . . ., n}. For k = 1, 2, . . . deﬁne Ik = I (b (k) ; r (k)) by rj = k for j = 1, 2, . . . , n; bj = k for j = j and bj = 4k . Then σ (Ik ) = (2k + 1)n and d(Ik ) = k; if a ∈ Ik then (k)

(k)

a 2α (2k)2|α|(4k + k)2αj and QS (a) (4k − k)2βj . Thus, remembering that mα,α (a) = a 2α /QS (a), we get lim

k→∞ Ik

mα,α (a) (2k)2|α| (4k + k)2αj dσ = lim = 0. k→∞ σ (Ik ) (4k − k)2βj

If mα,α satisﬁed the Wiener criterion then the limit would not depend on a particular choice of a sequence (Ik ). Since α ∈ S has been taken arbitrarily, we would have lim

d(I )→∞ I

mα,α (a) dσ = 0, σ (I )

α∈S

which contradicts the identity

α∈S mα,α

= 1.

It is an open, probably difﬁcult, problem to characterize these smoothnesses for which the Sobolev projection is of weak type (1, 1). First we enlist some “positive” results. T HEOREM 52 ([6]). Every two-dimensional smoothness is of weak type (1, 1). T HEOREM 53 ([72]). If the fundamental polynomial QS is h-elliptic, in particular if QS is elliptic, then PRn ,S is of weak type (1, 1). Recall that a mixed homogeneity is a vector h = (h1 , h2 , . . . , hn ) ∈ Zn with 1 h1 h2 · · · hn . A polynomial Q : Rn → C is said to be h-elliptic provided that there are | > R where ρh (ξ ) = 0 for C > 0 and R > 0 such that (ρh )degh (Q)(ξ ) C|Q(ξ )| for |ξ ξ = 0 and ρh (ξ ) is the unique positive root of the equation nj=1 ξj2 ρ −2hj = 1 for ξ = (ξj ) ∈ Rn , and degh (Q) is the maximum of h-degrees of monomials appearing in the shortest representation of Q as the sum of monomials; the h-degree of the monomial ξ α is h, α for α ∈ Zn+ . C OROLLARY 54 ([72]). Let S be an n-dimensional smoothness such that the set of maximal elements of S is a set of n pure derivatives. Then PRn ,S is of weak type (1, 1). The proofs of Theorems 52 and 53 are based upon the following (cf. [28], p. 28, [84]).

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T HE FABES –R IVIÈRE CRITERION . Let h be a mixed homogeneity. Let m : Rn → C be a n bounded function. Let s be an integer such that 2s > |h| = j =1 hj . Assume that for every |α| s the function m has continuous in Rn \ {0} partial derivative ∂ α m which satisﬁes

h,α α 2 R ∂ m(x) R −|h| dx < ∞.

sup R>0

R/2ρh (x)2R

Then m is a weak type (1, 1) multiplier. For h = (1, 1, . . . , 1) we get the classical Hörmander–Mikhlin criterion which we used to prove Theorem 4(a). The (1, 1, . . . , 1)-elliptic polynomials coincide with classical elliptic polynomials. A satisfactory characterization of smoothnesses S with PRn ,S of weak type (1, 1) is also known for smoothnesses of ord S 2, where ord S := max{|α|: α ∈ S}. A smoothness S is non-degenerated if all maximal elements of S are of order 1; it is reducible if ξj divides all symbols of all maximal elements of S for some j ∈ {1, 2, . . ., n}. T HEOREM 55 ([6]). Let S be a non-degenerated smoothness of order 2. Then (i) if ord S = 1 then S is isotropic hence PRn ,S is of weak type (1, 1); (ii) if S is reducible then PRn ,S is of weak type (1, 1); (iii) if S is irreducible and all maximal elements of S are of order 2 then the Sobolev projection is of weak type (1, 1) iff for every i, j ∈ {1, 2, . . . , n}, i = j , either ∂ 2 /∂xj2 ∈ S and ∂ 2 /∂xi2 ∈ S, or ∂ 2 /∂xi ∂xj ∈ S; (iv) if ∂/∂xi are the maximal elements of S for i ∈ A ⊂ {1, 2, . . . , n} and A = ∅ then the Sobolev projection is of weak type (1, 1) iff ∂ 2 /∂xj2 ∈ S for every j ∈ A. The proof of Theorem 55 is complicated and uses a variety of methods (cf. [6]). It follows from Theorem 55(iii) and (iv) that for n 3 there are examples of n-dimensional smoothnesses whose Sobolev projections fail to be of weak type (1, 1). In particular the Sobolev projections are not of weak type (1, 1) for the three-dimensional smoothnesses T3 and S with maximal elements T3# = {(1, 0, 0), (0, 1, 1)} and S # = {(2, 0, 0), (0, 1, 1)}. Yet another example is mentioned in remark (3) below. R EMARKS . (1) Theorem 51 is contained in [75] and (implicitly) in [87]. The arguments there as well as in [51] are rather complicated. The simple argument presented here is taken from [72]; it is based on the idea of J.-P. Kahane to apply Wiener criterion (private communication in 1987). (2) The counterparts of Theorems 52, 53, 55 and Corollary 54 for Tn holds. (3) A simple tool to study Sobolev projections . uses the quantity ap (PS ) = the norm of the projection regarded as an operator on α∈S Lp (Rn ). By duality ap (PS ) = ap (PS ) for p = p/(p − 1). Moreover, by the Marcinkiewicz interpolation theorem (cf. [107], XII (4.6)), if PS is of weak type (1, 1) then ap (PS ) C max(p, p ) for some constant C independent of p. This is not the case for some smoothness. For instance, if S is a fourdimensional smoothness being the Cartesian product of two two-dimensional isotropic smoothnesses, i.e., S # = {(1, 0, 1, 0), (1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 0, 1)} then PR4 ,S is the

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second tensor power of PR2 ,(1) thus ap (PS ) = [ap (PR2 ,(1))]2 [A(1, 2) max(p, p/(p − 1))]2 (cf. Section 2, remark (2) after Proposition 7). (4) If the n-dimensional smoothness Tn is deﬁned by Tn# = {(1, 0, . . . , 0), (0, 1, . . ., 1)} then ap (PTn ) Cn [max(p, p/(p − 1))]n−1 for some Cn > 0 independent of p (cf. [6], Proposition 2.47). One conjectures that for every n-dimensional smoothness S one has ap (PRn ,S ) CS [max(p, p/(p − 1))]n−1 for some CS > 0 independent of p. We know only that the above estimate is true if we replace n − 1 by n (cf. [100]). (5) Another open problem related to the theory of rational multipliers on Rn is the following. Is it true that for every smoothness S there exists a non-negative integer r such that C[max(p, p/(p − 1))]r ap (PS ) C −1 [max(p, p/(p − 1)]r for some C > 0 independent of p? Theorem 6 (cf. Section 2) extends with almost the same proof to arbitrary smoothnesses. T HEOREM 56. The multiplier transform T1/√QS extends to an isomorphism from Lp (Rn ) p onto LS (Rn ) for 1 < p < ∞ (k = 0, 1, . . . , n = 1, 2, . . .). Applying the Transference Theorem (cf. Section 2) we get the counterpart of Theorem 56 for Tn . We do not know the analogs of the extension theorems WET and JET (cf. Section 3) for anisotropic smoothnesses. We do not know whether C S (Tn ) and L1S (Tn ) are inﬁnitely divisible; in particular we do not know whether C S (Tn ) (resp. L1 (Tn )) is isomorphic to C S (Rn ) (resp. L1S (Rn )) for anisotropic smoothnesses. The next result sufﬁces to prove various properties of C S (Ω) and L1S (Ω) for arbitrary ∅ = Ω ⊂ Rn . p

T HEOREM 57. Let ∅ = Ω ⊂ Rn . Then LS (Tn ) for 1 p ∞ (resp. C S (Tn )) is isomorp phic to a complemented subspace of LS (Ω) for 1 p ∞ (resp. C S (Ω)). P ROOF. All the operators constructed below are well deﬁned for C ∞ functions. Examp ining the formulae we check that the operators are bounded in LS -norms. Thus they can be extended to bounded operators in appropriate Sobolev spaces in p-th norms. We put K n = (−1; 1)n , Qn = (0; 1)n , Hj− = {x = (xj ) ∈ Rn : xj < 0}, Hj0 = {x ∈ Rn : xj = 0}, Kj,n = K n ∩ Hj− , ej – the j -th coordinate versor, A – the closure of a set A ⊂ Rn (j = 1, 2, . . . , n; n = 1, 2, . . .). k r r Step I. Let F (z) = ∞ k=0 ak z be an entire function such that F (2 ) = (−1) for r = ∞ 0, 1, . . . . Let φ be a C non-negative function such that φ(t) = 1 for −1 < t < 0, and φ(t) = 0 for t > 4/3. Put ⎧ (x), for x ∈ Kj,n , ⎪ ⎨f k Ej f (x) = k ak φ(xj )f (x1 , x2 , . . . , xj −1 , −2 xj , xj +1 , . . . , xn ), ⎪ ⎩ for x ∈ K n \ Kj,n .

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Then Ej : C S (Kj,n ) → C S (K n ) is a linear extension operator (cf. [67] for details). The continuity of Ej in the Sobolev norm deﬁned by S follows from the deﬁnition of smoothness and the Leibniz formula cα,β ∂ α−β φ∂ β f, ∂ α (φf ) = 00β0α

where cα,β are appropriate “binomial” coefﬁcients. Step II. Put n

S Cj,0 Q = f ∈ C S Qn : lim ∂ α f (y) = 0 for x ∈ Hj0 ∩ Qn ; α∈S

C0S

n

Q

=

n

y→x

n

S Q . Cj,0

j =1

Then (+) C0S (Qn ) is a complemented subspace C S (K n ); (++) C0S (Qn ) is isomorphic to C S (Tn ). To establish (+) it is enough to construct a linear extension operator, say E : C S (K n \ Qn ) → C S (K n ). Then the desired projection is IdC S (K n ) −E; we can regard C0S (Qn ) as a subspace of C S (K n ) extending each function by 0. Let Rj denote the restriction operator to Kj,n for functions deﬁned on a superset of Kj,n . Put Λj := Ej Rj : C S (K n \ Qn ) → C S (K n ). Then the desired extension operator is E=

(−1)k Λjk ◦ Λjk−1 ◦ · · · ◦ Λj1 ;

the sum extends over all sequences of indices n jk > jk−1 > · · · > j1 1 (k = 1, 2, . . . , n). To verify (++) let Zj = Qn + ej and Qj,n = Qn ∪ Zj \ bd Qn ∪ Zj . Denote by Ej : C S (Qn ) → C S (Qj,n ) the linear extension operator which is an obvious modiﬁcation of Ej deﬁned in Step I. Note that limy→x (∂ α Ej f )(y) = 0 for α ∈ S, x ∈ Qj,n ∩ {Hj0 + 4/3ej }, f ∈ CS (Qn ). Let R|Zj denote the operator of restriction to Zj . Put Nj f (x) =

f (x) + R|Zj Ej f (x + ej ), f (x),

for x ∈ Qn with 0 < xj < 1/3, otherwise.

S (Qn ) onto the subspace of C S (Q ) consisting Then Nj is an isomorphism from Cj,0 n of functions which extend continuously together with their partial derivatives from S to 1-periodic function with respect to j -th coordinate. The inverse of Nj is the operator Mj deﬁned by

f (x) − R|Zj Ej f (x + ej ), for x ∈ Qn with 0 < xj < 1/3, Mj f (x) = f (x), otherwise.

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1415

Clearly, C S (Tn ) can be identiﬁed with the subspace of C S (Qn ) consisting of functions on Qn which extend continuously on Rn together with their partial derivatives from S to 1-periodic functions with respect to all coordinates. The desired isomorphism from C0S (Qn ) onto C S (Tn ) is deﬁned by N = Nn ◦ Nn−1 ◦ · · · ◦ N1 ; the inverse is deﬁned by M = M1 ◦ M2 ◦ · · · ◦ Mn . This concludes the proof of Step II. Finally observe that by a standard modiﬁcation of the extension operator constructed in Step I we construct an extension operator from C S (K n ) into C0S (2K n ). Thus C S (K n ) is isomorphic to a complemented subspace of C0S (2K n ). This sufﬁces to get the assertion of the theorem. Our next result is a generalization of Theorems 13 and 25. T HEOREM 58. Let S be an arbitrary n-dimensional smoothness for some n 2. Then the following are equivalent (v) S has exactly one maximal element; (a1 ) L1S (Tn ) (resp. L1S (Rn )) is isomorphic to L1 [0; 1]; (a2 ) L1S (Tn ) (resp. L1S (Rn )) is an L1 -space; (a3 ) every linear operator from L1S (Tn ) (resp. L1S (Rn )) to a Hilbert space is 1-summing; (b1 ) C S (Tn ) (resp. C S (Rn )) is isomorphic to C[0; 1]; (b2 ) C S (Tn ) (resp. C S (Rn )) is a L∞ -space; (b3 ) C S (Tn ) (resp. C S (Rn )) has GL. The proof of Theorem 58 is not straightforward. It bases on the following P ROPOSITION 59 (Solonnikov [89]). Let r, s be positive integers. Then there are positive constants C = C(r, s) and C = C (r, s) such that 2 (r,0) (0,s) ∂ ∂ (A) |ξ1 |r−1 |ξ2 |s−1 fˆ(ξ ) dξ C f (x) dx f (x) dx R2

R2

2

R2

for f ∈ S R ; r s 2 (0,l) r−1 t −1

(k,0) fˆ(a) |a1 | |a2 | ∂ C f L1 (T2 ) ∂ f L1 (T2 ) (B) a∈Z2

for f ∈ Trig T .

k=0 l=0

2

Applying the Plancherel identity to the left-hand sides of (A) and (B) one can view Proposition 59 as a Sobolev embedding type result. The case r = s = 1 is the classical Sobolev embedding for k = 1, n = 2, p = 1, q = 2. The proof of part (A) is easy for both r and s odd. We use the identity 1R− ×R+ $ ∂x∂ 1 h, ∂x∂ 2 g = h, g for h, g ∈ S(R2 ), which is a slight modiﬁcation of the identity (∗) in Section 4. Apply that identity for h = ∂ (r−1,0)f , g = ∂ (0,s−1)f , and use Plancherel identity and the Hausdorff–Young inequality. For other pairs of positive integers r, s the argument is more sophisticated (cf. [89,75,51]).

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Part (B) follows from (A) routinely. Fix φ ∈ D(Rn ) with φ(x) = 1 for x ∈ I 2 and φ(x) = 0 for x ∈ / 2I 2 . Apply (A) to f · φ where f is here understand as 1-periodic with respect to each coordinate function deﬁned on R2 . Use Leibniz formula for derivatives of the product. To reduce the proof of Theorem 58 to two-dimensional smoothnesses we need L EMMA 60 ([51]). Let n > 2. Then for every n-dimensional smoothness S which has more than one maximal element there exists a two-dimensional smoothness T which has more p than one maximal element such that LS (Rn ) for 1 p ∞ (resp. C S (Rn )) has a comp plemented subspace isomorphic LT (R2 ) (resp. C T (R2 )). The same is true for periodic models. O UTLINE OF THE PROOF. We consider the case of Rn ; the case of Tn can be proved similarly. If S has more than one maximal element then there is an ordering of coordinates and β, γ ∈ S # such that |β| = ord S and β1 < γ1 . Deﬁne Φ : Zn → Zn−1 by Φ(α) = (α1 , α2 , . . . , αn−2 , αn−1 + αn ). Then Φ(S) is an (n − 1)-dimensional smoothness with Φ(β) = Φ(γ ) and Φ(β), Φ(γ ) ∈ [Φ(S)]# . Denote by XS (resp. XΦ(S) ) one of the p p spaces LS (Rn ) for 1 p ∞ or C S (Rn ) (resp. LΦ(S) (Rn−1 ) or C Φ(S) (Rn−1 )). Fix a non-negative h ∈ D(R) with h(t) = h(−t) for t ∈ R and R h2 (t) dt = 1. Denote by z◦ the ﬁrst n − 2 coordinates of a vector z. Deﬁne U : XΦ(S) → XS and P : XS → XΦ(S) by

(Ug) x ◦ , xn−1 , xn = g x ◦ , xn−1 + xn h (xn−1 − xn )/2

g ∈ XΦ(S) , x = (xj ) ∈ Rn , ◦

(Pf ) y , yn−1 = f y ◦ , t, yn−1 − t h(yn−1 /2 + t) dt

R

f ∈ XS , y = (yj ) ∈ Rn−1 .

One veriﬁes that U and P are bounded operators satisfying P U = IdXΦ(S) (cf. [75], Lemma 5.3 for details). Now the desired conclusion follows by backward induction. P ROOF OF T HEOREM 58. Same argument as for isotropic spaces shows that L1S (Tn ) (resp. C S (Tn )) has a complemented subspace isomorphic to L1 [0; 1] (resp. to C(0; 1)). Thus the implications (v) ⇒ (a1 ) and (v) ⇒ (b1 ) follow from Theorem 51 by the standard decomposition method. The implications (a1 ) ⇒ (a2 ) ⇒ (a3 ) and (b1 ) ⇒ (b2 ) ⇒ (b3 ) are formal. Thus in view of Theorem 57 and Lemma 60 to complete the proof it is enough to work with periodic models and to establish the implications “non(v)” ⇒ “non(a3)” and “non(v)” ⇒ “non(b3)” under the additional assumption that S is a two-dimensional smoothness. If S is a two-dimensional smoothness with more than one maximal element then there are α and β in S # such that r = β1 − α1 1, s = α2 − β2 1, and the line passing through α and β supports conv S, equivalently rγ2 + sγ1 rα2 + sα1

for γ = (γ1 , γ2 ) ∈ S.

(35)

Sobolev spaces

1417

Apply the inequality (B) of Proposition 59 to f = ∂ (α1 ,β2 ) g for g ∈ Trig T2 . We get 2 g(a) ˆ |a1 |α1 +β1 −1 |a2 |α2 +β2 −1 C g2 1

LS (T2 )

a∈Z2

.

(36)

Deﬁne V : L1S (T2 ) → L2S (T2 ) by Vg =

7 g(a) ˆ

a∈Z2

|a1 |(α1 +β1 −1) |a2 |(α2 +β2 −1) ea QS (a)

for g ∈ L1S T2 .

Then using that ( √Qea (a) )a∈Z2 is an orthonormal system in L2S (T2 ) and (36), we infer that S

V is a bounded operator. Thus V I2,1 , where I2,1 : L2S (T2 ) → L1S (T2 ) is the natural embed% α +β −1 α +β −1 1 1 |a2 | 2 2 ding, is not Hilbert–Schmidt. Indeed V (ea ) = |a1 | ea for a ∈ Z2 . RevokQS (a) ing (35) we get |a1 |α1 +β1 −1 |a2 |α2 +β2 −1 #

V I2,1 ea / QS (a) 2 2 2 = = +∞. LS (T ) QS (a) 2 2

a∈Z

a∈Z

Thus V is not 1-summing. The proof of the implication “non(v) ⇒ non(b3)” for two-dimensional smoothnesses is a repetition with a few inessential changes of the proof of Theorem 25. First observe that Proposition 27 extends from isotropic case to the case of two-dimensional smoothnesses, because, by Theorem 52, PS is of weak type (1, 1). Next as in the proof of Theorem 13 we show that U = V I∞,1 (where I∞,1 : CS (T2 ) → L1S (T2 ) is the natural embedding and V : L1S (T2 ) → L2S (T2 ) is just constructed bounded not 1-summing operator) is an example of 1-summing operator which is not L1 -factorable. In the spirit of Theorem 58 is also T HEOREM 61 ([81], Section 8). An n-dimensional smoothness S has exactly one maximal element iff L1S (Rn ) is isomorphic to a complemented subspace of a Banach lattice. Proof of Theorem 61 is based on the same idea as the proof of Theorem 15. It requires a version of Peetre’s Theorem 20 for anisotropic Sobolev spaces spaces. It is natural to ask which anisotropic smoothnesses share properties of spaces of continuous functions discussed in Section 5. For an n-dimensional smoothness S put S∗ := S \ S # ; i.e., S∗ is the set of all nonmaximal elements of S. Clearly if S = {0} then S∗ is a smoothness and the inclusions C S (Ω) → C S∗ (Ω) and C0S (Ω) → C0S∗ (Ω) are bounded for Ω ⊂ Rn . If Ω = Rn then C S (Rn ) → C S∗ (Rn ) is compact for no S = {0}! Call an n-dimensional smoothness S of Ascoli type provided that the inclusion C S (I n ) → C S∗ (I n ) is compact. Clearly isotropic smoothnesses are of Ascoli type and S is of Ascoli type iff for every (equivalently for

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A. Pełczy´nski and M. Wojciechowski

some) open bounded ∅ = Ω ⊂ Rn or for Tn the corresponding inclusions are compact. One can show that S = {0} is Ascoli type iff α ∈ S∗ implies α + t (1, 1, . . . , 1) belongs to the interior of conv S for small t > 0. T HEOREM 62. If S is of Ascoli type then the duals of C S (Tn ), C S (Ω) and C0S (Ω) for ∅ = Ω ⊂ Rn are separable perturbations of the dual of C(0; 1). The proof reduces in view of Theorem 57 to the case of C(Rn ). The proof in this case can be found in [73], Theorem A; it is a slight modiﬁcation of the proof of Theorem 28 (cf. Section 5). Unfortunately in [73] in the formulation of Theorem A it is erroneously stated that “the inclusion C S (Rn ) → C S∗ (Rn ) is compact” instead of “S is of Ascoli type”. Slightly modifying the proof of Lemma 31 we get L EMMA 63.. If an n-dimensional smoothness S is of Ascoli type then J (C S (Tn )) is a rich subspace of α∈S C(Tn ) = C(Tn , E). Thus invoking Theorem 30 we get C OROLLARY 64. If an n-dimensional smoothness S is of Ascoli type then C S (Tn ) has properties (a), (b), (c) stated in Section 5. R EMARKS . (1) Theorem 58 is essentially due to Kislyakov and Sidorenko [51]; in [75] a similar result is proved with (a3 ) replaced by the weaker condition “C S (Rn ) (resp. C S (Tn )) is not isomorphic to a quotient of L∞ -space”. (2) We do not know the characterization of these smoothnesses that the dual of C S (Tn ) is a separable perturbation of an L1 -space. Every two-dimensional smoothness has this property (cf. [73], Proposition 3.2). On the other hand if S and S

are smoothnesses such that S = {0} and S

has more than one maximal element then the dual of C S (Tn ) is not a separable perturbation of an L1 -space where S = S ×S

and n = dim S +dim S

(cf. [73], Proposition 3.1). An interesting question concerning Banach space properties of anisotropic Sobolev spaces is whether L1S (Rn ) contains a complemented inﬁnite-dimensional Hilbertian subspace. The answer is unknown for isotropic spaces. Surprisingly, we are able to construct complemented invariant inﬁnite-dimensional Hilbertian subspaces of some anisotropic Sobolev spaces. Those invariant projections are related to the Paley projections in H 1 (cf. [17], p. 275). For an n-dimensional smoothness S and an m-element set A ⊂ {1, 2, . . . , n} put S|A = pA (S) where pA is the projection deﬁned by pA (x) = (xj )j ∈A for x = (xj )nj=1 ∈ Rn . T HEOREM 65 ([76]). Let S be an n-dimensional smoothness. The space L1S (Tn ) contains an invariant complemented inﬁnite-dimensional Hilbertian subspace iff (2) there is ∅ = A ⊂ {1, 2, . . . , n} such that (S|A )# contains two elements α, β such that |α| − |β| is an odd integer.

Sobolev spaces

1419

The projection is given by the idempotent multiplier which is a characteristic function of some Hadamard lacunary sequence of characters of Tn . We indicate the idea of the proof in the simplest case of the two-dimensional smoothness S = {(2, 0), (1, 0), (0, 1), (0, 0)}. Let (nk ) be sufﬁciently fast growing sequence of positive integers (for example, nk = k! for k = 1, 2, . . .). Put ak = (nk , n2k ) and let H be the (Hilbertian) subspace of L1S (T2 ) spanned by the characters {eak : k = 1, 2, . . .}. Then the characteristic function of {ak : k = 1, 2, . . .} induced a bounded multiplier transform on * L1S (T2 ). The proof of that statement goes as follows: put Rk (x) = kj =1 (1 + cosx, aj ) and let R be a weak-star limit of Rk . By the classical Riesz products theory, R is a prob k ) = 1 for k = 1, 2, . . . . One can prove (this is the place abilistic measure such that R(a where the property that (2, 0) and (0, 1) have different order modulo 2 is used) that for every f ∈ J (L1S (T2 )) the convolution R ∗ f belongs to the E-valued Hardy space in 2variables-H 1(T2 , E) where E is the four-dimensional Hilbert space. Then we apply the restriction to R ∗ J (L1S (T2 ) of the appropriate Paley projection (cf. [76] for details). If S fails (2), then a complete description of all complemented invariant subspaces of L1S (Tn ) is known, it is the same as for L1 (Tn ) (cf. [85]). T HEOREM 66 ([98]). The following dichotomy holds: for every smoothness S either L1S (Tn ) contains an inﬁnite-dimensional complemented invariant Hilbertian subspace, or every invariant projection in L1S (Tn ) is a convolution with an idempotent measure on Tn . In contrast for the sup-norm we have T HEOREM 67 ([98]). For each n-dimensional smoothness S every invariant projection in C S (Tn ) is a convolution with an idempotent measure on Tn . A consequence of Theorem 65 is C OROLLARY 68 ([77]). An n-dimensional smoothness S satisﬁes (2) iff there exists an 1-summing invariant surjection from C S (Tn ) onto an inﬁnite-dimensional subspace of L2S (Tn ). Note that the isotropic smoothnesses do not satisfy (2). Hence L1(k) (Tn ) does not have invariant inﬁnite-dimensional Hilbertian subspaces; the invariant projections in the space are only convolutions with idempotent measures (n, k = 1, 2, . . .).

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[67] Z. Ogrodzka, On simultaneous extension of inﬁnitely differentiable functions, Studia Math. 28 (1967), 193– 207. [68] D. Ornstein, A non-inequality for differential operators in the L1 norm, Arch. Rational Mech. Anal. 11 (1962), 40–49. [69] J. Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Mat. Univ. Padova 42 (1969), 15–26. [70] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser. 1, Durham (1976). [71] J. Peetre, A counterexample connected with Gagliardo’s trace theorem, Comment. Math., tomus specialis 2 (1979), 277–282. [72] A. Pełczy´nski, Boundedness of the canonical projection for Sobolev spaces generated by ﬁnite families of linear differential operators, Analysis at Urbana, Vol. I (Urbana, 1986–87), London Math. Soc. Lecture Note Ser. 137, Cambridge Univ. Press, Cambridge (1989), pp. 395–415. [73] A. Pełczy´nski, An analogue of the F. and M. Riesz theorem for spaces of differential functions, Banach Space Theory (Iowa City, 1987), Contemporary Math. 85, Amer. Math. Soc., Providence (1989), 405–425. [74] A. Pełczy´nski, Nonisomorphism of the disc algebra with spaces of differentiable functions, Miniconference on Probability and Analysis (Sydney, 1991), Proc. Centre Math. Appl. Austral. Nat. Univ. 29, Austral. Nat. Univ., Canberra (1992), 183–194. [75] A. Pełczy´nski and K. Senator, On isomorphisms of anisotropic Sobolev spaces with “classical Banach spaces” and a Sobolev type embedding theorem, Studia Math. 84 (1986), 169–215. [76] A. Pełczy´nski and M. Wojciechowski, Paley projection on anisotropic Sobolev spaces on tori, Proc. London Math. Soc. (3) 65 (1992), 405–422. [77] A. Pełczy´nski and M. Wojciechowski, Absolutely summing surjections from Sobolev spaces in the uniform norm, Progress in Functional Analysis (Peniscola, 1990), North-Holland Math. Stud. 170, North-Holland, Amsterdam (1992), 423–431. [78] A. Pełczy´nski and M. Wojciechowski, Molecular decompositions and embedding theorems for vectorvalued Sobolev spaces with gradient norm, Studia Math. 107 (1993), 61–100. [79] A. Pełczy´nski and M. Wojciechowski, Contribution to the isomorphic classiﬁcation of Sobolev spaces p L(k) (Ω) (1 p < ∞), Recent Progress in Functional Analysis, Proceedings Valdivia Conference, Valencia, July 2000, K.D. Bierstedt, J. Bonet, M. Maestre and J. Schmets, eds, North-Holland Math. Stud. (2001), 133–142. [80] A. Pełczy´nski and M. Wojciechowski, Sobolev spaces in several variables in L1 -type norms are not isomorphic to Banach lattices, Ark. Mat. 40 (2002), 363–382. [81] A. Pełczy´nski and M. Wojciechowski, Spaces of functions with bounded variation and Sobolev spaces without local unconditional structure, J. Reine Angew. Math., to appear. [82] G. Pisier, Interpolation between H p spaces and noncommutative generalizations, Paciﬁc J. Math. 155 (1992), 341–368. [83] S. Poornima, An embedding theorem for the Sobolev space W 1,1 , Bull. Sci. Math. (2) 107 (1983), 253–259. [84] N.M. Rivière, Singular integrals and multiplier operators, Arkiv Math. 9 (1971), 243–278. [85] W. Rudin, Fourier Analysis on Groups, Interscience, New York (1962). Reprint, Wiley, New York (1990). [86] G. Schechtman, Concentration, results and applications, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1603–1634 (this Handbook). [87] N.G. Sidorenko, Nonisomorphy of some Banach spaces of smooth functions with a space of continuous functions, Funktsional. Anal. i Prilozhen. 21 (4) (1987), 91–93 (in Russian). English transl.: Functional Anal. Appl. 21 (1987), 340–342. [88] S.L. Sobolev, On a theorem of functional analysis, Mat. Sb. (N.S.) 4 (1938), 471–497. → − [89] V.A. Solonnikov, Certain inequalities for functions from the classes W p (Rn ), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 194–210 (in Russian). [90] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton (1970). [91] E.M. Stein, Harmonic Analysis, Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton (1993). [92] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, Princeton (1971).

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[93] N. Tomczak-Jaegermann, Banach–Mazur Distances and Finite-dimensional Operator Ideals, Longman Scientiﬁc and Technical, Essex (1989). [94] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB, Berlin (1977). [95] N.Th. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, Cambridge (1992). [96] H. Whitney, Analytic extensions of differentiable functions deﬁned in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89. [97] M. Wojciechowski, Non-inequalities of Ornstein type in partial derivatives, Séminaire d’Initiation à l’Analyse, Paris VI (1992–93). [98] M. Wojciechowski, Characterizing translation invariant projections on Sobolev spaces on tori by the coset ring and Paley projections, Studia Math. 104 (1993), 181–193. [99] M. Wojciechowski, On the summing property of the Sobolev embedding operators, Positivity 1 (1997), 165–170. [100] M. Wojciechowski, A Marcinkiewicz type multiplier theorem for H 1 spaces on product domains, Studia Math. 140 (2000), 272–287. [101] P. Wojtaszczyk, Banach Space for Analysts, Cambridge Univ. Press, Cambridge (1991). [102] T. Wolff, A note of interpolation spaces, Harmonic Analysis (Minneapolis 1981), Lecture Notes in Math. 908, Springer, Berlin (1982), 199–204. [103] K. Wo´zniakowski, A new proof of the restriction theorem for weak type (1, 1) multipliers on Rn , Illinois J. Math. 40 (1996), 470–483. [104] W.P. Ziemer, Weakly Differentiable Functions, Springer (1989). [105] N. Zobin, Whitney’s problem of functions and intrinsic metric, Adv. Math. 133 (1998), 96–132. [106] N. Zobin, Extension of smooth functions from ﬁnitely connected planar domains, J. Geom. Anal. 9 (1999), 489–508. [107] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge (1978).

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CHAPTER 33

Operator Spaces Gilles Pisier∗ Équipe d’Analyse, Université Paris VI, Case 186, F-75252 Paris Cedex 05, France Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Minimal tensor product . . . . . . . . . . . . . . . . . . . . . . . 2. Ruan’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Quotient, ultraproduct and interpolation . . . . . . . . . . . . . . 5. Projective tensor product . . . . . . . . . . . . . . . . . . . . . . 6. Haagerup tensor product . . . . . . . . . . . . . . . . . . . . . . 7. Characterizations of operator algebras and modules . . . . . . . 8. The operator Hilbert space OH and non-commutative Lp -spaces 9. Local theory and exactness . . . . . . . . . . . . . . . . . . . . . 10. Applications to tensor products of C ∗ -algebras . . . . . . . . . . 11. Local reﬂexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Injective and projective operator spaces . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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0. Introduction The theory of “Operator Spaces” is quite recent. The starting point is the thesis of Ruan [94] who gave an “abstract” characterization of operator spaces. Soon after, Blecher and Paulsen [13] and Effros and Ruan [32] independently discovered that this characterization allows the introduction of a duality in the category of operator spaces and they developed the theory systematically from that point on (cf. [32–36,30,7–9,80]). The notion of operator space is intermediate between that of Banach space and that of C ∗ -algebra. They could also be called “non-commutative Banach spaces” (but the commutative case should be included!) or else “Quantum Banach spaces” (but “quantum” has been used so many times already. . .). An operator space (sometimes in short o.s.) is simply a closed subspace E ⊂ B(H ) of the space B(H ) of all bounded operators on a Hilbert space. This deﬁnition is a bit disconcerting: every Banach space E admits (for a suitable H ) an ⊂ B(H ), therefore all Banach spaces can appear as operator spaces. But isometric copy E the novelty is in the morphisms (and the isomorphisms) which are not those of the category of Banach spaces. Instead of bounded linear maps, we use as morphisms the completely bounded (in short c.b.) ones which appeared as a powerful tool in the early 80’s (see [79]) but were already implicit in the pioneering work of Stinespring (1955) and Arveson (1969) on completely positive maps, [4]. The underlying idea is the following: given two operator spaces: E1 ⊂ B(H1 ),

E2 ⊂ B(H2 ),

we want morphisms which respect the realizations of the Banach spaces E1 and E2 as operator spaces. For instance, if there exists a representation π : B(H1 ) → B(H2 ) (i.e., we have π(xy ∗) = π(x)π(y)∗ , π(1) = 1 whence π = 1) such that π(E1 ) ⊂ E2 , then the “restriction” π|E1 : E1 → E2 must clearly be accepted among morphisms, whence a ﬁrst type. Of course, the drawback is that this class does not form a vector space, but there is also a second type of natural morphisms: suppose given two bounded operators a : H1 → H2 and b : H1 → H2 , and consider the mapping Mab : B(H1 ) → B(H2 ) given by Mab x = axb ∗ . Then again, if Mab (E1 ) ⊂ E2 , it is natural to accept the restriction of Mab to E1 as a morphism. Completely bounded maps can be described as compositions of a morphism of the ﬁrst type followed by one of the second type. N OTATION . Let E ⊂ B(H ) be an operator space; we denote by Mn (E) the space of n × n matrices with coefﬁcients in E, equipped with the norm: ∀a = (aij ) ∈ Mn (E), 2 1/2 2 h a h ∈ H h 1 . aMn (E) = sup ij j j j i

(0.1)

j

In other words, we view the matrix a as acting on H ⊕ · · · ⊕ H and we compute its usual norm.

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D EFINITION 1. Let E ⊂ B(H ), F ⊂ B(K) be two operator spaces and let u : E → F be a linear map. We say that u is completely bounded if the mappings un : Mn (E) → Mn (F ) deﬁned by un ((aij )) = (u(aij )) are uniformly bounded in the usual sense for the norm deﬁned in (0.1) and we deﬁne: ucb = sup un . n1

We will denote by CB(E, F ) the Banach space of all c.b. maps from E into F , equipped with the c.b.-norm. The main interest of the preceding notion is the following fundamental factorization theorem, which appeared independently in the works of Wittstock [106], Haagerup [45] and Paulsen [78], following Arveson’s earlier work [4]. T HEOREM 2. Let E ⊂ B(H ), F ⊂ B(K) (H, K Hilbert), let u : E → F be a linear map and let C 0 be a constant. The following assertions are equivalent: (i) The mapping u is c.b. and satisﬁes ucb C. (ii) There exist a Hilbert space H1 , a representation π : B(H ) → B(H1 ) and two operators a, b : H1 → K with ab C such that: ∀x ∈ E

u(x) = aπ(x)b∗.

In other words we have: ucb = inf ab ,

(0.2)

where the inﬁmum runs over all factorizations as in (ii) and this inﬁmum is attained. The following extension property is crucial: it is the analog of the Hahn–Banach theorem for operator spaces. C OROLLARY 3. Let H, K be two Hilbert spaces. Consider an operator space F ⊂ B(H ) and a subspace E ⊂ F . Then every c.b. map u : E → B(K) admits a c.b. extension u˜ : F → B(K) such that u ˜ cb = ucb . The corresponding diagram is as follows: F −− −− u˜ − ∪ u→ E −−−−−→ B(H ). In another direction, Theorem 2 implies the decomposability of c.b. maps as linear combinations of completely positive ones. We say that u : E → F is completely positive (c.p. in short) if, with the preceding notation, all the mappings un : Mn (E) → Mn (F ) are positive with respect to the order structures induced by the positive cone of the C ∗ -algebras Mn (B(H )).

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C OROLLARY 4. Every c.b. map u : E → B(K) admits a decomposition u = u1 − u2 + i(u3 − u4 ) with uj c.p. such that: max uj cb max u1 + u2 cb , u3 + u4 cb ucb . j 4

P ROOF (sketch). We note that if a = b in Theorem 2, then u is completely positive. Thus this corollary follows simply from the polarization identity for skewlinear maps. We note that if u is c.p. on a C ∗ -algebra (or on an operator system), we have u = ucb and (in the unital case) u = u(1). We refer the reader to [79] or to [84] for more information on all these results Now that we know the morphisms, the notion of isomorphism is clear: we say that two operator spaces E, F are completely isomorphic (resp. completely isometric) if there exists an isomorphism u : E → F which is completely bounded with a completely bounded inverse (resp. with moreover ucb = u−1 cb = 1). We say that an isometry (not necessarily surjective) u : E → F is a complete isometry if ucb = u−1 |u(E) cb = 1. We say that a mapping u : E → F is a complete contraction if ucb 1. The reader will easily complete (!) this terminology. One of the great advantages of operator spaces over C ∗ -algebras is that they allow the use of ﬁnite-dimensional methods in operator algebra theory. More precisely, if E and F are two completely isomorphic operator spaces, we can measure their “degree of isomorphism” by the following distance: dcb (E, F ) = inf ucb u−1 cb | u : E → F ,

(0.3)

where the inf runs over all the complete isomorphisms u from E onto F . This deﬁnition is of course modelled on the “Banach–Mazur distance” between two Banach spaces, which is classically deﬁned as: d(E, F ) = inf uu−1 | u : E → F isomorphism .

(0.4)

By convention, we set dcb (E, F ) = ∞ or d(E, F ) = ∞ if E and F are not isomorphic. Consider now a Banach space X. There exists obviously many possible “operator space structures” (in short o.s.s.) on X. By deﬁnition, such a structure on X is the data of an isometric embedding j : X → B(H ). We will say that two such structures: j1 : X −→ B(H1 )

and j2 : X −→ B(H2 )

are equivalent if, for any operator space F ⊂ B(K) and any u : X → F , the c.b. norms of u are the same whether we use one of the embeddings j1 or the other j2 . Of course, this boils −1 −1 down to saying that j2 ◦ j1|j and j1 ◦ j2|j are complete isometries, or equivalently 1 (X) 2 (X) that the norms induced respectively by j1 and j2 on Mn (X) are the same for any n 1. Actually, there is no need to distinguish two equivalent operator spaces. In practice, we will always identify them. When it is really necessary, we might want to distinguish the

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“concrete” operator space E ⊂ B(H ) and the “abstract” associated o.s.s., i.e., the associated equivalence class (instead of a “concrete” representative of this class). If X is a C ∗ -algebra, there is of course a natural embedding of X into B(H ) as a ∗ C -subalgebra, by the Gelfand theory. In that case, if j 1 and j 2 are two C ∗ -algebraic embeddings, the preceding equivalence relation is automatic. Indeed, an injective C ∗ -algebra representation j : X → B(H ) is automatically isometric and since jn : Mn (X) → Mn (B(H )) is also an injective representation, it also is isometric, which boils down to saying that j is completely isometric. Therefore if, in the above, j 1 and j 2 are injective C ∗ -representations, then j 1 (j 2 )−1 and j 2 (j 1 )−1 are automatically complete isome|j 2 (X) |j 1 (X) tries. Hence, we may unambiguously speak of the natural structure of a C ∗ -algebra as an operator space. This last observation allows us to change our viewpoint: we can equivalently deﬁne an operator space as a subspace of a C ∗ -algebra, since we now know (by Gelfand’s theory) that there is a natural o.s.s. on any C ∗ -algebra. Recall, in particular, that any commutative unital C ∗ -algebra is isomorphic to the space C(T ) of all continuous functions on a compact set T , equipped with the uniform norm. Let B be an arbitrary Banach space. We can associate to it a compact set TB , namely the unit ball of the dual B ∗ equipped with the topology σ (B ∗ , B). We then have an isometric embedding j : B → C(TB ) which allows to equip B with an operator space structure (induced by the C ∗ -algebra C(TB )). We denote by min(B) (following [13]) the resulting operator space. This provides numerous examples. Of course, these examples are not too interesting since they are too “commutative”, but they have the merit of showing how the category of Banach spaces can be viewed as “embedded” into that of operator spaces. Indeed, if B1 , B2 are two Banach spaces, every bounded linear u : B1 → B2 deﬁnes a completely bounded map u : min(B1 ) → min(B2 ) with ucb = u. More generally, for any operator space E, every linear map u : E → B deﬁnes a c.b. map u : E → min(B) such that u = ucb . In particular, when u = IB , if E = B is equipped with any operator space structure (respecting the norm of B), we have a complete contraction E → min(B). This expresses the “minimality” of min(B). Following [13], we can also introduce the “maximal” structure on B. For that purpose, it is convenient to deﬁne ﬁrst a notion of direct sum in the category. of operator spaces. Let Ei ⊂ B(Hi ) (i ∈ I ) be a collection of operator spaces. We denote i∈I Ei the such that supi∈I xi < ∞ space formed of all families x = (xi )i∈I with xi ∈ Ei , ∀i ∈ I , . and equipped with the norm x = supi∈I xi . The space . .i∈I B(Hi ) is naturally a C ∗ -algebra (that can be seen as embedded into B( i∈I Hi ),. i∈I Hi meaning here the . E → B(H Hilbertian direct sum). Therefore, the isometric embedding i i ) ini∈I i∈I . duces an operator space structure on i∈I Ei . We thus have a notion of direct sum. Let B be an arbitrary Banach space. Let I be the class . of all the mappings u : B → B(Hu ) with u 1. We can deﬁne an embedding J : B → u∈I B(Hu ) by setting J (x) =

/

u(x).

u∈I

This embedding allows us to deﬁne an operator space structure on B. We denote by max(B) the associated operator space. By construction, we have the following “maximal-

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ity” property: for any operator space E and any u : max(B) → E, u bounded ⇒ u c.b. and u = ucb . In particular, if E = B is equipped with an arbitrary o.s.s. (respecting the norm of B), we have a complete contraction max(B) → E induced by the identity of B. In conclusion, let E = (B, j ) be an arbitrary o.s.s. on B associated to an isometric embedding j : B → B(H ); we then have completely contractive inclusions (= the identity map): max(B) −→ (B, j ) −→ min(B). Thus, maximal operator spaces provide us with another group of examples. Here are two more fundamental examples: we denote R = span[e1j |j 1] ⊂ B(2 ) and C = span[ei1 |i 1] ⊂ B(2 ). One often says C is formed of the column vectors and R of the row vectors in B(2 ). Note that we have: ∀x = (xi ) ∈ 2

xj e1j = |xj |2

1/2

= xi ei1

so that R and C are indistinguishable as Banach spaces: they are both isometric to 2 . In sharp contrast, they are not completely isomorphic and they provide us with two new o.s.s. on the Hilbert space 2 . Thus, at this point we already have four o.s.s. on 2 (which are known to be distinct): min(2 ), max(2 ), R and C. We will soon see that there are actually a whole continuum of such structures! For the moment the typical application of operator space theory is as follows: we have a C ∗ -algebra A equipped with a distinguished system of generators and we consider the operator space E which is the closure of the subspace linearly spanned by these generators (this space E is often isomorphic to a Hilbert space). Then, in many cases, one can “read” on the operator space structure of E several important properties of the C ∗ -algebra which it generates. See [91] for numerous examples illustrating this principle. Although Ruan’s 1988 thesis marks the real “birth” of operator space theory as such, many earlier contributions have had a strong and lasting inﬂuence. Among those, the factorization of multilinear completely bounded maps, due to Christensen and Sinclair [21] (and generalized to the operator space setting by Paulsen and Smith [81]) is fundamental (see Section 6). Even earlier, Effros and Haagerup [25] (inspired by Archbold and Batty’s previous work) discovered that operator spaces may fail to be locally reﬂexive in the c.b. setting in sharp contrast to the Banach space case. Their ideas are closely related to Kirchberg’s spectacular work (see [60,62,105,1]) on exact C ∗ -algebras. In addition, we should recall that “operator spaces” are the descendents of “operator systems”. An operator system is a unital self-adjoint operator space. The theory of operator systems was extensively developed by Arveson [4] and Choi and Effros [19] in the 70’s, using unital completely positive maps as morphisms. Although many of the subsequent ideas appeared already in germs for operator systems, the constant recourse to the order structure stood in the way of a total “linearization” of C ∗ -algebra theory, which operator space theory can now claim to have realized.

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This text is a sort of introduction to the subject. For more information we refer the reader to the books [41,91], or to [84,82] and to the recent proceedings volume [58]. N OTE . The present text is based on the author’s Bourbaki seminar report [87], ﬁrst translated into English, then expanded and updated.

1. Minimal tensor product Let H1 , H2 be two Hilbert spaces. We denote by H1 ⊗2 H2 their Hilbertian tensor product. Let E1 ⊂ B(H1 ), E2 ⊂ B(H2 ) be two operator spaces. We deﬁne a linear (injective) embedding j from the algebraic tensor product, denoted by E1 ⊗ E2 , into B(H1 ⊗2 H2 ) as follows: for x1 ∈ E1 , x2 ∈ E2 we set ∀h1 ∈ H1 , ∀h2 ∈ H2 ,

j (x1 ⊗ x2 )(h1 ⊗ h2 ) = x1 (h1 ) ⊗ x2 (h2 ),

then we extend by linearity. This embedding allows to deﬁne an o.s.s. on the completion of E1 ⊗ E2 relative to the induced norm. We denote by E1 ⊗min E2 the resulting operator space. Thus, by deﬁnition, we have a complete isometry; E1 ⊗min E2 ⊂ B(H1 ⊗2 H2 ). We denote by · min the norm induced by B(H1 ⊗2 H2 ) on E1 ⊗min E2 . One can then verify that, up to equivalence, the resulting operator space does not depend on the particular realizations of E1 and E2 in B(H1 ) and B(H2 ), but only on their o.s.s. This follows from the next well known observation, very simple but essential for the theory. P ROPOSITION 5. Let E1 , E2 be as above and let F1 ⊂ B(K1 ) and F2 ⊂ B(K2 ) be two other operator spaces. Let u1 ∈ CB(E1 , F2 ) and u2 ∈ CB(E2 , F2 ). Then u1 ⊗ u2 : E1 ⊗ E2 → F1 ⊗ F2 extends to a c.b. map (still denoted by u1 ⊗ u2 ) such that u1 ⊗ u2 CB(E1 ⊗min E2 ,F1 ⊗min F2 ) u1 cb u2 cb . Moreover, the minimal tensor product is injective, meaning that if u1 and u2 are both complete isometries, the same is true for u1 ⊗ u2 : E1 ⊗min E2 → F1 ⊗min F2 . R EMARK 6. Let E1 , E2 be two C ∗ -algebras, then E1 ⊗min E2 is a C ∗ -subalgebra of B(H1 ⊗2 H2 ). Thus we actually have a tensor product for the C ∗ -algebra category, which the preceding extends to operator spaces. By a classical theorem due to Takesaki (see Section 10 below), the norm · min is the smallest C ∗ -norm on the (linear) tensor product of two C ∗ -algebras. In the Banach space category, Grothendieck [44] showed that the injecˇ 2 of two Banach spaces realizes the smallest “reasonable” tensor tive tensor product B1 ⊗B norm on B1 ⊗ B2 . Blecher and Paulsen (cf. [13]) proved an analog of this for E1 ⊗min E2 .

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In passing, let us observe here that for any operator space E and any Banach space B, ˇ B. On the other hand, we have we have an isometric isomorphism E ⊗min min(B) = E ⊗ Mn (E) Mn ⊗min E

isometrically.

In the sequel, we will often identify Mn (E) with Mn ⊗ E. The minimal tensor product is commutative (i.e., E1 ⊗ min E2 E2 ⊗min E1 ) and associative (i.e., for instance (E1 ⊗min E2 ) ⊗min E3 E1 ⊗min (E2 ⊗min E3 )). Therefore, we may unambiguously deﬁne (either directly or by iteration) the minimal tensor product E1 ⊗min · · · ⊗min EN of an arbitrary number N of operator spaces, and we again denote by · min the corresponding norm. One easily veriﬁes that ∀xi ∈ Ei

x1 ⊗ · · · ⊗ xN min = x1 · · · xN .

(1.1)

2. Ruan’s theorem Before completion, a Banach space is just a vector space equipped with a norm. Ruan’s fundamental theorem allows to take an analogous viewpoint for operator spaces, but instead of a norm on V , we must consider a sequence of norms · n on Mn (V ) (or a single norm, but on n Mn (E)). Let E be a Banach space, or merely a vector space on C. Suppose given an operator space structure on E. Then, up to equivalence, this is the same as giving ourselves, for each n 1, a norm · n on the space Mn (E) (of n × n matrices with coefﬁcients in E). The problem solved by Ruan’s theorem is the inverse one: which sequences of norms come from an o.s.s. on E? We will ﬁrst identify two simple necessary conditions. So assume that V is embedded in B(H ) and that · n is the norm induced on Mn (V ) by Mn (B(H )). The following two properties are then easily veriﬁed: ∀n 1, ∀a, b ∈ Mn , ∀x ∈ Mn (V ) a · x · bn aMn xMn (V ) bMn , (R1 ) where we denoted a · x · b the matrix product of the matrix x ∈ Mn (V ) by the scalar matrices a and b. ⎧ ⎪ ∀n, m 1, ∀x ∈ Mn (V ), ∀y ∈ Mm (V ), ⎨ x 0 (R2 ) ⎪ = max xn , ym . ⎩ 0 y n+m

We can now state Ruan’s theorem: T HEOREM 7 ([94]). Let V be a complex vector space. For each n 1 we give ourselves a norm · n on Mn (V ). The following assertions are equivalent: (i) There exist a Hilbert space H and a linear injection j : V → B(H ) such that for all n:

IMn ⊗ j : Mn (V ), · n −→ Mn B(H )

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is an isometry. Equivalently, in other words, the sequence ( · n ) comes from the operator space structure on V associated to j . (ii) The sequence · n satisﬁes the axioms (R1 ) and (R2 ) above. Let K = K(2 ) be the space of all compact operators on 2 . K can be viewed as a space of bi-inﬁnite matrices, which allows us to consider Mn as “included” in K. We then set: K0 =

Mn .

n1

It is convenient in the preceding theorem to replace the sequence of norms ( · n ) by a single norm on K0 ⊗ E or (after completion) on K ⊗ E. Indeed, the axiom (R2 ) ensures that the embedding (Mn (E), · n ) ⊂ (Mn+1 (E), · n+1 ) is isometric, which allows to deﬁne a norm α on K0 ⊗ E as follows: for x ∈ K0 ⊗ E, choose n be such that x ∈ Mn ⊗ E, we then set: α(x) = xn .

(2.1)

Ruan’s theorem establishes a one-to-one correspondence between the set of operator space structures on V (up to equivalence) and the norms α on K0 ⊗ E (or on K ⊗ E) satisfying (R1 ) and (R2 ). R EMARK . Of course, if V is given to us equipped with a norm, we are mostly interested in the o.s.s. on V respecting the norm of V , i.e., such that (x)1 = x for all x ∈ V . It is then easy to check (this is obvious by (1.1) and the preceding theorem) that (R1 ) and (R2 ) imply a ⊗ xn = aMn x for all a in Mn and all x in V . Let α be the norm on K0 ⊗ E associated to this structure as deﬁned in (2.1), and let αmin , αmax be the norms associated respectively to the minimal and maximal structures, as above. We have then: αmin α αmax , which explains the use of the terms “minimal” and “maximal”. I MPORTANT R EMARK . It should be emphasized that the o.s.s. given by Ruan’s theorem are not explicit and, in most of the cases described below (duality, quotient, interpolation), we have no “concrete” description of them. Their existence follows from the Hahn–Banach theorem, cf. the simpliﬁed proof of Theorem 7 appearing in [34].

3. Duality Preliminary. Let E, F be two vector spaces. Let u ∈ F ⊗ E ∗ and let u˜ : E → F be the linear map associated to it. When E and F are Banach spaces, we know that u∨ = u ˜ ˇ E ∗ into the and u → u˜ is an isometric embedding of the injective tensor product F ⊗

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space B(E, F ) of all bounded linear maps from E into F . The duality of operator spaces is modeled on this, but the minimal tensor product replaces the injective one and “c.b.” replaces “bounded”. Let E ⊂ B(H ) be an operator space and let E ∗ be its Banach space dual. Then E ∗ can be equipped with a speciﬁc o.s.s. characterized by the following property: For any operator space F , the natural map u → u˜ from F ⊗min E ∗ into CB(E, F ) is an isometry.

(3.1)

We have thus an isometric embedding:

∗ E ⊗min F F ⊗min E ∗ −→ CB(E, F ). When dim(F ) < ∞, this embedding is surjective, whence isometric identiﬁcations: ∗

E ⊗min F F ⊗min E ∗ CB(E, F ). In the case F = Mn , we have in particular an isometric identiﬁcation: Mn ⊗min E ∗ CB(E, Mn ).

(3.2)

The basic idea (independently from [13] and [32]) to deﬁne this speciﬁc o.s.s. is to take the right side of (3.2) to deﬁne a sequence of norms on Mn (E ∗ ) and to verify the axioms (R1 ) and (R2 ). Ruan’s theorem then guarantees that there exists a structure on E ∗ verifying (3.2). One then rather easily deduces (3.1) from (3.2). The unicity of the corresponding structure (up to equivalence) is clear since (3.2) determines at most one o.s.s. on E ∗ . Note that for all u : E → F the transposed map t u : F ∗ → E ∗ satisﬁes ucb = t ucb . More generally, if F is another operator space, we can deﬁne an o.s.s. on CB(E, F ) for which, for each n, we have isometrically:

Mn CB(E, F ) CB E, Mn (F ) .

(3.3)

Indeed, there again the norms appearing on the right side of (3.3) satisfy the axioms (R1 ) and (R2 ). Thus, from now on we may consider CB(E, F ) as an operator space (and (3.1), (3.2) then become completely isometric). Examples. The following completely isometric identities can be checked (cf. [13,33]): R ∗ C,

C∗ R

and, for any Banach space B (cf. [13,8]):

min(B)∗ max B ∗ .

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G. Pisier

Let M be a von Neumann algebra with predual M∗ . By duality, the natural structure of M gives us an o.s.s. on M ∗ hence a fortiori on M∗ ⊂ M ∗ . This raises a “coherence” problem, but fortunately everything “ticks”: if we equip M∗ with the preceding o.s.s., its dual is completely isometric (actually equivalent) to M. Thus we still have existence and unicity of the predual of M in the category of operator spaces. In sharp contrast however, this is no longer true for general operator spaces: Le Merdy [67] showed that there exists an o.s.s. on B(H )∗ which is not the dual of any o.s.s. on B(H ).

4. Quotient, ultraproduct and interpolation We will deﬁne below some other operations (= functors) on operator spaces. It is worthwhile to emphasize that these operations extend the corresponding ones for Banach spaces. The principle is the same as for the duality: we ﬁrst work with the underlying Banach spaces to construct the new space (e.g., dual, quotient ultraproduct or interpolation space), and then equip the resulting space with an o.s.s. compatible with the norm and satisfying the “right” functorial properties, speciﬁc to each case. For instance, Ruan [94] deﬁned the quotient of two operator spaces E1 , E2 with E2 ⊂ E1 , as follows. We consider the norm · n on Mn (E1 /E2 ) naturally associated to the quotient of normed spaces Mn (E1 )/Mn (E2 ), then we verify (R1 ) and (R2 ). Theorem 7 then ensures that there exists an o.s.s. on E1 /E2 for which we have, for all n 1, an isometric identiﬁcation: Mn ⊗min (E1 /E2 ) = Mn (E1 )/Mn (E2 ). More generally, we have an isometric identiﬁcation: K ⊗min (E1 /E2 ) = (K ⊗min E1 )/(K ⊗min E2 ). Thus we now have a notion of “quotient” in the category of operator spaces. Let q : E1 → E1 /E2 be the canonical surjection and let E3 be another o.s. Then a linear map u : E1 /E2 → E3 is c.b. iff uq is c.b. and we have ucb = uqcb . Moreover this notion of quotient satisﬁes the usual duality rules: we have completely isometric identities (E1 /E2 )∗ = E2⊥

and E2∗ = E1∗ /E2⊥ .

In analogy with the Banach space case, a mapping u : E → F between o.s. is called a complete surjection (resp. a complete metric surjection) if it is onto and if the associated isomorphism E/ ker(u) → F is a complete (resp. completely isometric) isomorphism. Moreover, this is the case iff u∗ : F ∗ → E ∗ is a complete (resp. completely isometric) isomorphism from F ∗ to u∗ (F ∗ ).

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Let U be a non-trivial ultraﬁlter on a set I . Let (Ei )i∈I be a family of Banach spaces. We denote by the space of all families x = (xi )i∈I with xi ∈ Xi such that supi∈I xi < ∞. We equip this space with the norm x = supi∈I xi . Let nU ⊂ be the subspace formed of all families such that limU xi = 0. The quotient * /nU is called the ultraproduct of the family (Ei )i∈I with respect to U . We denote it by i∈I Ei /U . Now assume that each space Ei is given equipped with an operator space structure. It is very easy to extend the notion of ultraproduct to the operator space setting. We simply deﬁne $ $ Mn Xi /U = Mn (Xi )/U. (4.1) i∈I

i∈I

* This identity endows Mn ⊗ i∈I Xi /U * with a norm satisfying Ruan’s axioms (whence also after completion a norm on K ⊗ ( i∈I Xi /U)). Alternatively, we can view an operator space as a subspace of a C ∗ -algebra, then observing that C ∗ -algebras are stable by ultraproducts, we can realize any ultraproduct of operator spaces as a subspace of an ultraproduct of C ∗ -algebras, and we equip it with the induced operator space structure. This alternate route leads to the same operator space structure as (4.1). We now turn to interpolation. Let (E0 , E1 ) be a “compatible” couple of Banach spaces. This means that we are implicitly given two continuous injections E0 → X and E1 → X of E0 , E1 into a common topological vector space X , which allows us to view E0 and E1 as included in X . The typical example is E0 = L∞ , E1 = L1 and X = L0 . Note that actually one can always replace X by a Banach space namely the “sum” E0 + E1 . By this we mean the subspace of X formed of all elements x0 + x1 with x0 ∈ E0 , x1 ∈ E1 and equipped with the norm inf{x0 E0 + x1 E1 }. Similarly, the “intersection” E0 ∩ E1 is equipped with the Banach space norm x = max{xE0 , xE1 }. Furthermore, for each 0 < θ < 1 the complex interpolation method (due to Calderón and Lions independently) associates to each compatible couple (E0 , E1 ) an “interpolation space” denoted by (E0 , E1 )θ . We set Eθ = (E0 , E1 )θ . We thus obtain a “continuous family” (Eθ )0<θ<1 of Banach spaces included in X , satisfying the fundamental “interpolation property”, i.e., any map which is simultaneously bounded on E0 and E1 can be boundedly deﬁned also on Eθ for any 0 < θ < 1. Calderón also deﬁned a “dual” method, we will denote by E θ the corresponding interpolation space, which is usually larger but which always contains Eθ isometrically as a subspace (cf. [6,5]). Let us now assume that E0 , E1 are each equipped with an o.s.s. respecting their norm. Then we can deﬁne an o.s.s. on Eθ as follows: we equip Mn (Eθ ) with the norm of the interpolation space (Mn (E0 ), Mn (E1 ))θ (note: the inclusions Mn (Ei ) ⊂ Mn (X ), i = 0, 1, turn Mn (E0 ), Mn (E1 ) into a “compatible” couple). Then again, these norms satisfy the axioms (R1 ) and (R2 ), which ensures that there exists an o.s.s. on Eθ for which we have isometrically

Mn (Eθ ) = Mn (E0 ), Mn (E1 ) θ , or more generally K ⊗min Eθ = (K ⊗min E0 , K ⊗min E1 )θ .

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By the same method, we can deﬁne a natural o.s.s. on E θ . If E0 ∩ E1 is dense both in E0 and in E1 , the injective inclusions E0∗ → (E0 ∩ E1 )∗ and E1∗ → (E0 ∩ E1 )∗ allow to view the couple (E0∗ , E1∗ ) as compatible, so that we can deﬁne (E0∗ , E1∗ )θ and (E0∗ , E1∗ )θ . Among Calderón’s many classical results on these “methods”, the following isometric identity stands out:

θ (E0 , E1 )∗θ = E0∗ , E1∗ .

(4.2)

It can be described as the commutation of two functors: duality and interpolation. Here again this can be “completed”: when E0 , E1 are operator spaces, we proved in [85] that (4.2) actually becomes a complete isometry but this relies on a more involved argument than above where we only veriﬁed some “axioms”. We should mention that Xu [107] has also developed the real interpolation method (= the “Lions–Peetre method”) for operator spaces. E XAMPLES . Let (Ω, μ) be a measure space. We have already deﬁned a natural o.s.s. on L∞ (Ω, μ) and one on its predual L1 (Ω, μ). Thus, interpolation provides us with an o.s.s. on Lp (Ω, μ) which we will call “natural”. More generally, let M be a von Neumann algebra equipped with a normal, faithful and semi-ﬁnite trace τ . Provided M∗ is equipped with the appropriate o.s. structure, the “non-commutative” Lp -spaces can be deﬁned by interpolation as Lp (M, τ ) = (M, M∗ )θ

with θ = 1/p.

Again the natural o.s.s. on M and M∗ lead to one on Lp (M, τ ) which we call “natural” [89].

5. Projective tensor product Since the minimal tensor product is analogous to the injective tensor product (of Banach spaces), it is tempting to search for the analog of the projective tensor product. This question is studied independently in [13] and [32]. Effros and Ruan went further: in analogy with Grothendieck’s classical thesis [44], they introduced the o.s. analog of the approximation property (in short AP), of nuclear, integral or absolutely summing maps and of the “Dvoretzky–Rogers Theorem” (namely the identity is absolutely summing only in the ﬁnite-dimensional case). Their program meets several interesting obstacles (mainly due to the absence of local reﬂexivity for general operator spaces) but, essentially the theory works well. We refer the reader to [30,35,36]. See also [43] and [42] devoted more generally to a “non-commutative convexity” suggested by the o.s. version of the projective tensor product. For lack of space, we will limit ourselves here to a brief description of the projective tensor product of two operator spaces E, F , which we denote by E ⊗∧ F . This space is deﬁned in [13] as a predual. The deﬁnition in [32], is a bit more explicit, as follows: let u

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be an element of the algebraic tensor product E ⊗ F . Clearly for n 1 large enough we can write u in the following form: u=

αih xij ⊗ yh βj ,

(5.1)

ij hn

where x ∈ Mn (E), y ∈ Mn (F ) and α, β ∈ Mn . Then the norm in E ⊗∧ F can be deﬁned as uE⊗∧ F = inf α2 xMn (E) yMn (F ) β2 ,

(5.2)

where · 2 is the Hilbert–Schmidt norm and where the inﬁmum runs over all possible representations of u as in (5.1). We denote by E ⊗∧ F the completion of E ⊗ F for this norm. More generally, this space can be equipped with the o.s.s. associated to the norm · n deﬁned on Mn (E ⊗∧ F ) as follows: let u = (uij ) ∈ Mn (E ⊗ F ) and suppose: u = α · (x ⊗ y) · β, where the dot stands for the matrix product and where x ∈ M (E), y ∈ Mm (F ) and α (resp. β) is a matrix of size n × (m) (resp. (m) × n). Note that x ⊗ y is seen here as an element of Mm (E ⊗ F ) in the natural way (i.e., according to the isomorphism M ⊗ Mm Mm ). Then, following [32], we can set un = inf αMn,m xM (E) yMm (F ) βMm,n .

(5.3)

Note that, if n = 1, (5.3) reduces to (5.2). One more time, (5.3) satisﬁes the axioms (R1 ) and (R2 ), whence an o.s.s. on E ⊗∧ F for which we can write a posteriori un = uMn (E⊗∧ F ) for any u in Mn (E ⊗ F ). The key property is then: T HEOREM 8. We have completely isometric identities

∗ E ⊗∧ F = CB E, F ∗ = CB F, E ∗ . Moreover, the natural morphism E ⊗∧ F → E ⊗min F is a complete contraction. The projective tensor product is commutative and associative, but in general not injective. On the other hand, of course it is “projective”, i.e., if u1 : E1 → F1 and u2 : E2 → F2 are complete metric surjections (cf. Section 4), then this is true for u1 ⊗ u2 : E1 ⊗∧ E2 → F1 ⊗∧ F2 . In the Banach space setting, Grothendieck [44] observed the identity L1 (ν) = L1 (μ × ν). The o.s. version from [30] is as follows: Let M, N be L1 (μ) ⊗

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von Neumann algebras with preduals M∗ , N∗ . Let M ⊗ N be their tensor product as von Neumann algebras. We have then a completely isometric identity (M ⊗ N)∗ = M∗ ⊗∧ N∗ . Following [34], an o.s. E is said to have the OAP if there is a net of ﬁnite rank (c.b.) maps ui : E → E such that the net I ⊗ ui converges pointwise to the identity on K[E]. This is the o.s. analog of Grothendieck’s approximation property (AP) for Banach spaces. When the net (ui ) is bounded in CB(E, E), we say that E has the CBAP (this is analogous to the BAP for Banach spaces). To quote a sample result from [34]: E has the OAP iff the natural map E ∗ ⊗ ∧ E → E ∗ ⊗min E is injective. The class of groups G for which the reduced C ∗ -algebra of G has the OAP is studied in [46] (see also Section 9 in [59]). It should be emphasized that the AP for the underlying Banach space is totally irrelevant for the OAP: indeed, Arias [2] recently constructed an operator space isometric to 2 but failing the OAP! Building on previous unpublished work by Oikhberg, Oikhberg and Ricard obtained more dramatic examples of the same nature. In particular, they constructed a Hilbertian operator space X such that a linear map T on X is c.b. iff it is the sum of a multiple of the identity and a Hilbert–Schmidt map, or iff it is the sum of a multiple of the identity and a nuclear map in the o.s. sense (they can even produce ﬁnite-dimensional versions of the space X). In particular every T ∈ CB(X) has a non-trivial invariant subspace. The Banach space analog (with B(X) instead of CB(X)) is still open (but the ﬁnite-dimensional version is ruled out). The ideas revolving around the OAP or the CBAP are likely to lead to a simpler and more conceptual proof of the main result of [99], but unfortunately this challenge has resisted all attempts so far.

6. Haagerup tensor product The category of operator spaces possesses a truly remarkable tensor product which has no Banach space counterpart: the Haagerup tensor product. Inspired by [45], the paper [27] ﬁrst introduced it as a Banach space tensor product of two operator spaces, but its operator space structure was to become a cornerstone of the theory. Its importance, which cannot be overestimated, became clear only with the fundamental works of Christensen and Sinclair (cf. [21]) on multilinear c.b. maps on C ∗ -algebras and their extension to the general o.s. setting in [81]. Let E1 , E2 be two operator spaces. Let x1 ∈ K ⊗ E1 , x2 ∈ K ⊗ E2 . We denote (x1 , x2 ) → x1 ( x2 the bilinear form from (K ⊗ E1 ) × (K ⊗ E2 ) to K ⊗ (E1 ⊗ E2 ) deﬁned by (k1 ⊗ e1 ) ( (k2 ⊗ e2 ) = (k1 k2 ) ⊗ (e1 ⊗ e2 ). We set αi (xi ) = xi K⊗min Ei

(i = 1, 2).

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Then, for any x in K ⊗ E1 ⊗ E2 , we deﬁne αh (x) = inf

n

j j

α1 x1 α2 x2 ,

(6.1)

j =1

where the inﬁmum runs over all possible decompositions of x as a ﬁnite sum x=

n

j

j

x1 ( x2 ,

j =1 j

j

with x1 ∈ K ⊗ E1 , x2 ∈ K ⊗ E2 . Here there is a ﬁrst pleasant surprise: we may as well assume n = 1 in (6.1) above, i.e., αh (x) = inf{α1 (x1 )α2 (x2 )|x = x1 ( x2 }. Essentially, the reason behind this is Mn (K) K. It is easy to see that (6.1) satisﬁes Ruan’s axioms (R1 ) and (R2 ); hence, after completion, we obtain an operator space denoted by E1 ⊗h E2 . At ﬁrst it was unclear how to describe “concretely” the embedding E1 ⊗h E2 ⊂ B(H) in terms of the embeddings Ei ⊂ B(Hi ), but this became clear in [20], where it is shown that if Ei ⊂ Ai with Ai C ∗ -algebra, then E1 ⊗h E2 is naturally (completely isometrically) “realized” inside the free product A1 ∗ A2 of the two C ∗ -algebras (for a discussion of the unital free product, see [86]). The embedding is simply the mapping y1 ⊗ y2 → y1 y2 which takes y1 ⊗ y2 to the product (in the free product) y1 y2 . More precisely, the main result of [20] is as follows. T HEOREM 9. Let E1 , E2 be two operator spaces. Let Φ be the family of all the pairs σ = (σ1 , σ2 ) where σi : Ei → B(Hσ ) is a mapping with σi cb 1 (i = 1, 2) (and where, say, we restrict to dim Hσ max{card(E1 ), card(E2 )}). Then the mapping j : E1 ⊗h E2 →

/

B(Hσ )

σ ∈Φ

deﬁned by j

xj1 ⊗ xj2 =

n / σ ∈Φ j =1

is a complete isometry.

j j

σ1 x1 σ2 x2

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See [91] for a very direct proof. R EMARK 10. There is no problem to extend the above deﬁnition of E1 ⊗h E2 to the product E1 ⊗h · · · ⊗h EN of any member of operator spaces. One then easily checks that the Haagerup tensor product is associative. Moreover, it is both injective and projective, which is a rarity! Its projectivity follows from the deﬁnition and injectivity from Theorem 9 with Corollary 3. C OROLLARY 11. Let E1 , . . . , EN be operator spaces. There are completely isometric embeddings ϕi : Ei → B(H ) into a common B(H ) such that the mapping ϕ1 · ϕ2 · · · ϕN : E1 ⊗h · · · ⊗h EN → B(H ) deﬁned by ϕ1 · · · ϕN (x1 ⊗ · · · ⊗ xN ) = ϕ1 (x1 ) · · · ϕN (xN ) is a completely isometric embedding. From this, we can deduce the Christensen–Sinclair factorization for c.b. multilinear maps (cf. [21] and references there) in the version given in [81]: C OROLLARY 12. Let A1 , . . . , AN be C ∗ -algebras, and let Ei ⊂ Ai (i = 1, . . . , N) be operator subspaces. Let u : E1 ⊗ · · · ⊗ EN → B(H ) a linear map (= multilinear on E1 × · · · × EN ). The following assertions are equivalent (i) The map u extends to a complete contraction from E1 ⊗h · · · ⊗h EN into B(H ). (ii) There are Hilbert spaces Hi , representations πi : Ai → B(Hi ) and operators Ti : Hi+1 → Hi with norm 1 such that HN+1 = H0 = H and: ∀xi ∈ Ei ,

u(x1 ⊗ · · · ⊗ xN ) = T0 π1 (x1 )T1 π2 (x2 ) · · · πN (xN )TN .

R EMARK . A multilinear map u : E1 × · · · × EN → B(H ) is called completely contractive if it satisﬁes (i) above. (And of course we say that u is c.b. with ucb c if u/c is completely contractive.) Another very striking property of the Haagerup tensor product is its self-duality (which explains why it is both injective and projective) for which we refer to [33] (but the ﬁnitedimensional case is due to Blecher, see also [7]). C OROLLARY 13. Let E1 , E2 be operator spaces. If one of them is ﬁnite-dimensional, we have (E1 ⊗h E2 )∗ E1∗ ⊗h E2∗

completely isometrically.

Moreover, in the general case, we have a completely isometric embedding E1∗ ⊗h E2∗ ⊂ (E1 ⊗h E2 )∗ . We refer to [16,40] for a study of the Haagerup tensor product of dual operator spaces.

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R EMARK 14. We will illustrate this by a few examples. The ﬁrst ones show the central role of the Haagerup tensor product. The whole theory could be reconstructed from it. Indeed we have completely isometrically Cn ⊗h Rn Mn

and C ⊗h R K,

where the correspondence is deﬁned by ei1 ⊗ e1j → eij . More generally, we have (completely isometrically) for any operator space E Cn ⊗h E ⊗h Rn Mn (E)

and C ⊗h E ⊗h R K ⊗min E.

Let H be an arbitrary Hilbert space. We denote by Hc (resp. Hr ) the space obtained after equipping H with the operator space structure associated to H B(C, H ) (resp. H B(H , C)). Let K be another Hilbert space. We have then completely isometrically (cf. [33]): Hc ⊗h Kc (H ⊗2 K)c

and Hr ⊗h Kr (H ⊗2 K)r .

7. Characterizations of operator algebras and modules In the Banach algebra literature, an operator algebra is just a closed subalgebra (not necessarily self-adjoint) of B(H ). A uniform algebra is a subalgebra of the space C(T ) of all continuous functions on a compact set T . (One sometimes assumes that A is unital and separates the points of K, but we dont, unless stated otherwise.) In short, an operator algebra is a (closed) subalgebra of a general C ∗ -algebra, while a uniform algebra is a subalgebra of a commutative C ∗ -algebra. Let A ⊂ B(H ) be an operator algebra and let I ⊂ A be a closed two-sided ideal. Then the quotient is clearly a Banach algebra, but is it an operator algebra? Curiously, the answer is yes. This is due to Cole when A is a uniform algebra and to Lumer and Bernard in general: there always exists (for a suitable H) an isometric homomorphism A/I → B(H). This is of course reminiscent of the o.s.s. on quotients (cf. Section 4 above) which appeared much later. In the 70’s, a lot of work was devoted to attempt to characterize operator algebras, notably by Craw, Davie, Varopoulos (and his students Tonge and Charpentier). Varopoulos’s work lead to the conjecture that there should exist a tensor norm γ satisfying the following: a Banach algebra A is (isomorphic to) an operator algebra iff its product map p : A ⊗ A → A is bounded on A ⊗γ A. Indeed, Varopoulos discovered that when γ is the norm of factorization through Hilbert space, i.e., γ = γ2 (denoted by · H in [44]), the boundedness of p is sufﬁcient (but not necessary). Ultimately however, Carne [17] proved that there is no reasonable tensor norm γ satisfying this conjecture, thus concluding negatively these investigations. Quite surprisingly, these same questions were revived some 20 years later in the setting of operator spaces with c.b. multilinear maps instead of bounded ones, and in sharp contrast with the above, the answers are all positive, as follows:

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T HEOREM 15 ([15]). Let A be an algebra with a unit denoted by 1A . Assume A equipped with an operator space structure (compatible with its norm) for which 1A = 1. The following assertions are equivalent: (i) The product map p : A ⊗h A → A is completely contractive (i.e., pcb 1). (ii) There exists, for a suitable H , a unital homomorphism j : A → B(H ) which is a completely isometric embedding. Surprisingly, the “isomorphic” version of this result resisted for a while, until Blecher [9] proved it: T HEOREM 16 ([9]). Let A be an algebra equipped with an o.s.s. The following are equivalent: (i) p : A ⊗h A → A is c.b. (ii) There exists a homomorphism j : A → B(H ) which also is a complete isomorphism from A to j (A). The proofs of these results do not use the above mentioned results of Cole, Lumer and Bernard, which can be recovered as corollaries. However, more recently (cf. [91]) we have found a converse path in which the above theorems are deduced from the original Lumer– Bernard result on quotients of operator algebras, but while simpler this proof is still not so easy. We refer the reader to [68] for an extension of the Cole–Lumer–Bernard result to quotients of subalgebras of B(X) with X Banach, as well as to [11] for a detailed study of the possible operator algebra structures on p or on various classical Banach spaces. R EMARK . Several examples of unital operator algebras A enjoy the following property: any bounded unital homomorphism π : A → B(H ) “automatically” is c.b. This is the case for any ﬁnite-dimensional A, but also if A is a commutative C ∗ -algebra, or a nuclear one (see Section 9 below for the deﬁnition). However, it is unknown whether this holds when A is an arbitrary C ∗ -algebra (this is equivalent to a well known conjecture of Kadison). The same problem for the disc algebra A = A(D) has been recently solved negatively by the author (see [84]). Equivalently, this means that there are “exotic” operator algebra structures on A(D) (or on H ∞ ), i.e., unusual embeddings A(D) ⊂ B(H) which are not “equivalent” to the classical ones obtained using multiplication (or Toeplitz) operators on H 2 . Yet another reformulation is that there is a polynomially bounded operator on Hilbert space which is not similar to a contraction, thus answering a well known question of Halmos. The construction combines ideas from the theory of Hankel operators (Peller) with the ideology of c.b. maps and operator spaces. Operator spaces which are also modules over an operator algebra (in other words “operator modules”) can be characterized in a similar way (see [20] and [29], see also [70] for dual modules), as follows.

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T HEOREM 17 ([20]). Let A, B be C ∗ -algebras and let X be an (A, B)-bimodule, that is to say X is both a left A-module and a right B-module, so that we have a well deﬁned action map m : A × X × B → X. Assume X given with an o.s.s. The following are equivalent: (i) m deﬁnes a complete contraction from A ⊗h X ⊗h B to X. (ii) There exists a completely isometric embedding j : X → B(H ) and representations π : A → B(H ) and ρ : B → B(H ) such that, for all x in X, a in A and b in B, we have

j m(a, x, b) = π(a)j (x)ρ(b). Suitably modiﬁed versions of the Haagerup tensor product are available for operator modules. Operator modules play a central rôle in [12] where the foundations of a Morita theory for non-self-adjoint operator algebras are laid. There Blecher, Muhly and Paulsen show that operator modules are an appropriate “metric” context for the C ∗ -algebraic theory of strong Morita equivalence, and the related theory of C ∗ -modules. For example, Rieffel’s C ∗ -module tensor product is exactly the Haagerup module tensor product of the C ∗ -modules with their natural operator space structures. See [12,10], Blecher’s survey in [58] and references contained therein for more on this. Operator spaces have been fruitfully used in several other directions. On one hand, they were used by Effros and Ruan in their work on quantum groups [37] and by Ruan [96,97] in his work on the amenability of Kac algebras.

8. The operator Hilbert space OH and non-commutative Lp -spaces For convenience, we will say that an o.s. is Hilbertian if the underlying Banach space is isometric to a Hilbert space. For instance, R or C are Hilbertian. Numerous examples of this kind (sometimes only isomorphically Hilbertian) can be found in the literature, especially in connection with quantum physics: generators of the Fermion or Clifford algebras, random matrices, generators of the Cuntz algebra or of the reduced C ∗ -algebra of the free group, free semi-circular systems in Voiculescu’s sense (see [103]). . . . However, with the above duality, none of these turns out to be self-dual, which tends to indicate that operator spaces do not admit a true analog of Hilbert space. But actually the next result says that they do. T HEOREM 18. Let H be any Hilbert space. For a suitable H, there is an operator space EH ⊂ B(H) isometric to H and such that the canonical identiﬁcation (derived from the ∗ → E is a complete isometry. Moreover, this space E is unique up scalar product) EH H H to a complete isometry. (Note: for any operator space E ⊂ B(H ), we denote by E the complex conjugate equipped with the o.s.s. induced by the embedding E ⊂ B(H ) B(H ).) If

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H = 2 , we denote EH by OH and we call it the operator Hilbert space. Similarly, we denote it by OH n if H = n2 and by OH(I ) if H = 2 (I ). The preceding theorem suggests to systematically review the many situations of Banach space theory where Hilbert spaces play a central role to search for their operator space counterpart. This program is pursued in [85]. For instance, it is proved there that OH possesses some remarkable properties with respect to complex interpolation: we have completely isometric identities

min(2 ), max(2 )

1 2

= OH

and (R, C) 1 = OH. 2

In the second case, we view (R, C) as a “compatible” couple using the injection x → t x from R to C which allows to view R, C as both continuously injected into (say) X = C. The unicity of OH and (4.2) imply that all the o.s.s. deﬁned above in Section 4 on the spaces L2 (μ) and L2 (M, τ ) are completely isometric, provided they have the same underlying Hilbertian dimension. Finally, we note a nice property from [85] relative to the Haagerup tensor product: for any sets I and J we have completely isometrically: OH(I ) ⊗h OH(J ) OH(I × J ). In the above Section 4, we introduced the operator space structure of non-commutative Lp -spaces. One can even go further and develop (cf. [89]) a theory of vector valued noncommutative Lp -spaces. There the measure space is replaced by an injective von Neumann algebra M equipped with a faithful normal semi-ﬁnite trace τ , and the space of vectorial “values” is an operator space E. The resulting space is denoted by Lp (M, τ ; E). When (M, τ ) = (B(2 ), tr), the space Lp (M, τ ) is the non-commutative analog of p , namely the Schatten class Sp . The space Lp (M, τ ; E) then appears as the non-commutative analog of the vector valued sequence space p (E). We should emphasize that to have a satisfactory theory we must assume that the underlying von Neumann algebra is injective. This is required to have the non-commutative analogue of the fact that if F is a closed subspace of E then Lp (μ; F ) is a closed subspace of Lp (μ; E). See [32], Proposition 3.3 for the case p = 1. This leads naturally to the notion of “completely p-summing map” for which a natural analogue of the Pietsch factorization is proved in [89]. (For the particular case p = 1, see [36].) We say that a mapping u : E → F (between two operator spaces) is completely p-summing if ISp ⊗ u deﬁnes a bounded mapping from Sp ⊗min E into Sp [F ]. This framework also yields ([89]) a characterization of “operators factoring completely boundedly through OH” entirely analogous to the Grothendieck–Kwapie´n [44,64] characterization of operators factoring through a Hilbert space. [89] also considers operators factoring completely boundedly through non-commutative Lp -spaces. When dealing with ultraproducts,

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the lack of “exactness” (see Section 9 below) raises difﬁculties which have been largely elucidated by Junge in [51]. We refer to [56,92,52] for several operator space versions of Grothendieck’s theorem. Note that in the Banach literature, this is closely linked to the Khinchine inequalities (or their Gaussian variant), which express that the span in Lp of the Rademacher functions (or independent Gaussian random variables) is the same for all values of 1 p < ∞, and is isomorphic to 2 . In the o.s. case, this is more complicated: the spans have been computed as operator spaces (thanks mainly to Lust-Piquard’s work, see [89]), but they do depend on p and they are not completely isomorphic to OH unless p = 2. Nevertheless, Junge [52] proved very recently that OH can be embedded completely isomorphically into a non-commutative L1 -space associated to a von Neumann algebra of type III. By [90], type III is unavoidable. In a series of recent papers, the “local theory” of non-commutative Lp as operator spaces, and more generally the study of the operator space version of Lp -spaces has started to be investigated ([39,65,54,55,57]). 9. Local theory and exactness Let E, F be two normed spaces (resp. two operator spaces); recall that we have deﬁned the distances d(E, F ) and dcb (E, F ) in (0.3) and (0.4). We set δ(E, F ) = log d(E, F ),

δcb (E, F ) = log dcb (E, F ).

Let n 1. Let OSn (resp. Bn ) denote the set of all n-dimensional operator spaces (resp. Banach spaces) where, by convention, we declare that two spaces are the same if they are completely isometric (resp. isometric). Equipped with the distance δcb (resp. δ), OSn (resp. Bn ) becomes a complete metric space. In the normed (= Banach) case, the space (Bn , δ) even is compact, this is the famous “Banach–Mazur compactum”, to which a good part of the “local theory of Banach spaces” is devoted. Thus it is natural to investigate the analogous properties for (OSn , δcb ): for instance, what is its diameter? Is it compact? If not, is it separable? Here are some answers (we prefer to avoid the Log’s and use dcb instead of δcb ). T HEOREM 19. For any E ∈ OSn , we have dcb (E, OH n )

√ n

(9.1)

whence for all E, F in OSn dcb (E, F ) n. The latter estimate is optimal since dcb (Rn , Cn ) = n. Let us denote by λcb (E) the projection constant of an operator space E, i.e., the smallest C such that whenever E is embedded completely isometrically into a larger space F ,

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there is a c.b. projection √ P : F → E with P cb C. Just like in the Banach space case, we have λcb (E) n#for any E ∈ OSn , but in sharp contrast (see [92,52]) λcb (OH n ) is exactly of the order of n/Log(n) when n → ∞. Let E ∈ Bn . By the compactness of the unit ball of E ∗ , it is well known that for any ⊂ N ε > 0, there is N = N(ε, n) and E ∞ such that d(E, E) < 1 + ε. The analogous property dramatically fails for operator spaces with MN in the place of N ∞ , which suggests to introduce the following parameter for any E ∈ OSn

|E ⊂ MN , N 1 . dS K (E) = inf dcb E, E |E ⊂ K}, By a simple perturbation argument (see [89], p. 74) this is equal to inf{dcb (E, E) which explains the notation. For any operator space X, we denote dS K (X) = sup dS K (E) where the supremum runs over all the ﬁnite-dimensional subspaces E ⊂ X. Kirchberg [61] discovered that, when X is a C ∗ -algebra, the ﬁniteness of this number is equivalent to the fact that the functor A → A ⊗min X preserves the exactness of short exact sequences in the category of C ∗ -algebras, and this number dS K (X) actually must be equal to 1. He called “exact” the C ∗ -algebras satisfying this. In [83] the notion is extended to operator spaces, in which case the parameter dS K (X) needs no longer be equal to 1 when it is ﬁnite. It provides a useful measurement of the “degree of exactness” of an operator space. Exact operator spaces are characterized by the property that their ﬁnite dimensional subspaces can be “realized” in ﬁnite-dimensional (= matricial) C ∗ -algebras, with dimension free bounds. = λ such that for any ultraproduct Y * Equivalently (see [83]), X is exact iff there is , with xk ∈ X and Yi /U of operator spaces and for any ﬁnite sum xk ⊗ yk in X ⊗ Y associated to (yk (i))i∈I , we have yk ∈ Y lim λ . xk ⊗ yk (i) xk ⊗ yk xk ⊗ yk X⊗min Y

U

X⊗min Yi

X⊗min Y

(Here the ﬁrst inequality always holds.) Curiously however, exactness turns out to be a rather rare property, as shown by the following estimates, which show in particular that 1 and OH are not exact. T HEOREM 20 ([85,83]). For any E in OSn , we have √ dS K (E) n.

(9.2)

Moreover, for any n > 2, the space n1 = (n∞ )∗ equipped with the o.s.s. dual to the “natural” structure of n∞ , satisﬁes √ √ n n ∼ √ dS K n1 n, 2 2 n−1 and for OH n we have √ √

1/2 n1/4 / 2 ∼ n/ 2 n − 1 dS K (OH n ) n1/4 .

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Kirchberg observed early on that (OSn , δcb ) is not compact in general (and actually for any n > 2, but the case n = 2 remains open) and he raised the question of the separability of (OSn , δcb ). But again the answer is negative. T HEOREM 21 ([56]). The space (OSn , δcb ) is not separable if n > 2. More precisely, even the subset of all operator spaces isometric to n2 is not separable. The best asymptotic estimate of this “non-separability” is as follows: let δ(n) be the inﬁmum of the numbers δ > 0 for which (OSn , δcb ) admits a countable δ-net in the following sense (we use dcb rather than δcb ): there is a countable subset of D ⊂ OSn such that ∈ D with dcb (E, E) < δ. With this notation, we have, for n = p + 1 with p ∀E ∈ OSn ∃E prime 3 (see [56]): √ n n ∼ √ δ(n). 2 2 n−1 √ On the other hand, (9.1) (or (9.2)) obviously implies that δ(n) n for all n 1. Using √ the (quite delicate) random matrix bounds in [47], one can show that δ(n) n/[2(1 + n)] for all n 1, see [91] for details and an update.

10. Applications to tensor products of C ∗ -algebras Whenever A1 , A2 are C ∗ -algebras, their algebraic tensor product carries a natural structure of a ∗-algebra. By a C ∗ -norm, we mean a norm on A1 ⊗ A2 such that x = x ∗ ,

xy xy and xx ∗ = x2

for all x, y in A1 ⊗ A2 . Since the works of Takesaki (1958) and Guichardet (1965), it has been known that there is a minimal C ∗ -norm · min and a maximal one · max , so that any C ∗ -norm · on A1 ⊗ A2 must satisfy ∀x ∈ A1 ⊗ A2 ,

xmin x xmax .

We denote by A1 ⊗min A2 (resp. A1 ⊗max A2 ) the completion of A1 ⊗ A2 for the norm · min (resp. · max ). A C ∗ -algebra A1 is called nuclear if, for any C ∗ -algebra A2 , we have · min = · max , or equivalently there is a unique C ∗ -norm on A1 ⊗ A2 . For instance, all commutative C ∗ -algebras are nuclear, as well as K(H ) but, by a result due to Wassermann [104], B(H ) is not nuclear when dim H = ∞. From the combined works of Choi and Effros and Connes, a C ∗ -algebra is nuclear iff its bidual A∗∗ is an injective von Neumann algebra. Moreover (Choi and Effros, Kirchberg) this holds iff the identity is approximable pointwise by a net

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of completely positive maps of ﬁnite rank. Kirchberg’s recent work [60] revived the study of pairs of C ∗ -algebras A, B for which A ⊗ B admits a unique C ∗ -norm, i.e., such that A ⊗min B = A ⊗max B

isometrically.

(10.1)

Kirchberg gave the ﬁrst example of a C ∗ -algebra A for which (10.1) holds when B = Aop (the opposite C ∗ -algebra, i.e., A with the inverse product x · y = yx), but which is not nuclear. Moreover, he showed that (10.1) holds when B = B(H ) and A = C ∗ (F∞ ) the C ∗ -algebra of the free group (denoted by F∞ ) with (say) inﬁnitely many generators, i.e., the C ∗ -algebra generated by the universal representation of the discrete group F∞ . Using operator space theory, it is possible to give a very simple proof of this result (see [86]). Moreover, that same theory also led the authors of [56] to the solution of a long standing problem discussed at length by Kirchberg in [60], as follows T HEOREM 22 ([56]). If dim H = ∞, we have B(H ) ⊗min B(H ) = B(H ) ⊗max B(H ). The paper [56] describes three different approaches to this result with quantitative estimates of variable sharpness. The best estimate (see [100] for more on this) uses the delicate number theoretic results of Lubotzky, Phillips and Sarnak. T HEOREM 23. Let umax λ(n) = sup umin

u ∈ B(H ) ⊗ B(H ), r(u) n ,

where r(u) denotes the rank of u. Then, for any n of the form n = p + 1 with p prime 3, we have √ √ n n ∼ √ λ(n) n. 2 2 n−1 √ R EMARK . The above upper bound λ(n) n is valid for all√n and follows for instance from (9.2). But the delicate point is the proof that λ(n) n(2 n − 1)−1 , which is a consequence of the lower bounds on the number δ(n) appearing in Section 9. We will now merely outline the link between δ(n) and λ(n) to point out which tensors are “responsible” for the large values of λ(n) found in Theorem 23. In order to do that, let n = p + 1 with p prime 3. Let S ⊂ R3 be the Euclidean sphere equipped with its normalized surface measure. We let L20 ⊂ L2 (S, μ) be the subspace of functions with mean zero (i.e., the orthogonal complement of the constant functions) and let ρ : SO(3) → B(L20 ) be the natural representation deﬁned by

ρ(t)f (ω) = f t −1 (ω) .

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. Let ρ = m0 πm be the decomposition of ρ into irreducible components, each ﬁnitedimensional since SO(3) is compact. For any subset Ω ⊂ N we set ρΩ =

/

πm

and HΩ =

m∈Ω

/

Hm .

m∈Ω

By [69], for n = p + 1 as above, there are t1 , . . . , tn in SO(3) such that n √ ρ(ti ) = 2 n − 1 1

whence n √ sup πm (ti ) ⊗ πm (ti ) 2 n − 1. m=m

(10.2)

i=1

The construction in [56] shows as a by-product that the n-tuples ρΩ (t1 ), . . . , ρΩ (tn ) are “often” linearly independent. Let EΩ be their linear span, and (assuming EΩ n-dimen1 , . . . , ξ n be the biorthogonal functionals in E ∗ . By the duality of operator sional) let ξΩ Ω Ω spaces (see Section 2 above), we know that there is a Hilbert space HΩ and a speciﬁc isometric embedding ∗ ⊂ B(HΩ ). EΩ

i represents the identity map on EΩ , we have necessarily Moreover, since n1 ρΩ (ti ) ⊗ ξΩ by (3.1) and by the injectivity of the minimal tensor product n i ρΩ (ti ) ⊗ ξΩ (10.3) = IEΩ CB(EΩ ,EΩ ) = 1. i=1

min

But, on the other hand, the method used in [56] (see also [102]) shows (here we skip all details) that (10.2) implies that, for any ε > 0, there is an inﬁnite subset Ω ⊂ N such that ρΩ (t1 ), . . . , ρΩ (tn ) are linearly independent and satisfy: n n ρΩ (ti ) ⊗ ξiΩ . (10.4) √ −ε 2 n−1 i=1 B(H )⊗ B(H ) Ω

max

Ω

In conclusion, one obtains Theorem 23 by combining (10.3) and (10.4).

11. Local reﬂexivity In Banach space theory, the “principle of local reﬂexivity” says that every Banach space B satisﬁes B(F, B)∗∗ = B(F, B ∗∗ ) isometrically for any ﬁnite-dimensional (normed)

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space F . Consequently, B ∗∗ is always ﬁnitely representable in B. This useful principle goes back to Lindenstrauss and Rosenthal with roots in Grothendieck’s and Schatten’s early work (see [24], p. 178 and references there). Similarly, an o.s. E is called “locally reﬂexive” if we have CB(F, E)∗∗ = CB(F, E ∗∗ ) isometrically for any ﬁnite-dimensional o.s. F (and when this holds for all F , it actually holds completely isometrically). This property was “exported” ﬁrst to C ∗ -algebra theory by Archbold and Batty, then for operator spaces in [25]. In the o.s. setting, it is unclear how this notion relates to the o.s.-ﬁnite representability of E ∗∗ into E. More precisely, we will say that an o.s. E is os-ﬁnitely representable into another F , if for any ε > 0 and any ﬁnite-dimensional subspace E1 ⊂ E there is a subspace F1 ⊂ F such that dcb (E1 , F1 ) 1 + ε. As the reader can guess, not every o.s. is locally reﬂexive, so the “principle” now fails to be universal: as shown in [25], C ∗ (F∞ ) is not locally reﬂexive. Local reﬂexivity passes to subspaces (but not to quotients) and is trivially satisﬁed by all reﬂexive o.s. (a puzzling fact since reﬂexivity is a property of the underlying Banach space only!). It is known that all nuclear C ∗ -algebras are locally reﬂexive (essentially due to Archbold and Batty, see [25]). More generally, by Kirchberg’s results, exactness ⇒ local reﬂexivity for C ∗ -algebras (see [59] or [105]), but the converse remains open. Actually, it might be true that exact ⇒ locally reﬂexive for all o.s. but the converse is certainly false since there are reﬂexive but non-exact o.s. (such as OH). It was proved recently in [28] that if an operator space X is 1-exact (meaning that dS K (X) = 1) then it is locally reﬂexive. All this shows that local reﬂexivity is a rather rare property. Therefore, it came as a big surprise (at least to the author) when, in 1997, Effros, Junge and Ruan [26] managed to prove that every predual of a von Neumann algebra (a fortiori the dual of any C ∗ -algebra) is locally reﬂexive. This striking result is proved using a non-standard application of Kaplansky’s classical density theorem, together with a careful comparison of the various notions of “integral operators” relevant to o.s. theory (see [53] for an alternate proof). Actually, [26] contains a remarkable strengthening: for any von Neumann algebra M, the dual M ∗ = (M∗ )∗∗ is o.s.-ﬁnitely representable in M∗ . This is already non-trivial when M = B(H )! More recently, Ozawa [77] proved that the space of all n-dimensional subspaces of a non-commutative L1 -space (= the predual of a von Neuman algebra) is compact for the dcb -distance. Since the ﬁnite representability is always clear in some “weak sense” this compactness contains the preceding statement.

12. Injective and projective operator spaces Injective objects have always played a major role both in Banach space and operator algebra theory. One reason was the quest for generalizations of the Hahn–Banach extension theorem to maps with ranges of dimension more than 1, or inﬁnite. Moreover, in von Neumann algebra theory, injective factors are of crucial importance because of Connes’ landmark paper [23] where he proves (in addition to the equivalence between “injective” “semi-discrete” and “hyperﬁnite”) that there is only one injective factor on 2 with a ﬁnite faithful normal trace (such algebras are called of type II1 ). This can be viewed as a noncommutative analogue of the fact that the Lebesgue interval is the only inﬁnite non-atomic

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countably generated measure space, or equivalently that L∞ ([0, 1]) is the only inﬁnitedimensional L∞ -space over a non-atomic countably generated measure space. A Banach space X is called injective (isometrically) if for any diagram Y ∪ u S −→ X there is an extension u˜ : Y → X with u ˜ = u. An operator space X is called injective if for any such diagram (with Y an o.s. and u c.b.) there is an extension u˜ : Y → X with u ˜ cb = ucb . The basic examples are X = ∞ (Γ ) or X = L∞ (Ω, Σ, μ) with Γ an arbitrary set or (Ω, Σ, μ) an arbitrary measure space. Since any Banach space X embeds isometrically into ∞ (Γ ), it is easy to see that X is injective iff X is the range of a contractive (= of norm 1) projection on ∞ (Γ ). Analogously, any o.s. X embeds completely isometrically into B(H ) for some suitable Hilbert space H . Thus the same reasoning (recall Corollary 3) shows that X is injective iff X is the range of a completely contractive projection on B(H ). In particular, in addition to B(H ), the operator spaces B(C, H ) and B(H, C), or the column and row Hilbert spaces C and R are injective. We note in passing that, by a result due to Tomiyama a contractive linear projection P on a C ∗ -algebra A is automatically completely contractive. However, the examples of C and R (with the projections x → xe11 and x → e11 x) show that the range of P need not be completely isometric to a C ∗ -algebra (see, e.g., [95], p. 97). We will say that a C ∗ -algebra (or a von Neumann algebra) is injective if it is injective as an operator space (with its natural structure). In [95] the following nice characterization of injective operator spaces is given. T HEOREM 24. An operator space X is injective iff there exists an injective C ∗ -algebra A and two projections p, q in A such that X is completely isometric to pAq. Moreover, Roger Smith observed (unpublished) that if X is ﬁnite-dimensional, A too can be chosen ﬁnite-dimensional. The preceding theorem is closely connected to the important notion of injective envelope of an operator space due to Hamana (see [48] and also an unpublished manuscript). Given an operator space X, we say that an operator space X with X ⊃ X (completely isometrically) is an injective envelope if X is injective and if is the only completely contractive map extending the inclumoreover the identity of X sion X → X. Hamana [48,49] (and Ruan independently) proved that every operator space admits a unique injective envelope. The notion of injectivity also makes sense in the isomorphic setting: a Banach (resp. operator space) X is called λ-injective if for any diagram as before we have an extension u˜ with u ˜ λu (resp. u ˜ cb λucb ). Of course X is then the range of a projection P on ∞ (Γ ) (resp. B(H )) with P λ (resp. P cb λ), but when λ > 1, the structure of these projections can be quite complicated and much less is known. However, there is another notion, “separable injectivity”, on which a lot of work has been done. A separable Banach space (resp. operator space) X is called “separably λ-injective” if for any subspace S ⊂ Y of a separable space Y there is an extension u˜ : Y → X with

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u ˜ λu (resp. u ˜ cb λucb ). We will say that X is separably injective if it is separably λ-injective for some λ < ∞. In 1941, Sobczyk [98] (see also [101]) proved that the Banach space c0 is separably injective as a Banach space, and as a corollary c0 is complemented in any separable superspace. Of course this is also true of any space isomorphic to c0 . Since no other example was found, this raised the question whether actually c0 was the only possible example up to isomorphism. This remained open for a long time until Zippin, in a deep paper [108], proved the converse to Sobczyk’s theorem: X is separably injective only if X is isomorphic to c0 . Since the space K of compact operators on 2 is the natural analog of c0 , it was natural to expect that K should be separably injective as an operator space. However, a very interesting example due to Kirchberg . [59]shows that it is. notthe case. NevertheC ) or ( less, Rosenthal [93] showed that the spaces ( n c 0 n1 n1 Rn )c0 are both separably 2-injective as operator spaces. (Note that, as o.s., these spaces are different, and also distinct from c0 .) In the same paper [93], Rosenthal studied the many possible variants of the extension property. This work was continued with Oikhberg in [75]. Simpler approaches appear in [76] and [3], see also [50]. One of the main results of [75] is that if we restrict the extension property to subspaces S ⊂ Y of a locally reﬂexive separable operator space Y , and if we assume that u : S → K is a complete isomorphism, then u : S → K admits a c.b. extension u˜ : Y → K. Thus if we restrict to Y locally reﬂexive we do have the c.b.-analogue of the corollary to Sobczyk’s theorem: K is completely complemented in any locally reﬂexive separable operator space containing it. Nevertheless, even assuming Y locally reﬂexive, the c.b.-version of the separable extension property fails for K in general. It remains open whether K is complemented (by a merely bounded projection) in any separable operator space containing it. We now turn to “projective objects”, that is to say spaces satisfying certain lifting properties. We will say that a Banach space (resp. operator space) X is λ-projective if for any ε > 0, any map u : X → Y/S into a quotient of Banach (resp. operator) space admits a lifting u˜ : X → Y with u ˜ (λ + ε)u (resp. u ˜ cb (λ + ε)ucb ). It is an elementary fact that X = 1 (Γ ) satisﬁes this with λ = 1 as a Banach space. Consequently, X = max(1 (Γ )) satisﬁes the same as an operator space. It is known ([63]) that these are the only Banach spaces which are λ-projective for some λ. However, in the o.s. setting, there is a larger class of projective spaces. Indeed Blecher [8] proved that although S1 itself is not projective, the direct sum in the sense of 1 of a family of spaces of the form S1ni for some integers {ni | i ∈ I } is 1-projective. He also proved that any operator space is (completely isometric to) a quotient of a space of this form for suitable I and (ni ); he could thus observe that a 1-projective operator space X is 1-projective iff for any ε > 0, X is (1 + ε)-completely isomorphic to a (1 + ε)-completely complemented subspace of a space of this form. Actually, by the more recent results of [28], we can take ε = 0. While there are rather few projective Banach spaces, many more spaces satisfy the “local” version of projectivity (or equivalently a local form of lifting property). The resulting class of Banach spaces is the class of L1 spaces (see [66]) which can be deﬁned in many equivalent ways. One of these is: X is L1 iff X∗∗ is isomorphic to a complemented subspace of an L1 -space. In sharp contrast, the operator space versions of the various de-

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ﬁnitions of L1 -spaces (or more generally Lp -spaces) lead to possibly distinct classes of operator spaces, see [39]. This difﬁculty is of course related to the lack of local reﬂexivity in general. One of the possible variants is studied in [65] under the name of “λ-local lifting property” (in short λ-LLP): an operator space X has the λ-LLP if for any map u : X → Y/S, any ε > 0 and any ﬁnite-dimensional subspace E ⊂ X, the restriction of u to E admits a lifting u˜ : E → Y with u ˜ cb (λ + ε)ucb . It is proved in [65] that X has the λ-LLP iff X∗ is λ-injective. (As a corollary, X λ-projective implies X∗ λ-injective.) See also [32,35] for related results. More recently, in [28] the authors prove that this happens for λ = 1 iff there is an injective von Neumann algebra R and a (self-adjoint) projection p in R such that X∗ (1 − p)Rp

(completely isometrically). ai

It follows that X has the 1-LLP iff there is a net of ﬁnite-rank maps of the form X −→ n

bi

S1 i −→ X with ai cb , bi cb 1 which tend pointwise to the identity on X. In another direction, the results of [28] provide an extension to operator spaces of the classical work of Choi, Effros and Connes (see [19]) on nuclear C ∗ -algebras. We will say ai

bi

that an operator space X is λ-nuclear if there is a net of maps of the form X −→ Mni −→ X with ai cb bi cb λ which tends pointwise to the identity on X. It is known (see [19]) that a C ∗ -algebra A is 1-nuclear iff A∗∗ is injective (equivalently is a 1-injective operator space). The o.s. version of this result proved in [28] now reads like this: an operator space X is 1-nuclear iff X is locally reﬂexive and WEP. We say that X is WEP if the canonical inclusion X → X∗∗ factors completely contractively through B(H ).

References [1] C. Anantharaman–Delaroche, Classiﬁcation des C ∗ -algèbres purement inﬁnies nucléaires (d’après E. Kirchberg), Sém. Bourbaki, 1995–96, n◦ 805, Astérisque 241 (1997). [2] A. Arias, Operator Hilbert spaces without the OAP, Proc. Amer. Math. Soc. 130 (2002), 2669–2677. [3] A. Arias and H.P. Rosenthal, M-complete approximate identities in operator spaces, Studia Math. 141 (2000), 143–200. [4] W. Arveson, Subalgebras of C ∗ -algebras, Acta Math. 123 (1969), 141–224; Part II, Acta Math. 128 (1972), 271–308. [5] J. Bergh, On the relation between the two complex methods of interpolation, Indiana Univ. Math. J. 28 (1979), 775–777. [6] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, New York (1976). [7] D. Blecher, Tensor products of operator spaces II, Canad. J. Math. 44 (1992), 75–90. [8] D. Blecher, The standard dual of an operator space, Paciﬁc J. Math. 153 (1992), 15–30. [9] D. Blecher, A completely bounded characterization of operator algebras, Math. Ann. 303 (1995), 227–240. [10] D. Blecher, A new approach to Hilbert C ∗ -modules, Math. Ann. 307 (1997), 253–290. [11] D. Blecher and C. Le Merdy, On quotients of function algebras and operator algebra structures on p , J. Operator Theory 34 (1995), 315–346. [12] D. Blecher, P. Muhly and V. Paulsen, Categories of Operator Modules (Morita Equivalence and Projective Modules), Mem. Amer. Math. Soc. 143 (681) (2000).

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[13] D. Blecher and V. Paulsen, Tensor products of operator spaces, J. Funct. Anal. 99 (1991), 262–292. [14] D. Blecher and V. Paulsen, Explicit constructions of universal operator algebras and applications to polynomial factorization, Proc. Amer. Math. Soc. 112 (1991), 839–850. [15] D. Blecher, Z.J. Ruan and A. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1990), 188–201. [16] D. Blecher and R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. 45 (1992), 126– 144. [17] T.K. Carne, Not all H -algebras are operator algebras, Math. Proc. Cambridge Philos. Soc. 86 (1979), 243–249. [18] M.D. Choi and E. Effros, Nuclear C*-algebras and injectivity: the general case, Indiana Univ. Math. J. 26 (1977), 443–446. [19] M.D. Choi and E. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977), 156–209. [20] E. Christensen, E. Effros and A. Sinclair, Completely bounded multilinear maps and C ∗ -algebraic cohomology, Invent. Math. 90 (1987), 279–296. [21] E. Christensen and A. Sinclair, A survey of completely bounded operators, Bull. London Math. Soc. 21 (1989), 417–448. [22] Y. Colin de Verdiere, Distribution de points sur une sphère (d’apres Lubotzky, Phillips and Sarnak), Astérisque 177–178 (1989), 83–93. [23] A. Connes, Classiﬁcation of injective factors, Cases II1 , II∞ , IIIλ , λ = 1, Ann. Math. 104 (1976), 73–116. [24] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press (1995). [25] E. Effros and U. Haagerup, Lifting problems and local reﬂexivity for C ∗ -algebras, Duke Math. J. 52 (1985), 103–128. [26] E. Effros, M. Junge and Z.J. Ruan, Integral mappings and the principle of local reﬂexivity for noncommutative L1 -spaces, Ann. of Math. 151 (2000), 59–92. [27] E. Effros and A. Kishimoto, Module maps and Hochschild–Johnson cohomology, Indiana Univ. Math. J. 36 (1987), 257–276. [28] E. Effros, N. Ozawa and Z.J. Ruan, On injectivity and nuclearity for operator spaces, Duke Math. J. 110 (2001), 489–521. [29] E. Effros and Z.J. Ruan, Representations of operator bimodules and their applications, J. Operator Theory 19 (1988), 137–157. [30] E. Effros and Z.J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990), 163–187. [31] E. Effros and Z.J. Ruan, On non-self adjoint operator algebras, Proc. Amer. Math. Soc. 110 (1990), 915– 922. [32] E. Effros and Z.J. Ruan, A new approach to operator spaces, Canad. Math. Bull. 34 (1991), 329–337. [33] E. Effros and Z.J. Ruan, Self-duality for the Haagerup tensor product and Hilbert space factorization, J. Funct. Anal. 100 (1991), 257–284. [34] E. Effros and Z.J. Ruan, On the abstract characterization of operator spaces, Proc. Amer. Math. Soc. 119 (1993), 579–584. [35] E. Effros and Z.J. Ruan, Mapping spaces and liftings for operator spaces, Proc. London Math. Soc. 69 (1994), 171–197. [36] E. Effros and Z.J. Ruan, The Grothendieck–Pietsch and Dvoretzky–Rogers theorems for operator spaces, J. Funct. Anal. 122 (1994), 428–450. [37] E. Effros and Z.J. Ruan, Discrete quantum groups I. The Haar measure, Internat. J. Math. 5 (1994), 681– 723. [38] E. Effros and Z.J. Ruan, On the analogues of integral mappings and local reﬂexivity for operator spaces, Indiana Univ. Math. J. 46 (1997), 1289–1310. [39] E. Effros and Z.J. Ruan, OLp -spaces, Contemp. Math. 228 (1998), 51–77. [40] E. Effros and Z.J. Ruan, Operator convolution algebras: an approach to quantum groups, Preprint. [41] E. Effros and Z.J. Ruan, Operator Spaces, Oxford Univ. Press (2000). [42] E. Effros and C. Webster, Operator analogues of locally convex spaces, to appear. [43] E. Effros and S. Winkler, Matrix convexity: operator analogues of the bipolar and Hahn–Banach theorems, J. Funct. Anal. 144 (1997), 117–152.

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[76] N. Ozawa, A short proof of the Oikhberg–Rosenthal theorem, Preprint (1999). [77] N. Ozawa, On the set of ﬁnite dimensional subspaces of preduals of von Neumann algebras, C.R. Acad. Sci. Paris Sér. I Math. 331 (2000), 309–312. [78] V. Paulsen, Completely bounded maps on C ∗ -algebras and invariant operator ranges, Proc. Amer. Math. Soc. 86 (1982), 91–96. [79] V. Paulsen, Completely Bounded Maps and Dilations, Pitman Res. Notes 146, Pitman Longman (Wiley) (1986). [80] V. Paulsen, Representation of function algebras, abstract operator spaces and Banach space geometry, J. Funct. Anal. 109 (1992), 113–129. [81] V. Paulsen and R. Smith, Multilinear maps and tensor norms on operator systems, J. Funct. Anal. 73 (1987), 258–276. [82] G. Pisier, Espaces de Banach quantiques: une introduction a la théorie des espaces d’opérateurs, Soc. Math. France (1994). [83] G. Pisier, Exact operator spaces, Colloque sur les algèbres d’operateurs, Orléans 1992, Astérisque 232 (1995), 159–187. [84] G. Pisier, Similarity Problems and Completely Bounded Maps, Lecture Notes in Math. 1618, Springer, Heidelberg (1995). [85] G. Pisier, The Operator Hilbert Space OH, Complex Interpolation and Tensor Norms, Mem. Amer. Math. Soc. 122 (585) (1996). [86] G. Pisier, A simple proof of a theorem of Kirchberg and related results on C ∗ -norms, J. Operator Theory 35 (1996), 317–335. [87] G. Pisier, Espaces d’opérateurs: une nouvelle dualité, Séminaire Bourbaki 95–96, Astérisque 241 (1997), 243–273. [88] G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), 351–369. [89] G. Pisier, Non-commutative vector valued Lp -spaces and completely p-summing maps, Astérisque 247 (1998), 1–131. [90] G. Pisier, The Operator Hilbert Space OH and TYPE III von Neumann Algebras, Preprint (September 2002). [91] G. Pisier, An Introduction to Operator Space Theory, Cambridge Univ. Press, to appear. [92] G. Pisier and D. Shlyakhtenko, Grothendieck’s theorem for operator spaces, Invent. Math. 150 (2002), 185–217. [93] H.P. Rosenthal, The complete separable extension property, J. Operator Theory 43 (2000), 329–374. [94] Z.J. Ruan, Subspaces of C ∗ -algebras, J. Funct. Anal. 76 (1988), 217–230. [95] Z.J. Ruan, Injectivity and operator spaces, Trans. Amer. Math. Soc. 315 (1989), 89–104. [96] Z.J. Ruan, The operator amenability of A(G), Amer. J. Math. 117 (1995), 1449–1474. [97] Z.J. Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), 466– 499. [98] A. Sobczyk, Projection of the space m on its subspace c0 , Bull. Amer. Math. Soc. 47 (1941), 938–947. [99] A. Szankowski, B(H ) does not have the approximation property, Acta Math. 147 (1981), 89–108. [100] A. Valette, An application of Ramanujan graphs to C ∗ -algebra tensor products, Discrete Math. 167/168 (1997), 597–603. [101] W. Veech, Short proof of Sobszyk’s theorem, Proc. Amer. Math. Soc. 28 (1971), 627–628. [102] D. Voiculescu, Property T and approximation of operators, Bull. London Math. Soc. 22 (1990), 25–30. [103] D.V. Voiculescu, K.J. Dykema and A. Nica, Free Random Variables, CRM Monogr. Ser., Vol. 1, Centre de Recherches Mathématiques, Université Montréal. [104] S. Wassermann, On tensor products of certain group C ∗ -algebras, J. Funct. Anal. 23 (1976), 239–254. [105] S. Wassermann, Exact C ∗ -algebras and Related Topics, Lecture Notes Ser., Seoul Nat. Univ. (1994). [106] G. Wittstock, Ein operatorwertigen Hahn–Banach Satz, J. Funct. Anal. 40 (1981), 127–150. [107] Q. Xu, The real interpolation in the category of operator spaces, J. Funct. Anal. 139 (1996), 500–539. [108] M. Zippin, The separable extension problem, Israel J. Math. 26 (3–4) (1977), 372–387.

CHAPTER 34

Non-Commutative Lp -Spaces Gilles Pisier∗ Équipe d’Analyse, Université Paris VI, Case 186, F-75252 Paris Cedex 05, France Texas A&M University, College Station, TX 77843, USA E-mail: [email protected]

Quanhua Xu Laboratoire de Mathématiques, Université de Franche-Comté, UFR des Sciences et Techniques, 16, Route de Gray, 25030 Besançon Cedex, France E-mail: [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. General von Neumann algebras, including type III . . . . . . . . . . . 4. From classic Lp to non-commutative Lp : similarities and differences 5. Uniform convexity (real and complex) and uniform smoothness . . . 6. Non-commutative Khintchine inequalities . . . . . . . . . . . . . . . . 7. Non-commutative martingale inequalities . . . . . . . . . . . . . . . . 8. Non-commutative Hardy spaces . . . . . . . . . . . . . . . . . . . . . 9. Hankel operators and Schur multipliers . . . . . . . . . . . . . . . . . 10. Isomorphism and embedding . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Introduction This survey is devoted to the theory of non-commutative Lp -spaces. This theory (in the tracial case) was laid out in the early 50’s by Segal [181] and Dixmier [47] (see also [110, 183]). Since then the theory has been extensively studied, extended and applied, and by now the strong parallelism between non-commutative and classical Lebesgue integration is well-known. We will see that on the one hand, non-commutative Lp -spaces share many properties with the usual Lp -spaces (to which we will refer as commutative Lp -spaces), and on the other, they are very different from the latter. They provide interesting (often “pathological”) examples which cannot exist among the usual function or sequence spaces. They are also used as fundamental tools in some other directions of mathematics (such as operator algebra theory, non-commutative geometry and non-commutative probability), as well as in mathematical physics. Some tools in the study of the usual commutative Lp -spaces still work in the noncommutative setting. However, most of the time, new techniques must be invented. To illustrate the difﬁculties one may encounter when studying non-commutative Lp -spaces, we mention here three well-known facts. Let H be a complex Hilbert space, and let B(H ) denote the algebra of all bounded operators on H . The ﬁrst fact states that the usual triangle inequality for the modulus of complex numbers is no longer valid for the modulus of operators, namely, in general, we do not have |x + y| |x| + |y| for x, y ∈ B(H ), where |x| = (x ∗ x)1/2 is the modulus of x. However, there is a useful substitute, obtained in [1], which reads as follows. For any x, y ∈ B(H ) there are two isometries u and v in B(H ) such that |x + y| u|x|u∗ + v|y|v ∗ . The second fact is about operator monotone functions. Let α be a positive real number. In general, the condition that 0 x y (x, y ∈ B(H )) does not imply x α y α . This implication holds only in the case of α 1. The last fact concerns the convexity of the map x → x α on the positive part B(H )+ of B(H ). For α < 1 this map is concave (actually, the function (x, y) → x α ⊗ y 1−α is concave on B(H )+ × B(H )+ , [112]), but for α 1 convexity holds only if 1 α 2. The reader can ﬁnd more results of this nature in [17]. Some even worse phenomena may happen. It is well known that composed with the usual trace Tr on B(H ), all the preceding maps have the usual desired properties. For instance, the function x → Tr(x α ) becomes convex for all α 1, as one can expect. Now consider the function

1/α

(x1 , . . . , xn ) → Tr x1α + · · · + xnα on B(H )n+ . In the commutative setting, the convexity of this function for all α 1 and n 1 is extremely useful in many situations. Again, in the non-commutative case, this convexity is not guaranteed, at least for α > 2 (cf. [36]; see also [10] for some related results).

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Despite the difﬁculty caused by the lack of these elementary properties, we feel the theory has now matured enough for us to be able to present the reader with a rather satisfactory picture. Of course much remains to be done, as shown by the many open problems which we will encounter. We now brieﬂy describe the organization of this survey. After a preliminary section, we discuss the interpolation of non-commutative Lp -spaces (associated with a trace) in Section 2. This is one of the oldest subjects in the ﬁeld. The main result there allows to reduce all interpolation problems on non-commutative Lp -spaces to the corresponding ones on commutative Lp -spaces. Section 3 can be still considered as a preliminary one. There we introduce the noncommutative Lp -spaces associated with a state or weight. This section also contains two useful results. The ﬁrst one says that the non-commutative Lp -spaces over the hyperﬁnite II1 factor are the smallest ones among all those over von Neumann algebras not of type I. The second one is Haagerup’s approximation theorem. In the short Section 4 we discuss very brieﬂy some similarities and differences between the commutative Lp -spaces and their non-commutative counterparts. One remarkable result in the early stage of the non-commutative Lp -space theory is the Gordon–Lewis theorem on local unconditional structure of the Schatten classes. This (negative) result shows that compared with the usual function spaces, the Schatten classes (and so the general non-commutative Lp -spaces) are, in a certain sense, “very non-commutative”. Section 5 discusses the uniform convexities and smoothness, and the related type and cotype properties. Although the problem on the uniform (real) convexity of the noncommutative Lp -spaces goes back to the 50’s, the best constant for the modulus of convexity was found only at the beginning of the 90’s. Two uniform complex convexities (the uniform PL-convexity and Hardy convexity) are also discussed in this section. The central object in Section 6 is the non-commutative Khintchine inequalities, of paramount importance in this theory. Like in the commutative case, they are the key to a large part of non-commutative analysis, including of course the type and cotype properties of non-commutative Lp -spaces, and closely linked to the non-commutative Grothendieck theorem. Section 7 presents some very recent results on non-commutative martingale inequalities. In view of its close relations with quantum (= non-commutative) probability, this direction, which is still at an early stage of development, is likely to get more attention in the near future. Section 8 deals with the non-commutative Hardy spaces. We present there some noncommutative analogues of the classical theorems on the Hardy spaces in the unit disc, such as the boundedness of the Hilbert transformation, Szegö and Riesz factorizations. The ﬁrst result in Section 9 is Peller’s characterization of the membership of a Hankel operator in a Schatten class. This result is related to Schur multipliers. The rest of this section gives an outline of the recent works by Harcharras on Schur multipliers and noncommutative Λ(p)-sets. The last section concerns the embedding and isomorphism of non-commutative Lp -spaces. Almost all results given there were obtained just in the last few years. This is still a very active direction.

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We end this introductory section by pointing out that we will freely use standard notation and notions from operator algebra theory, for which we refer to [48,104,178,184,185,190].

1. Preliminaries In this section we give some necessary preliminaries on non-commutative Lp -spaces associated with a trace. This requires that the underlying von Neumann algebra be semiﬁnite (see below the deﬁnition). In Section 3, we will consider the non-tracial case. M will always denote a von Neumann algebra, and M+ its positive part. We recall that a trace on M is a map τ : M+ → [0, ∞] satisfying (i) τ (x + y) = τ (x) + τ (y), ∀x, y ∈ M+ ; (ii) τ (λx) = λτ (x), ∀λ ∈ [0, ∞), x ∈ M+ ; (iii) τ (u∗ u) = τ (uu∗ ), ∀u ∈ M. τ is said to be normal if supα τ (xα ) = τ (supα xα ) for any bounded increasing net (xα ) in M+ , semiﬁnite if for any non-zero x ∈ M+ there is a non-zero y ∈ M+ such that y x and τ (y) < ∞, and faithful if τ (x) = 0 implies x = 0. If τ (1) < ∞ (1 denoting the identity of M), τ is said to be ﬁnite. If τ is ﬁnite, we will assume almost systematically that τ is normalized, that is, τ (1) = 1. We often think of τ as a non-commutative (= quantum) probability. A von Neumann algebra M is called semiﬁnite if it admits a normal semiﬁnite faithful (abbreviated as n.s.f.) trace τ , which we assume in the remainder of this section. Then let S+ be the set of all x ∈ M+ such that τ (supp x) < ∞, where supp x denotes the support of x (deﬁned as the least projection p in M such that px = x or equivalently xp = x). Let S be the linear span of S+ . It is easy to check that S is a ∗-subalgebra of M which is w∗ -dense in M, moreover for any 0 < p < ∞, x ∈ S implies |x|p ∈ S+ (and so τ (|x|p ) < ∞), where |x| = (x ∗ x)1/2 is the modulus of x. Now we deﬁne

1/p xp = τ |x|p ,

x ∈ S.

One can show that · p is a norm on S if 1 p < ∞, and a quasi-norm (more precisely, a p-norm) if 0 < p < 1. The completion of (S, · p ) is denoted by Lp (M, τ ). This is the non-commutative Lp -space associated with (M, τ ). For convenience, we set L∞ (M, τ ) = M equipped with the operator norm. The trace τ can be extended to a linear functional on S, which will be still denoted by τ . Then τ (x) x1 ,

∀x ∈ S.

Thus τ extends to a continuous functional on L1 (M, τ ). The elements in Lp (M, τ ) can be viewed as closed densely deﬁned operators on H (H being the Hilbert space on which M acts). We recall this brieﬂy. A closed densely deﬁned operator x on H is said to be afﬁliated with M if xu = ux for any unitary u in the commutant M of M. An afﬁliated operator x is said to be τ -measurable or simply measurable if

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τ (eλ (|x|)) < ∞ for some λ > 0, where eλ (|x|) denotes the spectral resolution of |x| (corresponding to the indicator function of (λ, ∞)). For any measurable operator x we deﬁne the generalized singular numbers by

μt (x) = inf λ > 0: τ eλ |x| t ,

t > 0.

It will be convenient to denote simply by μ(x) the function t → μt (x). Note that μ(x) is a non-increasing function on (0, ∞). This notion is the generalization of the usual singular numbers for compact operators on a Hilbert space (see [72]). It was ﬁrst introduced in a Bourbaki seminar note by Grothendieck [77]. It was studied in details in [132,62] and [64]. Let L0 (M, τ ) denote the space of all measurable operators in M. Then L0 (M, τ ) is a ∗-algebra, which can be made into a topological ∗-algebra as follows. Let V (ε, δ) = x ∈ L0 (M, τ ): με (x) δ . Then {V (ε, δ): ε, δ > 0} is a system of neighbourhoods at 0 for which L0 (M, τ ) becomes a metrizable topological ∗-algebra. The convergence with respect to this topology is called the convergence in measure. Then M is dense in L0 (M, τ ). We refer to [131] and [191] for more information. The trace τ is extended to a positive tracial functional on the positive part L0+ (M, τ ) of 0 L (M, τ ), still denoted by τ , satisfying

∞

τ (x) =

μt (x) dt, 0

x ∈ L0+ (M, τ ).

Then for 0 < p < ∞,

1/p . Lp (M, τ ) = x ∈ L0 (M, τ ): τ |x|p < ∞ and xp = τ |x|p Also note that x ∈ Lp (M, τ ) iff μ(x) ∈ Lp (0, ∞), and xp = μ(x)Lp (0,∞) . Recall that μ(x) = μ(x ∗ ) = μ(|x|); so x ∈ Lp (M, τ ) iff x ∗ ∈ Lp (M, τ ), and we have xp = x ∗ p . The usual Hölder inequality extends to the non-commutative setting. Let 0 < r, p, q ∞ be such that 1/r = 1/p + 1/q. Then x ∈ Lp (M, τ ), y ∈ Lq (M, τ ) 4⇒ xy ∈ Lr (M, τ ) and xyr xp yq . In particular, if r = 1, τ (xy) xy1 xp yq ,

x ∈ Lp (M, τ ), y ∈ Lq (M, τ ).

This deﬁnes a natural duality between Lp (M, τ ) and Lq (M, τ ): x, y = τ (xy). Then for any 1 p < ∞ we have

Lp (M, τ )

∗

= Lq (M, τ ) (isometrically).

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Thus, L1 (M, τ ) is the predual of M, and Lp (M, τ ) is reﬂexive for 1 < p < ∞. Note that the classical theorem of Day on the dual of Lp for 0 < p < 1 was extended to the non-commutative setting by Saito [176]: the dual of Lp (M, τ ), 0 < p < 1, is trivial iff M has no minimal projection. R EMARK . [114] contains a different construction of non-commutative Lp -spaces via a non-commutative upper integral. Although we will concentrate on non-commutative Lp -spaces in this survey, the more general so-called “symmetric operator spaces” are worth mentioning: let E be a rearrangement invariant (in short r.i.) function space on (0, ∞), the symmetric operator space associated with (M, τ ) and E is deﬁned by and xE(M,τ ) = μ(x)E . E(M, τ ) = x ∈ L0 (M, τ ): μ(x) ∈ E In particular, if E = Lp (0, ∞), we recover Lp (M, τ ). These symmetric operator spaces have been extensively studied, see, e.g., [51,53,54,133,134] and [203] for more information. We end this section by some examples. (i) Commutative Lp -spaces. Let M be an Abelian von Neumann algebra. Then M = ∞ L (Ω, μ) for a measure space (Ω, μ), integration with respect to the measure μ gives us an n.s.f. trace, and Lp (M, τ ) is just the commutative Lp -space Lp (Ω, μ). (ii) Schatten classes. Let M = B(H ), the algebra of all bounded operators on H , and τ = Tr, the usual trace on B(H ). Then the associated Lp -space Lp (M, τ ) is the Schatten class S p (H ). If H is separable and dim H = ∞ (resp. dim H = n), we denote S p (H ) p by S p (resp. Sn ). Note that in our notation S ∞ (H ) is not the ideal of all compact operators on H but B(H ) itself. [72,128] and [182] contain elementary properties of S p (H ). (iii) The hyperﬁnite II1 factor. Let Mn denote the full algebra of all complex n × n matrices, equipped with the normalized trace σn . Let (R, τ ) =

8

(An , τn ),

(An , τn ) = (M2 , σ2 ), n ∈ N,

n1

be the von Neumann algebra tensor product. Then R is the hyperﬁnite II1 factor and τ is the (unique) normalized trace on R. There is another useful description of R. Let (εn )n1 be a sequence of self-adjoint unitaries on a Hilbert space, satisfying the following canonical anticommutation relations εi εj + εj εi = 2δij ,

i, j ∈ N.

(CAR)

Let R0 be the C ∗ -algebra generated by the εi ’s. Then R0 admits a unique faithful tracial state, denoted by τ , which is deﬁned as follows. For any ﬁnite subset A = {i1 , . . . , in } ⊂ N with i1 < · · · < in we put wA = εi1 · · · εin , and w∅ = 1. Then the trace τ is uniquely determined by its action on the wA ’s: τ (wA ) = 1 (resp. = 0) if A = ∅ (resp. = ∅). Consider R0 as a C ∗ -algebra acting on L2 (τ ) by left multiplication. Then the von Neumann algebra

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generated by R0 in B(L2 (τ )) can be (isomorphically) regarded as the hyperﬁnite II1 factor R. Note that the family of all linear combinations of the wA ’s are w∗ -dense in R and dense in Lp (R) for all 0 < p < ∞; also note that {wA : A ⊂ N} is an orthonormal basis of L2 (R) (= L2 (τ )). Finally, we mention that the von Neumann subalgebra generated by {ε1 , . . . , ε2n } is isomorphic to M2n , and then the restriction of τ to this subalgebra is just the normalized trace of M2n . We refer to [25] and [158] for more information. (iv) Group algebras. Consider a discrete group Γ . Let vN(Γ ) ⊂ B(2 (Γ )) be the associated von Neumann algebra generated by the left translations. Let τΓ be the canonical trace on vN(Γ ), deﬁned as follows: τΓ (x) = x(δe ), δe for any x ∈ vN(Γ ), where (δg )g∈Γ denotes the canonical basis of l2 (Γ ), and where e is the identity of Γ . This is a normal faithful normalized ﬁnite trace on vN(Γ ). A particularly interesting case is when Γ = Fn , the free group on n generators. We refer to [67] and [196] for more on this theme.

2. Interpolation This section is devoted to the interpolation of non-commutative Lp -spaces. It is well known that the non-commutative Lp -spaces associated with a semiﬁnite von Neumann algebra form an interpolation scale with respect to both the real and complex interpolation methods (see (2.1) and (2.2) below). This result not only is useful in applications but also can be taken as a starting point to deﬁne non-commutative Lp -spaces associated to a von Neumann algebra of type III (which admits no n.s.f. trace). This is indeed the viewpoint taken by Kosaki [106] (see also [192]). We will discuss this point in the next section. Here we restrict ourselves only to semiﬁnite von Neumann algebras. Thus throughout this section, M will always denote a semiﬁnite von Neumann algebra equipped with a faithful normal semiﬁnite trace τ . We refer to [15] for all notions and notation from interpolation theory used below. Let 1 p0 , p1 ∞ and 0 < θ < 1. It is well known that

Lp (M, τ ) = Lp0 (M, τ ), Lp1 (M, τ ) θ (with equal norms),

Lp (M, τ ) = Lp0 (M, τ ), Lp1 (M, τ ) θ,p (with equivalent norms),

(2.1) (2.2)

where 1/p = (1−θ )/p0 +θ/p1 , and where (·, ·)θ , (·, ·)θ,p denote respectively the complex and real interpolation methods. It is not easy to retrace the origin of these interpolation results. Some weaker or particular forms go back to the 50’s (cf., e.g., [47,110,172]). The results in the full generality were achieved by Ovchinikov [133,134] (see also [135] for the real interpolation, and [141] in the case of Schatten classes). (2.1) and (2.2) easily follow from the following result. Recall that μ(x) denotes the generalized singular number of x (see Section 1) and that a map T : X → Y is called contractive (or a contraction) if T 1. T HEOREM 2.1. For any ﬁxed x ∈ L1 (M, τ ) + L∞ (M, τ ) there are linear maps T and S (which may depend on x) satisfying the following properties: (i) T : L1 (M, τ ) + L∞ (M, τ ) → L1 (0, ∞) + L∞ (0, ∞), T is contractive from Lp (M, τ ) to Lp (0, ∞) for p = 1 and p = ∞, and T x = μ(x);

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(ii) S : L1 (0, ∞) + L∞ (0, ∞) → L1 (M, τ ) + L∞ (M, τ ), S is contractive from Lp (0, ∞) to Lp (M, τ ) for p = 1 and p = ∞, and Sμ(x) = x. Although not explicitly stated, Theorem 2.1 is implicit in the literature. It is essentially contained in [4] for Schatten classes, and in some different (weaker) form in [53] for the general case. We will include a proof at the end of the section. R EMARK . In interpolation language, Theorem 2.1 implies that the pair (L1 (M, τ ), L∞ (M, τ )) is a (contractive) partial retract of (L1 (0, ∞), L∞ (0, ∞)). We should emphasize the usefulness of such a result: it reduces all interpolation problems on (L1 (M, τ ), L∞ (M, τ )) to those on (L1 (0, ∞), L∞ (0, ∞)). Recall that (L1 (0, ∞), L∞ (0, ∞)) is one of the best understood pairs in interpolation theory. We now illustrate this by some examples. More applications can be found in [4,53,54] and [133,134]. First let us show how to get (2.1) and (2.2) from their commutative counterparts. P ROOF OF (2.1) AND (2.2). Let x ∈ Lp (M, τ ) (noting that Lp (M, τ ) ⊂ L1 (M, τ ) + L∞ (M, τ )). Let S be the map associated to x given by Theorem 2.1. Then by interpolation

S : L1 (0, ∞), L∞ (0, ∞) θ → L1 (M, τ ), L∞ (M, τ ) θ is a contraction. However, it is classical that 1

L (0, ∞), L∞ (0, ∞) θ = Lp (0, ∞) (with equal norms). Thus we deduce xθ = Sμ(x)θ μ(x)θ = xp ; whence

Lp (M, τ ) ⊂ L1 (M, τ ), L∞ (M, τ ) θ ,

a contractive inclusion.

The inverse inclusion is proved similarly by means of the map T . Therefore, we have shown (2.1). In the same way, we get (2.2). The above argument works in a more general setting as well. C OROLLARY 2.2. Let F be an interpolation functor. Then

F L1 (M, τ ), L∞ (M, τ ) = F L1 (0, ∞), L∞ (0, ∞) (M, τ ). More generally, for any r.i. function spaces E0 , E1 on (0, ∞)

F E0 (M, τ ), E1 (M, τ ) = F (E0 , E1 )(M, τ ).

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This corollary is contained in [53] (and in [4,141] for Schatten classes). R EMARK . As a consequence of Corollary 2.2, we have that for any r.i. function space E the map T (resp. S) in Theorem 2.1 is contractive from E(M, τ ) (resp. E) to E (resp. E(M, τ )). In particular, T and S are contractions between the Lp -spaces in consideration for all 1 p ∞. The following particular case of Corollary 2.2 is worth being mentioned explicitly. Here Kt denotes the usual K-functional from interpolation theory. C OROLLARY 2.3. Let 1 p0 , p1 ∞. Then for any x ∈ Lp0 (M, τ ) + Lp1 (M, τ ) and any t > 0

Kt x; Lp0 (M, τ ), Lp1 (M, τ ) = Kt μ(x); Lp0 (0, ∞), Lp1 (0, ∞) . In particular,

Kt x; L1 (M, τ ), L∞ (M, τ ) =

t

μs (x) ds. 0

R EMARKS . (i) Using a factorization argument, one can easily extend Corollary 2.3 to the case of quasi-Banach spaces, so that p0 , p1 are now allowed to be in (0, ∞]. Then the equality there has to be replaced by an equivalence with relevant constants depending only on p0 , p1 (see also [135]). (ii) As a consequence of the preceding remark, the indices p0 , p1 in (2.2) can vary in (0, ∞]. (iii) (2.1) also extends to the quasi-Banach space case (cf. [201]). P ROOF OF T HEOREM 2.1. Fix an x ∈ L1 (M, τ ) + L∞ (M, τ ). We may assume x 0. Indeed, by polar decomposition, it is easy to reduce the proof to this case. First we suppose x is an elementary operator, i.e., of the form x=

n

a k ek ,

k=1

where for all 1 k n, ak ∈ (0, ∞), and where the ek ’s are disjoint projections with τ (ek ) ∈ (0, ∞). Then we deﬁne Py =

n τ (yek ) k=1

τ (ek )

ek ,

y ∈ L1 (M, τ ) + L∞ (M, τ ).

Note that P is the orthogonal projection of L2 (M, τ ) onto its subspace generated by {e1 , . . . , en }. In particular, P is selfadjoint. Let y ∈ L∞ (M, τ ). Then |τ (yek )| y∞ τ (ek ) sup = y∞ . τ (ek ) 1kn τ (ek ) 1kn

P y∞ sup

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Therefore, P is a contraction on L∞ (M, τ ). By duality, P is a contraction on L1 (M, τ ) as well. Let N be the subalgebra generated by {e1 , . . . , en } (the identity of N being e = e1 + n and τ induces a weighted counting measure ν on · · · + en ). Then N is isomorphic to l∞ n l∞ , namely, ν({k}) = τ (ek ) for all 1 k n. It is clear that for any p Lp (N , τ |N ) = lpn (ν) isometrically. n , ν). With this identiﬁcation, μ(x) is exactly the Thus we can identify (N , τ |N ) with (l∞ usual non-increasing rearrangement of x with respect to the measure ν. On the other hand, it is classical (and easy to prove in our special case) that there are linear maps R and Q satisfying (cf. [33]) n (ν) → L1 (0, ∞) + L∞ (0, ∞), R is contractive from l n (ν) to (i) R : l1n (ν) + l∞ p p L (0, ∞) for p = 1, ∞, and Rx = μ(x); n (ν), Q is contractive from Lp (0, ∞) to (ii) Q : L1 (0, ∞) + L∞ (0, ∞) → l1n (ν) + l∞ n lp (ν) for p = 1, ∞, and Qμ(x) = x. n (ν) Then we set T = RP and S = iQ, where i is the natural inclusion of l1n (ν) + l∞ (= L1 (N , τ |N ) + L∞ (N , τ |N )) into L1 (M, τ ) + L∞ (M, τ ). One easily checks that T and S satisfy all requirements of Theorem 2.1. Therefore, Theorem 2.1 is proved for elementary operators. Before passing to general (positive) operators, we note that T and S constructed above are positive in the sense that y 0 (resp. f 0) implies T y 0 (resp. Sf 0). Now for a positive x ∈ L1 (M, τ ) + L∞ (M, τ ), using the spectral decomposition of x, we may choose an increasing sequence {xn } of elementary positive operators such that xn x for all n 1, limn→∞ μt (xn ) = μt (x) for all t > 0 and limn→∞ xn = x in the topology σ (L1 (M, τ ) + L∞ (M, τ ), L1 (M, τ ) ∩ L∞ (M, τ )). See [64], pp. 277– 278. By the ﬁrst part of the proof, for each n there are Tn and Sn associated with xn as in Theorem 2.1. Thus (Tn ) is a bounded sequence in B(L∞ (M, τ ), L∞ (0, ∞)). Since B(L∞ (M, τ ), L∞ (0, ∞)) is a dual space with predual L∞ (M, τ ) ⊗∧ L1 (0, ∞), passing to a subsequence if necessary, we may assume that Tn converges to T in B(L∞ (M, τ ), L∞ (0, ∞)) with respect to the w∗ -topology. Thus T is a contraction from L∞ (M, τ ) to L∞ (0, ∞). To show that T also deﬁnes a contraction from L1 (M, τ ) to L1 (0, ∞) let y ∈ L1 (M, τ ) ∩ L∞ (M, τ ) and f ∈ L1 (0, ∞) ∩ L∞ (0, ∞). Then

∞

T (y)f = lim

∞

n→∞ 0

0

Tn (y)f ;

whence

∞ 0

T (y)f lim sup Tn (y)1 f ∞ y1 f ∞ , n→∞

which implies that T y ∈ L1 (0, ∞) and T y1 y1 . Hence T extends to a contraction from L1 (M, τ ) into L1 (0, ∞).

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On the other hand, by the positivity of Tn , we have μ(xn ) = Tn xn Tn x. Taking limits, we ﬁnd μ(x) T x. Hence μ(x) = wT x for some w ∈ L∞ (0, ∞) with w∞ 1. Then one sees that the map T deﬁned by T y = wT y has the property (i) of Theorem 2.1. Similarly, from the sequence (Sn ) we get the desired map S. R EMARK 2.4. The result of interpolation applied to a compatible pair (X0 , X1 ) of Banach spaces depends in general very much on the way in which we view this pair as compatible. There is however an elementary “invariance” property which we will invoke in the sequel, as follows: let (X0 , X1 ) be a compatible pair of Banach spaces. Now let (Y0 , Y1 ) be another compatible pair of Banach spaces and let u0 : Y0 → X0 and u1 : Y1 → X1 be isometric isomorphisms, which coincide on Y0 ∩ Y1 (in that case it is customary in interpolation theory to think of u0 and u1 as the “same” map!). Equivalently, we have an isometric isomorphism u : Y0 + Y1 → X0 + X1 such that the restrictions u0 = u|Y0 and u1 = u|Y1 are isometric isomorphisms respectively from Y0 to X0 and from Y1 to X1 . Then u deﬁnes an isometric isomorphism from (Y0 , Y1 )θ to (X0 , X1 )θ , so that (Y0 , Y1 )θ (X0 , X1 )θ

(0 < θ < 1).

This follows from the interpolation property applied separately to u and its inverse. If we assume that the pairs are made compatible with respect to continuous injections J : X0 → X1 and j : Y0 → Y1 . Then to say that u0 and u1 are the “same” map means that u1 j = J u0 . In particular, if X0 = Y0 and if u0 is the identity on X0 = Y0 , then this reduces to u1 j = J .

3. General von Neumann algebras, including type III The construction of non-commutative Lp -spaces based on n.s.f. traces outlined in Section 1 does not apply to von Neumann algebras of type III, which do not admit n.s.f. traces. However, it is known that any von Neumann algebra has an n.s.f. weight (a weight is simply an additive and positively homogeneous functional on the positive cone with values in [0, ∞]). This section is devoted to the non-commutative Lp -spaces associated with a von Neumann algebra equipped with an n.s.f. weight. There are several ways to construct the latter spaces (cf., e.g., [3,78,87,106,113,192]). We will present two of them. The ﬁrst one is to reduce von Neumann algebras of type III to semiﬁnite von Neumann algebras with the help of crossed products, as proposed by Haagerup [78]. The second way is via the complex interpolation; so it can be considered as a continuation of the results established in the previous section for the semiﬁnite case. This was developed by Kosaki [106] and Terp [192] (see also [88,89] for related results). We begin with the construction via interpolation. Let M be a von Neumann algebra. We know that M is a dual space with a unique predual, denoted by M∗ . We deﬁne, as usual, L1 (M) = M∗ and L∞ (M) = M. Now we are confronted with the problem of deﬁning Lp (M) for any 1 < p < ∞. For simplicity and clarity we will consider only the case where M is σ -ﬁnite, as in [106]. The reader is referred to [192] for the general case. Fix a

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1471

distinguished normal faithful state ϕ on M. Then we embed M into M∗ by the following left injection j : M → M∗ ,

j (x) = xϕ

(here xϕ(y) = ϕ(yx) ∀y ∈ M).

It is clear that j is a contractive injection with dense range. Thus we obtain a “compatibility” for the pair (M, M∗ ) with respect to which we may consider interpolation spaces between M and M∗ . Now let 1 < p < ∞. Following Kosaki, we introduce the corresponding non-commutative Lp -space as Lp (M, ϕ) = (M, M∗ )1/p . To show the so-deﬁned non-commutative Lp -spaces possess all properties one can expect, one should ﬁrst note the important fact that L2 (M, ϕ) is a Hilbert space, more precisely, L2 (M, ϕ) = Hϕ , where Hϕ is the Hilbert space in the GNS construction induced by ϕ (obtained after completion of M equipped with the inner product x, y = ϕ(y ∗ x)). The proof of this fact given in [106] uses the modular theory. Here, we would like to point out that it directly follows from a general result in interpolation theory, that we describe as follows. Let X be a complex Banach space. Let X denote the conjugate space of X, i.e., X is just ¯ for any λ ∈ C X itself but equipped with the conjugate complex multiplication: λ · x = λx and x ∈ X. For x ∈ X, x¯ denotes the element x considered as an element in X. Given a linear map v : X → Y , we denote by v¯ : X → Y the same map acting on the “conjugates”. Now suppose that there is a bounded linear map J : X∗ → X which is injective and of dense range. This allows us to consider (X∗ , X) as a compatible pair of Banach spaces. Suppose further that J is positive, i.e., ξ(J (ξ )) 0 for any ξ ∈ X∗ . Then ξ, η = ξ(J (η)) deﬁnes ¯ = ξ(x).) Let H be the a scalar product on X∗ . (Note: for ξ ∈ X∗ and x ∈ X we write ξ(x) completion of X∗ with respect to the above scalar product. Note that H contains X∗ as a dense linear subspace. Thus we can deﬁne a bounded linear injection v : H → X by simply setting (on an element of X∗ ) v(ξ ) = J (ξ ), and extending by density to the whole of H . Identifying H ∗ with H as well as (X)∗ with X∗ , and denoting by t v : (X)∗ → H ∗ = H the adjoint of v (in the Banach space sense) we see that J = v t v. These facts are well known (and easy to check). The general theorem referred to above is the following T HEOREM 3.1. With the above assumptions, (X∗ , X)1/2 = H with equal norms. R EMARK . This is well-known ([116]) with the additional assumption that X is reﬂexive. The general form as above was observed in [153], p. 26 (see also [197] and [42] for related results). C OROLLARY 3.2. L2 (M, ϕ) = Hϕ with equal norms. P ROOF. We let X = M∗ , X∗ = M. Recall that the involution on M∗ is deﬁned by ψ ∗ (x) = ψ(x ∗ ) (ψ ∈ M∗ , x ∈ M). Let J : M → M∗ be the map taking x to j (x)∗ = ϕx ∗ and let u1 : M∗ → M∗ be the (linear) isometry taking ψ to ψ ∗ . We have ξ, η =

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ξ(J (η)) = ξ(ϕη∗ ) = ξ(ϕη∗ ) = ϕ(η∗ ξ ), thus we ﬁnd H = Hϕ with equal norms and we have J = u1 j , so that the result follows by invoking Remark 2.4 (here u0 is simply the identity of M). Using this corollary and the reiteration theorem, we see that the dual space of Lp (M, ϕ)

is (isometrically) equal to Lp (M, ϕ) for any 1 < p < ∞ (1/p + 1/p = 1). The duality is induced by the scalar product of Hϕ , that is, x, y = ϕ(y ∗ x), x, y ∈ M. Corollary 3.2 also yields the Clarkson inequalities in Lp (M, ϕ) for any 1 < p < ∞ (see Theorem 5.1 below). Thus, Lp (M, ϕ) (1 < p < ∞) is uniformly convex. We will see more precise results on this in Section 5. R EMARK . Instead of the left injection considered previously, one could equally take the right injection of M into M∗ , i.e., x → ϕx (here ϕx(y) = ϕ(xy) ∀y ∈ M). Then the resulting interpolation spaces are isometric to those obtained previously. In view of the results in the last section, one is naturally led to consider the real interpolation as well. Set, for 1 < p < ∞ Lp,p (M, ϕ) = (M, M∗ )1/p,p . The problem now is whether Lp,p (M, ϕ) and Lp (M, ϕ) are isomorphic. For the special case of p = 2, the answer is afﬁrmative. Indeed, Theorem 3.1 admits a counterpart for the real interpolation as well (see [116] in the case of reﬂexive spaces; [205], p. 519 for the general case; see also [42] for more related results). Thus L2,2 (M, ϕ) = L2 (M, ϕ) with equivalent norms. However, this is no longer true for all other values of p, as shown by the following example, due to Junge and the second named author. E XAMPLE 3.3. Let ϕ be the state of B(l 2 ) given by a diagonal operator D of trace 1 and whose diagonal entries are all positive. Then, for any 1 < p = 2 < ∞, the two spaces (B(l 2 ), B(l 2 )∗ )1/p and (B(l 2 ), B(l 2 )∗ )1/p, p do not coincide. Indeed, let R (resp. R∗ ) be the subspace of B(l 2 ) (resp. B(l 2 )∗ ) consisting of matrices whose all rows but the ﬁrst are zero. It is clear that R = l 2 and R∗ = l 2 (d) isometrically, where d = (dn )n is the sequence of the diagonal entries of D, and where l 2 (d) is the weighted l 2 -space with the norm xl 2 (d) =

1/2 |xn dn |2

.

n

On the other hand, let P : B(l 2 ) → R be the natural projection. P is contractive on B(l 2 ). It is easy to check that under the left injection associated with ϕ, P is also a contractive projection from B(l 2 )∗ onto R∗ . Now assume that for some 1 < p < ∞ the two interpolation spaces (B(l 2 ), B(l 2 )∗ )1/p and (B(l 2 ), B(l 2 )∗ )1/p, p have equivalent norms. Then we deduce that (R, R∗ )1/p and (R, R∗ )1/p, p have equivalent norms too. However, it is wellknown that the ﬁrst space is still a weighted l 2 -space (and hence a Hilbert space), while the second one is isomorphic to a Hilbert space only when p = 2. Thus we have proved our assertion.

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This construction by interpolation has several disadvantages: there is no natural notion of positive cone, no reasonably handy bimodule action by multiplication of M on Lp (M, ϕ), and ﬁnally the case p < 1 is excluded. However, these difﬁculties disappear in Haagerup’s construction, to which we now turn. Our main reference for Haagerup’s Lp -spaces is [191]. Let M be a von Neumann algebra equipped with a distinguished n.s.f weight ϕ. Let σt = ϕ σt , t ∈ R, denote the one parameter modular automorphism group of R on M associated with ϕ. We consider the crossed product R = M σ R. Recall brieﬂy the deﬁnition of R. If M acts on a Hilbert space H , R is a von Neumann algebra acting on L2 (R, H ), generated by the operators π(x), x ∈ M, and the operators λ(s), s ∈ R, deﬁned by the following conditions: for any ξ ∈ L2 (R, H ) and t ∈ R and λ(s)(ξ )(t) = ξ(t − s).

π(x)(ξ )(t) = σ−t (x)ξ(t)

Note that π is a normal faithful representation of M on L2 (R, H ). Thus we may identify M with π(M). Then the modular automorphism group {σt }t ∈R is given by σt (x) = λ(t)xλ(t)∗ ,

x ∈ M, t ∈ R.

There is a dual action {σˆ t }t ∈R of R on R. This is a one parameter automorphism group of R on R, implemented by the unitary representation {W (t)}t ∈R of R on L2 (R, H ): σˆ t (x) = W (t)xW (t)∗ ,

t ∈ R, x ∈ R,

where W (t)(ξ )(s) = e−it s ξ(s),

ξ ∈ L2 (R, H ), t, s ∈ R.

Note that the dual action σˆ t is also uniquely determined by the following conditions σˆ t (x) = x

and σˆ t λ(s) = e−ist λ(s),

∀x ∈ M, s, t ∈ R.

Thus M is invariant under {σˆ t }t ∈R . In fact, M is exactly the space of the ﬁxed points of {σˆ t }t ∈R , namely, M = x ∈ R: σˆ t (x) = x, ∀t ∈ R . Recall that the crossed product R is semiﬁnite. Let τ be its n.s.f. trace satisfying τ ◦ σˆ t = e−t τ,

∀ t ∈ R.

Also recall that any n.s.f. weight ψ on M induces a dual n.s.f. weight ψ˜ on R. Then ψ˜ admits a Radon–Nikodym derivative with respect to τ . In particular, the dual weight ϕ˜ of our distinguished weight ϕ has a Radon–Nikodym derivative D with respect to τ . Then ϕ(x) ˜ = τ (Dx),

x ∈ R+ .

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Recall that D is an invertible positive selfadjoint operator on L2 (R, H ), afﬁliated with R, and that the regular representation λ(t) above is given by λ(t) = D it ,

t ∈ R.

Now we deﬁne the Haagerup non-commutative Lp -spaces (0 < p ∞) by Λp (M, ϕ) = x ∈ L0 (R, τ ): σˆ t (x) = e−t /p x, ∀t ∈ R . (Recall that L0 (R, τ ) denotes the topological ∗-algebra of all operators on L2 (R, H ) measurable with respect to (R, τ ).) It is clear that Λp (M, ϕ) is a vector subspace of L0 (R, τ ), invariant under the ∗-operation. The algebraic structure of Λp (M, ϕ) is inherited from that of L0 (R, τ ). Let x ∈ Λp (M, ϕ) and x = u|x| its polar decomposition. Then u ∈ M and |x| ∈ Λp (M, ϕ). Recall that Λ∞ (M, ϕ) = M and Λ1 (M, ϕ) = M∗ . The latter equality is understood as follows. As mentioned previously, for any ω ∈ M+ ∗, the dual weight ω˜ has a Radon–Nikodym derivative, denoted by hω , with respect to τ : ω(x) ˜ = τ (hω x),

x ∈ R+ .

Then hω ∈ L0 (R, τ )

and σˆ t (hω ) = e−t hω ,

∀t ∈ R.

1 Thus hω ∈ Λ1 (M, ϕ)+ . This correspondence between M+ ∗ and Λ (M, ϕ)+ extends to 1 a bijection between M∗ and Λ (M, ϕ). Then for any ω ∈ M∗ , if ω = u|ω| is its polar decomposition, the corresponding hω ∈ Λ1 (M, ϕ) admits the polar decomposition

hω = u|hω | = uh|ω| . Thus we can deﬁne a norm on Λ1 (M, ϕ) by hω 1 = |ω|(1) = ωM∗ ,

ω ∈ M∗ .

In this way, Λ1 (M, ϕ) = M∗ isometrically. Now let 0 < p < ∞. Since x ∈ Λp (M, ϕ) iff |x|p ∈ Λ1 (M, ϕ), we deﬁne 1/p xp = |x|p 1 ,

x ∈ Λp (M, ϕ).

Then if 1 p < ∞, ·p is a norm (cf. [78] and [191]), and if 0 < p < 1, ·p is a p-norm (cf. [108]). Equipped with · p , Λp (M, ϕ) becomes a Banach space or a quasi-Banach space, according to 1 p < ∞ or 0 < p < 1. Clearly, xp = x ∗ p = |x|p , x ∈ Λp (M, ϕ).

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R EMARKS . (i) Using [191] Lemma II.5, one easily checks that Λp (M, ϕ) is isometric to a subspace of the non-commutative weak Lp -space Lp,∞ (R, τ ). Also note that in Λp (M, ϕ) the topology deﬁned by · p coincides with the topology induced by that of L0 (R, τ ) (cf. [191]). (ii) One weak point of the Haagerup non-commutative Lp -spaces is the fact that for any p = q the intersection of Λp (M, ϕ) and Λq (M, ϕ) is trivial. In particular, these spaces do not form an interpolation scale. This causes some difﬁculties in applications (especially when interpolation is used).

As usual, for 1 p < ∞ the dual space of Λp (M, ϕ) is Λp (M, ϕ), 1/p + 1/p = 1. To describe this duality, we need to introduce a distinguished linear functional on Λ1 (M, ϕ), called trace and denoted by tr, which is deﬁned by tr(x) = ωx (1),

x ∈ Λ1 (M, ϕ),

where ωx ∈ M∗ is the unique normal functional associated with x by the above identiﬁcation between M∗ and Λ1 (M, ϕ). Then tr is a continuous functional on Λ1 (M, ϕ) satisfying

tr(x) tr |x| = x1 ,

x ∈ Λ1 (M, ϕ).

The usual Hölder inequality also holds for these non-commutative Lp -spaces. Let 0 < p, q, r ∞ such that 1/r = 1/p + 1/q. Then x ∈ Λp (M, ϕ) and y ∈ Λq (M, ϕ) 4⇒ xy ∈ Λr (M, ϕ) and xyr xp yq . In particular, for any 1 p ∞ we have tr(xy) xy1 xp yp ,

x ∈ Λp (M, ϕ), y ∈ Λp (M, ϕ).

Thus, (x, y) → tr(xy) deﬁnes a duality between Λp (M, ϕ) and Λp (M, ϕ), with respect to which p

∗

Λ (M, ϕ) = Λp (M, ϕ)

isometrically, 1 p < ∞.

This functional tr on Λ1 (M, ϕ) plays the role of a trace. Indeed, it satisﬁes the following tracial property tr(xy) = tr(yx),

x ∈ Λp (M, ϕ), y ∈ Λp (M, ϕ).

The reader is referred to [191] for more information.

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T HEOREM 3.4. Let M be a von Neumann algebra. (i) Let 0 < p < ∞. If τ is an n.s.f. trace on M, then Lp (M, τ ) (the non-commutative Lp -space described in Section 1) is isometric to Λp (M, ϕ). (ii) Let 0 < p < ∞. Then Λp (M, ϕ) is independent of ϕ, i.e., if ϕ and ψ are two n.s.f. weights on M, then Λp (M, ϕ) and Λp (M, ψ) are isometric. (iii) Let ϕ be a normal faithful state on M and 1 < p < ∞. Then Lp (M, ϕ) and Λp (M, ϕ) are isometric. The ﬁrst two parts of Theorem 3.4 are due to Haagerup [78] (see also [191]), and the third one to Kosaki [106]. As can be expected, the proof of Theorem 3.4 heavily depends on the modular theory. The preceding statement allows a considerable simpliﬁcation of the notation, as follows: C ONVENTION . From now on, given a von Neumann algebra M, Lp (M) will denote any one of the non-commutative Lp -spaces associated with M appearing in Theorem 3.4. (The latter shows that these spaces are all “the same”.) However, if M is semiﬁnite, we will always assume that Lp (M) is the Lp -space constructed from an n.s.f. trace as in Section 1. The following basic result is very useful to reduce the failure of certain properties of to the special case of the hyperﬁnite factor Lp (R). Recall that R denotes the hyperﬁnite II1 factor (see Section 1).

Lp -spaces

T HEOREM 3.5. Let M be a von Neumann algebra not of type I. Then for any 0 < p ∞ (resp. 1 p ∞) Lp (R) is isometric to a (resp. 1-complemented) subspace of Lp (M). The proof of Theorem 3.5 combines several more or less well-known facts. The key point is that if M is not of type I, then R is isomorphic, as von Neumann algebra, to a w∗ -closed ∗-subalgebra of M which is the range of a normal conditional expectation on M. The reader is referred to [122] for more details and precise references. We end this section with Haagerup’s approximation theorem of an Lp (M) associated with an algebra M of type III by those associated with semiﬁnite von Neumann algebras (cf. [79]). T HEOREM 3.6. Let M be a von Neumann algebra equipped with an n.s.f. weight ϕ. Let Λp (M, ϕ) be the associated Haagerup Lp -space (0 < p < ∞). Then there are a Banach space X (a p-Banach space if 0 < p < 1), a directed family {(Mi , τi )}i∈I of ﬁnite von Neumann algebras Mi (with normal faithful ﬁnite traces τi ), and a family {ji }i∈I of isometric embeddings ji : Lp (Mi , τi ) → X such that (i) ji (Lp (Mi , τi )) ⊂ ji (Lp (Mi , τi )) for all i, i ∈ I with i i ; (ii) i∈I ji (Lp (Mi , τi )) is dense in X; (iii) Λp (M, ϕ) is isometric to a (complemented for 1 p < ∞) subspace of X.

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4. From classic Lp to non-commutative Lp : similarities and differences A good part of the early theory consisted in extending commutative results over to the non-commutative case; this usually required speciﬁc new methods, but without too many surprises. For instance, we have already seen that a non-commutative Lp -space Lp (M) is reﬂexive for any 1 < p < ∞. Moreover, just like in the commutative case it is easy to check that L1 (M) has the RNP iff M is atomic. Indeed, if M is not atomic, L1 (M) contains a 1-complemented subspace isometric to L1 (0, 1), hence fails the RNP. On the other hand, L1 (M) is weakly sequentially complete for any M. Moreover, there are characterizations of weakly compact subsets in L1 (M), analogous to those in the commutative setting (cf. [190, III.5] and the references given there; see also [140] for more recent results). Moreover, we will see later in Section 5 that any non-commutative Lp -space (0 < p 1) has the analytic RNP. However, the differentiability of the norms of non-commutative Lp -spaces has not been well understood yet. This problem was considered only for the Schatten classes in [194]. It was announced there (with a sketch of proof) that the norm of S p had the same differentiability as that of l p (1 < p < ∞). It seems unclear how to extend this to the general case (or, at least, to the semiﬁnite case). In a different direction, the papers [55,56] are devoted to the problem of characterizing the symmetric spaces of measurable operators for which the absolute-value mapping x → |x| is Lipschitz continuous. In the case of non-commutative L1 -spaces, Kosaki proves in [108] the following useful inequality: for any ϕ and ψ in such a space, we have √

|ϕ| − |ψ| 2 ϕ + ψ1 ϕ − ψ1 1/2 . 1 The passage from the Schatten classes to von Neumann algebras with semiﬁnite traces, i.e., from the discrete to the continuous case, can sometimes be quite substantial. See, for instance, Brown’s extension of Weyl’s classical inequalities: ( |λn (T )|p )1/p T S p (here λn (T ) are the eigenvalues of T repeated according to multiplicity). Brown [26] had to invent a new kind of spectral measure (now called Brown’s measure) to extend this, together with Lidskii’s trace theorem, to the semiﬁnite case. The study of non-commutative Lp -spaces, or more generally, of symmetric operator spaces, goes mainly in two closely related directions: lift topological or geometrical properties from the commutative setting to the non-commutative one, and reduce problems in the non-commutative case to those in the commutative one. We have already seen several examples in both directions. To discuss more illustrations, it is better to place ourselves in the context of symmetric operator spaces. Let M be a semiﬁnite von Neumann algebra equipped with an n.s.f. trace τ , and let E be an r.i. function space on (0, ∞). One naturally expects that properties of E(M, τ ) should be reﬂected by those of E. Works already done in this direction are too numerous to enumerate. Here we content ourselves with only three examples. The ﬁrst one concerns the (uniform) Kadets–Klee properties. The lifting of these properties from E to E(M, τ ) has been extensively studied (cf., e.g., [6,37,38,41,50,57]). The second example is about the reduction of weakly compact subsets in E(M, τ ) to those in E. This was

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achieved in [52] and [58] (see also the references there for previous works on this problem). Finally, the geometry of the unit ball of E(M, τ ) was studied in [5] and [40]. More examples and references of this kind will be given in appropriate places in the subsequent sections (see also [49]). Despite the strong analogy between the commutative and non-commutative settings, non-commutative Lp -spaces behave, in some aspects, very differently from their commutative counterparts. One of the most spectacular differences concerns unconditional bases or “unconditional structures”. Already in [111], it was proved that S 1 cannot be embedded into any space with an unconditional basis, in sharp contrast with 1 . But the big surprise came when Gordon and Lewis [74] proved that the Schatten class S p fails to have any unconditional basis when p = 2 in sharp contrast with p or Lp . More generally they proved that S p fails “local unconditional structure” in their sense (abbreviated as GL-l.u.st.; see [94] for the precise deﬁnition). This was the ﬁrst example of a reﬂexive Banach space which was not isomorphic to any complemented subspace of a Banach lattice. More precisely, let lu(X) denote the GL-l.u.st. constant of a Banach space X (lu(X) is equal to the norm of factorization through a Banach lattice of the identity of X∗∗ ). The following theorem was proved by Gordon and Lewis [74] using a criterion (necessary but not sufﬁcient) for the GL-l.u.st. of a space X: any 1-absolutely summing operator on X must factor through L1 (this is now called the GL-property). More precisely, they obtained the ﬁrst part of the next statement (the second part comes from [143], see also [180] and [146], 8.d for related results): T HEOREM 4.1. There is a constant C > 0 such that for any 1 p ∞ and any n 1 p

Cn|1/p−1/2| lu Sn n|1/p−1/2|. Consequently, S p does not have the GL-l.u.st. for p = 2. More generally, let X be any Banach lattice of ﬁnite cotype (resp. of type > 1), then there is a constant C > 0 such that, if E is any n2 -dimensional subspace (resp. subspace of a quotient) of X, we have p d(Sn , E) C n|1/p−1/2| . Combining Theorem 3.5 and Theorem 4.1, we immediately obtain C OROLLARY 4.2. A non-commutative Lp (M), 1 p < ∞ and p = 2, has the GL-l.u.st. iff M is isomorphic, as Banach space, to L∞ (Ω, μ) for some measure space (Ω, μ). Moreover, this happens iff Lp (M) is isomorphic to a subspace of a commutative Lp -space. Note that M is isomorphic, as Banach space, to a commutative L∞ iff M is the direct sum (∞ sense) of ﬁnitely many algebras of the form L∞ (μ; B(H )) (= L∞ (μ) ⊗ B(H )) with dim(H ) < ∞. Another striking divergence from the classical case is provided by the uniform approximation property (UAP in short): by an extremely complicated construction, Szankowski proved that B(2 ) fails the approximation property (AP in short), and moreover ([189])

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that S p (or S p ) fails the UAP for p > 80. It remains a challenging open problem to prove this for any p = 2. We will describe another striking difference in Section 7, that is, a non-commutative Lp -space Lp (M), 0 < p 1, is never an analytic UMD space except when M is isomorphic, as Banach space, to a commutative L∞ -space. Surprisingly, by [122], “stability” provides us with one more sharp contrast. Recall that a Banach space X is stable (in Krivine–Maurey’s sense) if for any bounded sequences {xm }m1 , {yn }n1 in X and any ultraﬁlters U , V on N lim lim xm + yn = lim lim xm + yn .

m∈U n∈V

n∈V m∈U

It is well known that any commutative Lp -space (1 p < ∞) is stable (cf. [109]). This is no longer true in the non-commutative setting. In fact, we have the following characterization of stable non-commutative Lp -spaces. T HEOREM 4.3. Let 1 p < ∞, p = 2. Then Lp (M) is stable iff M is of type I. The “if” part of Theorem 4.3 was independently proved by Arazy [8] and Raynaud [166]. The “only if” part is due to Marcolino [122]. Marcolino’s proof is divided into two steps. The ﬁrst one (the proof of which is relatively easy) is that Lp (R), p = 2, is not stable (recalling that R is the hyperﬁnite II1 factor). The second step is the above Theorem 3.5.

5. Uniform convexity (real and complex) and uniform smoothness The fact that Lp (M), 1 < p < ∞, is uniformly convex and smooth immediately follows from the following Clarkson type inequalities. T HEOREM 5.1. Let 1 < p, p < ∞ with 1/p + 1/p = 1. Then (i) if 1 p 2, &

'1/p 1 p p

x + yp + x − yp 2 p p 1/p xp + yp , x, y ∈ Lp (M);

(5.1)

(ii) if 2 p ∞, &

'1/p 1 p p

x + yp + x − yp 2 p p 1/p xp + yp , x, y ∈ Lp (M).

(5.2)

Inequalities (5.1) and (5.2), of course, have their origin in the classical Clarkson inequalities for commutative Lp -spaces. In the non-commutative setting, some partial or particular

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cases of (5.1) and (5.2) were obtained in [47,128] (see also [49] for additional references). (5.1) and (5.2), as stated above were proved in [78] and [106] (see also [191] and [64]). P ROOF OF T HEOREM 5.1. The proof is almost obvious via the complex interpolation. Indeed, (5.1) (resp. (5.2)) is trivially true for p = 1, 2 (resp. p = 2, ∞). Then the complex interpolation yields (5.1) and (5.2). We also note that (5.1) and (5.2) are dual to each other. Let δX (resp. ρX ) denote the modulus of convexity (resp. smoothness) of a Banach space X. Theorem 5.1 implies the following C OROLLARY 5.2. Let 1 0;

1 p t , p

t > 0.

and ρLp (M) (t)

(ii) for 2 < p < ∞ δLp (M) (ε)

1 p ε , p2p

0 < ε < 2,

and ρLp (M) (t)

The reader can ﬁnd some applications of the uniform convexity of Lp (M), e.g., in [107,108]. Let us comment on the estimate for the modulus of convexity given by Corollary 5.2 (the same comment, of course, applies to the modulus of smoothness as well). This estimate is best possible only in the case of 2 < p < ∞. We should also point out that in this case the relevant constant 1/(p2p ) is optimal (for it is already so in the commutative case; see [115], p. 63). Keeping in mind the well-known result on the modulus of convexity of commutative Lp -spaces, one would expect that the order of δLp (M) (ε) for 1 < p < 2 be O(ε2 ). This is indeed the case (cf. [193]). In fact, we have a more precise result as follows. T HEOREM 5.3. Let 1 < p < ∞. Then (i) for 1 < p 2

1/2 x2p + (p − 1)y2p '1/p & 1 p p

x + yp + x − yp , 2

∀x, y ∈ Lp (M);

(5.3)

(ii) for 2 < p < ∞, &

'1/p 1 p p

x + yp + x − yp 2

1/2 x2p + (p − 1)y2p ,

∀x, y ∈ Lp (M).

Moreover, the constant p − 1 is optimal in both (5.3) and (5.4).

(5.4)

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This theorem was proved in [14] for Schatten classes. As pointed out by the authors, the arguments there work for semiﬁnite von Neumann algebras as well. Then the general case follows by Theorem 3.6. We should emphasize that the optimality of the constant p − 1 in (5.3) and (5.4) has important applications to hypercontractivity. We will later illustrate this by discussing the Fermionic hypercontractivity. Note that if one does not care about the best constants, one can deduce Theorem 5.3 from the optimal order of δLp (M) (ε) and ρLp (M) (t) obtained in [193] (at least, for Schatten classes). Note also that (5.3) and (5.4) are equivalent by duality. We will include a very simple proof of (5.4) for p = 2n (n ∈ N), and so by interpolation for all 2 0 p−1 2 ε , 8 p−1 2 t , ρLp (M) (t) 2 δLp (M) (ε)

1 < p 2,

and

2 p < ∞.

R EMARK . The constants (p − 1)/8 and (p − 1)/2 in the above estimates are optimal (see [115], p. 63 for the commutative case). Corollaries 5.2 and 5.4 yield the type and cotype of Lp (M) for 1 < p < ∞. C OROLLARY 5.5. Let 1 < p < ∞. Then Lp (M) is of type min(2, p) and cotype max(2, p). The type and cotype of Lp (M) were determined in [193] for Schatten classes, and in [63] for the general case. We will see later that Lp (M) is of cotype 2 for 0 < p 1. Now we turn to the application of the optimality of the constant p − 1 in (5.3) and (5.4) to the Fermionic hypercontractivity. Before starting our discussion, we should point out, however, that in the scalar case (i.e., in the case where M = C) Theorem 5.3 is exactly Nelson’s celebrated hypercontractivity inequality for the two point space (cf. [20] and [130]). This two point hypercontractivity inequality easily yields the optimal hypercontractivity for the classical Ornstein–Uhlenbeck semigroup. Carlen and Lieb used Theorem 5.3 (in the case of Schatten classes) to obtain the optimal Fermionic hypercontractivity, thus solving a problem left open since Gross’ pioneer works in the domain (cf. [75]). Let R be the hyperﬁnite II1 factor. We recall that R is generated by a sequence (εn )n1 of self-adjoint unitaries satisfying (CAR) (see Section 1). We also recall that {wA : A ⊂ N, A ﬁnite} is an orthonormal basis of L2 (R). We deﬁne the number operator N by NwA = |A|wA (|A| denoting the cardinality of A). N is an unbounded positive self-adjoint operator on L2 (R). It generates the Fermionic Ornstein–Uhlenbeck semigroup Pt : Pt = e−t N , t 0. One can show that Pt is a contraction on Lp (R) for all 1 p ∞. The optimal Fermionic hypercontractivity is contained in the following

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T HEOREM 5.6. Let 1 < p < q < ∞. Then Pt is a contraction from Lp (R) to Lq (R) iff e−2t (p − 1)/(q − 1). Let us brieﬂy comment on the proof of Theorem 5.6. First, since linear combinations of the wA ’s are dense in Lp (R), it sufﬁces to prove Theorem 5.6 in the ﬁnite-dimensional case, that is, when Pt is restricted to the Lp -spaces based on the von Neumann algebra generated by {ε1 , . . . , εn } (n ∈ N). Second, by standard arguments as for the classical Ornstein–Uhlenbeck semigroup, one can reduce Theorem 5.6 to the special case where 2 = p < q. Assuming these reductions, one can use (5.4) to prove Theorem 5.6 by induction on n (noting that the case n = 1 corresponds to Nelson’s two point hypercontractivity). We refer to [35] for the details. R EMARKS . (i) Theorem 5.6 implies, and in fact, is equivalent to the optimal Fermionic logarithmic Sobolev inequality, see [76] and [35]. (ii) Biane [18] obtained the analogue of Theorem 5.6 for the free Ornstein–Uhlenbeck semigroup. Note that this latter semigroup is also ultracontractive (cf. [23,24]). We end the discussion on the uniform convexity and smoothness by providing a simple proof for Theorem 5.3 (except for the best constant). We need only to consider (5.4). We are going to show that for 2 p < ∞ there is a constant Cp , depending only on p, such that &

'1/p 1 p p

x + yp + x − yp 2

1/2 x2p + Cp y2p , ∀x, y ∈ Lp (M).

(5.4p )

To that end, by Theorem 3.6, we can assume that M is semiﬁnite and equipped with a faithful normal semiﬁnite trace τ . The key step in the proof of (5.4p ) is the implication “(5.4p ) ⇒ (5.42p )”. Let us show this. Assume (5.4p ). Let x, y ∈ L2p (M), and set a = x ∗ x + y ∗ y, b = x ∗ y + y ∗ x. Then a, b ∈ Lp (M) and 1 1 2p 2p

p p

x + y2p + x − y2p = a + bp + a − bp 2 2

p/2

by (5.4p ) a2p + Cp b2p

2 p/2

x22p + y22p + 4Cp x22p y22p

p x22p + (2Cp + 1)y22p ; whence (5.42p ) with C2p 2Cp + 1. Therefore, starting with the trivial case p = 2 (noting that C2 = 1), and by iteration, we get C2n 2n − 1 (in fact, C2n = 2n − 1). Thus for these special values of p we obtain the best constant in (5.4). Then for any other value of p, say, 2n < p < 2n+1 , by complex interpolation, we deduce (5.4p ) with

1−θ n+1

θ 2 −1 , Cp 2n − 1 where 1/p = (1 − θ )/2n + θ/2n+1 .

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Now we pass on to the uniform PL-convexity and Hardy convexity of non-commutative Lp -spaces. This time, we admit quasi-normed spaces (so p < 1 is allowed). Let X be a (complex) quasi-Banach space. Let T be the unit circle equipped with normalized Lebesgue measure. For 0 < p < ∞ we denote by Lp (T, X) the usual Lp -space of Bochner measurable functions with values in X. Note that Lp (T, Lp (M)) is just the noncommutative Lp -space based on the von Neumann algebra tensor product L∞ (T) ⊗ M. Let P(X) denote the family of all complex polynomials with coefﬁcients in X: P(X) =

n

xk z : xk ∈ X, 0 k n, n ∈ N . k

k=0

D EFINITION . Let X be a quasi-Banach space. Let 0 0. We deﬁne HX (ε) = inf x + zyL1 (T,X) − 1: x = 1, y ε, x, y ∈ X and p hX (ε) = inf f Lp (T,X) − 1: f (0) = 1, f − f (0)Lp (T,X) ε, f ∈ P(X) . p

X is said to be uniformly PL-convex (resp. H p -convex) if HX (ε) > 0 (resp. hX (ε) > 0) for p all ε > 0. HX (ε) (resp. hX (ε)) is called the modulus of PL-convexity (resp. H p -convexity) of X. The uniform PL-convexity was introduced and studied in [45]. It was shown there that in the deﬁnition of HX (ε) above, if the L1 -norm is replaced by an Lp -norm, then the resulting modulus is equivalent to HX (ε). The uniform H p -convexity was explicitly introduced in [199]; however, it is already implicit in [80]. It was proved in [202] that if X is uniformly H p -convex for one p ∈ (0, ∞), then so is it for all p ∈ (0, ∞). Thus we say that X is uniformly H-convex if it is uniformly H p -convex for some p. The uniform PLconvexity (resp. H-convexity) is closely related to inequalities satisﬁed by analytic (resp. Hardy) martingales with values in X. The Enﬂo–Pisier renorming theorem about the uniform (real) convexity admits analogues for these uniform complex convexities. We refer to [45,199,201,202] and [149] for more information. R EMARKS . (i) For any given 0 0 such that for all quasiBanach spaces X p

HX (ε) αp hX (αp ε),

0 < ε 1.

Consequently, the uniform H-convexity implies the uniform PL-convexity. (ii) If a Banach space X is uniformly convex, it is uniformly H-convex.

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T HEOREM 5.7. Assume 0 < p, q < ∞. Let M be a von Neumann algebra. Then p

hLq (M) (ε) αεr ,

0 < ε 1,

where r = max(2, p, q) and α > 0 is a constant depending only on p, q. R EMARKS . (i) In the case q > 1, Theorem 5.7 easily follows from Corollaries 5.2 and 5.4. Thus the non-trivial part of Theorem 5.7 lies in the case q 1. (ii) Theorem 5.7 implies, of course, that the same estimate holds for the modulus of PL-convexity. (iii) In the case of q = 1, Theorem 5.7 is contained in [80]. In fact, it is this result which motivated the introduction of the uniform H-convexity. Theorem 5.7, as stated above, was proved in [201]. The ingredient of the proof is the Riesz type factorization for Hardy spaces of analytic functions with values in noncommutative Lp -spaces. In Section 8 below we will discuss such a factorization in a more general context. The following corollary completes Corollary 5.5. Thus the non-commutative Lp -spaces have the same type and cotype as the commutative Lp -spaces. C OROLLARY 5.8. Lp (M) is of cotype 2 for any 0 < p 1 and any von Neumann algebra M. This corollary was proved in [193] for p = 1 and in [201] for 0 < p < 1. We recall that a quasi-Banach space X has the analytic Radon–Nikodym property (abbreviated as analytic RNP) if any bounded analytic function F : D → X has a.e. radial limits in X, where D denotes the unit disc (cf. [30,59], and also [32] for additional references). It is known that the uniform H-convexity implies the analytic RNP. Thus we get the C OROLLARY 5.9. Lp (M) has the analytic RNP for any 0 < p 1 and any von Neumann algebra M. The results discussed in this section have all been extended to symmetric operator spaces. We refer to [70,195] for the cotype, uniform convexity, PL-convexity and smoothness in the unitary ideals, and in the general case, to [200] for the uniform convexity and smoothness, to [203,204] for the uniform H-convexity, RNP and analytic RNP (see also [129]). Finally, we mention that [39] contains related results, especially those on the local uniform convexity for symmetric operator spaces.

6. Non-commutative Khintchine inequalities This section is devoted to the non-commutative Khintchine inequalities and the closely related Grothendieck-type factorization theorems. Although all results in this section hold

Non-commutative Lp -spaces

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for the general non-commutative Lp -spaces, we will restrict ourselves to the semiﬁnite ones, i.e., those constructed from an n.s.f. trace. Letters Ap , Bp , . . . , will denote positive constants depending only on p. Let (εn )n1 be a Rademacher (or Bernoulli) sequence, i.e., a sequence of independent random variables on some probability space (Ω, F , P ) such that P (εn = 1) = P (εn = −1) = 1/2 for all n 1. We ﬁrst recall the classical Khintchine inequalities. Let 0 < p < ∞. Then for all ﬁnite sequences (an ) of complex numbers A−1 a ε a ε B a ε . (6.1) n n n n p n n p Lp (Ω,P )

n1

L2 (Ω,P )

n1

n1

Lp (Ω,P )

(Note that obviously n1 an εn L2 (Ω,P ) = ( n1 |an |2 )1/2 .) These inequalities remain valid (suitably modiﬁed) when the coefﬁcients an ’s are vectors from a Banach space X. In that case they are due to Kahane, and are usually called “Khintchine–Kahane inequalities”: for all ﬁnite sequences (an ) in X −1 Ap an εn an εn Lp (Ω,P ;X)

n1

L2 (Ω,P ;X)

n1

Bp an εn n1

(6.2)

.

Lp (Ω,P ;X)

In particular, if X is a commutative Lp -space, say X = Lp over (0, 1), (6.2) implies that for all ﬁnite sequences (an ) in Lp (0, 1) 1/2 2 A−1 |a | n p n1

Lp

a ε n n n1

L2 (Ω,P ;Lp )

1/2 2 Bp |an | n1

(6.3)

.

Lp

It is (6.3) that we will extend to the non-commutative setting. Now let M be a semiﬁnite von Neumann algebra equipped with an n.s.f. trace τ . Let a = (an ) be a ﬁnite sequence in Lp (M) (recalling that by our convention, Lp (M) = Lp (M, τ )). Deﬁne 1/2 2 , aLp (M;l 2 ) = |a | n C

n0

p

∗ 2 1/2 . aLp (M;l 2 ) = |a | n R

n0

p

This gives two norms (or quasi-norms if p < 1) on the family of all ﬁnite sequences in Lp (M). The corresponding completions (relative to the w∗ -topology for p = ∞) are denoted by Lp (M; lC2 ) and Lp (M; lR2 ), respectively. The reader is referred to [156] for a discussion of these norms.

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Now we can state the non-commutative Khintchine inequalities. T HEOREM 6.1. Let 1 p < ∞, and let M be a semiﬁnite von Neumann algebra. Let a = (an )n0 be a ﬁnite sequence in Lp (M). (i) If 2 p < ∞, there is a constant Bp (depending only on p) such that εn an aLp (M;l 2 )∩Lp (M;l 2 ) C R n0

Lp (Ω,P ;Lp (M))

Bp aLp (M;l 2 )∩Lp (M;l 2 ) . C

(6.4)

R

(ii) If 1 p < 2, there is an absolute constant A > 0 (independent of p and a) such that AaLp (M;l 2 )+Lp (M;l 2 ) εn an C

R

n0

Lp (Ω,P ;Lp (M))

aLp (M;l 2 )+Lp (M;l 2 ) . C

(6.5)

R

For the convenience of the reader we recall the norms in Lp (M; lC2 ) ∩ Lp (M; lR2 ) and

Lp (M; lC2 ) + Lp (M; lR2 ):

aLp (M;l 2 )∩Lp (M;l 2 ) = max aLp (M;l 2 ) , aLp (M;l 2 ) C

R

C

R

and aLp (M;l 2 )+Lp (M;l 2 ) = inf bLp (M;l 2 ) + cLp (M;l 2 ) , C

R

C

R

where the inﬁmum runs over all decompositions a = b + c with b ∈ Lp (M; lC2 ) and c ∈ Lp (M; lR2 ). This result was ﬁrst proved in [117] for 1 < p < ∞ in the case of the Schatten classes. The general statement as above (including p = 1) is contained in [121]. Modulo the classical fact that in all preceding inequalities the sequence (εn ) can be replaced by a lacunary n sequence, say, by (z2 )n1 on the unit circle T, the main ingredient of the proof in [121] is a Riesz type factorization theorem (see Theorem 8.3 below). R EMARKS . (i) Like in the classical Khintchine inequalities (6.1), the constant Bp in (6.4) √ is of order p (the best possible) as p → ∞ (cf. [154, p. 106]). (ii) We have already mentioned that in Theorem 6.1, the sequence (εn ) can be replaced by a lacunary sequence. It is also classical that (εn ) can be replaced by a sequence of independent standard Gaussian variables. (iii) More generally, Theorem 6.1 holds when (εn ) is replaced by certain sequences in a non-commutative Lp -space Lp (N ) and εn an is replaced by εn ⊗ an in Lp (N ⊗ M), for instance, this holds for the generators of a free group, for a free semi-circular system (in

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Voiculescu’s sense [196]) and for a sequence of CAR operators (as in Section 1). Note that in the free cases, (6.4) even holds for p = ∞! The reader is referred to [81,154] for more information, and also to [27,29] for some related results and for the best constants in these inequalities. (iv) Theorem 6.1 also holds for non-commutative Lp -spaces associated with a general von Neumann algebra (cf. [100,101]). Note that [100,101] contains more inequalities related to (6.4) and (6.5). C OROLLARY 6.2. Let (εij ) be an independent collection (indexed by N × N) of mean zero ±1-valued random variables on (Ω, F , P ). For any 2 p < ∞, there is a constant Cp such that for any ﬁnitely supported function x : N2 → C, we have |||x|||p εij x(i, j )eij Cp |||x|||p , (6.6) Lp (Ω,P ;S p )

i,j

where 2 p/2 1/p 2 p/2 1/p x(i, j ) x(i, j ) , . |||x|||p = max i

j

j

i

(6.7) A fortiori this implies εij x(i, j )eij

Lp (Ω,P ;S p )

Cp

inf

ε(i,j )=±1

ε(i, j )x(i, j )eij p . S

(6.8)

P ROOF. Take Lp (M) = S p . Let aij = x(i, j )eij . Then ( ij aij∗ aij )1/2 = j λj ejj and ( aij aij∗ )1/2 = μi eii where λj = ( i |x(i, j )|2 )1/2 and μi = ( j |x(i, j )|2)1/2 . Thus (6.6) is a special case of (6.4). R EMARK 6.3. The preceding result remains valid with the same proof when 1 p < 2 provided one changes the deﬁnition of |||x|||p to the following one (dual to the other): |||x|||p = inf yp (2 ) + t zp ( ) , 2

where the inﬁmum runs over all possible decompositions of the form x = y + z. R EMARK . [101] contains more inequalities of type (6.6). Here we just mention one of them, which is an extension of (6.6). Let (fij ) be an independent collection of mean zero random variables in Lp (Ω, F , P ) (2 p < ∞). Then 1/p p/2 1/p p 2 e ≈ max f , f , f ij ij p ij p ij 2 p L (Ω,P ;S )

i,j

j

where the equivalence constants depend only on p.

i

i

j

p/2 1/p fij 22 ,

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In the case of 0 < p < 1, it is easy to check that the second inequality of (6.5) still holds. However, this is not clear for the ﬁrst one. P ROBLEM 6.4. Does the ﬁrst inequality of (6.5) hold for 0 < p < 1 (with some constant depending on p)? Does Remark 6.3 extend to p < 1? Like in the commutative setting, the non-commutative Khintchine inequalities are closely related to non-commutative Grothendieck type factorization theorems. Indeed, it was shown in [121] that (6.5) in the case of p = 1 is equivalent to the non-commutative little Grothendieck theorem. To go further, we need one more deﬁnition. D EFINITION . Let 1 p ∞, 0 < q r < ∞. Let Y be a Banach space, and let u : Lp (M) → Y be an operator (M being a semiﬁnite von Neumann algebra). u is said to be (r, q)-concave if there is a constant C such that for all ﬁnite sequences (an ) in Lp (M)

uan r

1/r

q C |an |s

1/q

, p

where |a|s = ((a ∗ a +aa ∗)/2)1/2 denotes the symmetric modulus of an operator a. If q = r, u is simply said to be q-concave. In the case of p = ∞ (then M can be any C ∗ -algebra), the above notion reduces to that of (r, q)-C ∗ -summing operators introduced in [144] and [147]. The following is an easy consequence of the Hahn–Banach theorem (cf. [144] for a proof). P ROPOSITION 6.5. Let M be a semiﬁnite von Neumann algebra. Let 1 q < p ∞ and s = p/q. Then for any operator u : Lp (M) → Y the following assertions are equivalent (i) u is q-concave; (ii) there are a constant C and f ∈ (Ls (M))∗ , f 0, such that q

1/q ua C f |a|s ,

∀a ∈ Lp (M).

The following Grothendieck-type factorization theorem (when Y is a Hilbert space) is equivalent to (6.5) with p in place of p. T HEOREM 6.6. Let 2 < p ∞, and let Y be a Banach space of cotype 2. Then any operator u : Lp (M) → Y is 2-concave, equivalently (via Proposition 6.5), there are a constant C and f ∈ (Lp/2 (M))∗ with f 0 such that

1/2 ua Cu f |a|2s ,

∀a ∈ Lp (M).

Moreover, C can be chosen to depend only on the cotype 2 constant of Y . R EMARK . The basic case p = ∞ (= non-commutative Grothendieck theorem), is proved in [147] (see also [144,146]). In this case, M can actually be any C ∗ -algebra. In the case of

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p < ∞, Theorem 6.6 is essentially the main result in [118]. More generally, [118] proves this for operators u : E(M, τ ) → H , where H is a Hilbert space and E is a 2-convex r.i. space with an additional mild condition. This, together with [121], implies that Theorem 6.1 can be extended to some symmetric operator spaces. The main difﬁculty in [118] is to obtain Theorem 6.6 with a constant C independent of p, or equivalently which remains bounded when p → ∞. If we ignore this important point, it is very easy to deduce Theorem 6.6 from (6.4), as follows. P ROOF OF T HEOREM 6.6 FOR p < ∞ WITH C = Cp . Since Lp (M) is of type 2, by Kwapie´n’s theorem (cf. [146], Theorem 3.2), u factors through a Hilbert space. Thus we may assume Y itself is a Hilbert space. Let (an ) be a ﬁnite sequence in Lp (M). Then u(an )2

1/2

= u(an )εn

L2 (Ω,P ;Y )

u an εn Cp |an |2s

L2 (Ω,P ;Lp (M)) 1/2

p

by (6.4) .

Therefore, u is 2-concave.

Unfortunately, the preceding proof does not work for p = ∞. The main difﬁculty in this case is to show that an operator u from M into a space of cotype 2 factors through a Hilbert space. This was done in [147]. The proof given there relies on another result of independent interest, that we state as follows. T HEOREM 6.7. Let 1 < q < ∞. Let u : A → Y be an operator from a C ∗ -algebra A into a Banach space Y . Then the following assertions are equivalent (i) u is (q, 1)-C ∗ -summing; (ii) there are a constant C and a state f on A such that

1/q a1−1/q , ua Cu f |a|s

∀a ∈ A;

(iii) for any 1 r < q there are a constant C and a state f on A such that

1/q ua Cu f |a|rs a1−r/q ,

∀a ∈ A;

(iv) u is (q, r)-C ∗ -summing for any 1 r < q. Thus Theorem 6.7 gives a characterization of (q, r)-C ∗ -summing operators deﬁned on a C ∗ -algebra. (ii) and (iii) above can be reformulated as a Pietsch-type factorization of u through a non-commutative Lorentz space Lq,1 , constructed from the state f via the real interpolation in the spirit of Kosaki’s construction presented in Section 3. The resulting spaces, denoted by Lq,1 (f ), possess properties similar to the usual Lorentz spaces. The reader is referred to [147] for more information.

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R EMARK . There does not seem to be a known characterization similar to that in Theorem 6.7 for (q, r)-concave operators deﬁned on Lp (M) (p < ∞). Let us close this section by an application of Theorem 6.7. T HEOREM 6.8. Let M be any von Neumann algebra and X ⊂ M∗ a reﬂexive subspace. Then there are a normal state f of M and p > 1 such that X embeds isomorphically into Lp,p (f ), where Lp,p (f ) is the non-commutative Lp -space referred to above. This theorem, proved in [147], is a non-commutative version of a classical theorem due to Rosenthal in the commutative setting. Its proof uses Theorem 6.7 and a previous result in [92] that any reﬂexive subspace of M∗ is superreﬂexive. Note that the real interpolation space Lp,p (f ) can be replaced by the corresponding complex interpolation space. R EMARK . Let A be a C ∗ -algebra, and let T : A → 2 be absolutely summing (in the usual sense). If A is commutative, it is well known that T factors as T = T1 T2 , where T2 : A → 2 is bounded and T1 ∈ S 2 . In [161] it is shown that for a general C ∗ -algebra A, one can get a factorization T = T1 T2 , where T2 : A → 2 is bounded and T1 : 2 → 2 belongs to the Schatten class S4 (the exponent 4 is optimal). A fortiori T is compact. In particular, there is no embedding of 2 into a non-commutative L1 -space with absolutely summing adjoint. See [146, p. 68] for background on embeddings of this kind. 7. Non-commutative martingale inequalities This section deals with non-commutative martingale inequalities. The reader is referred to [71] for the classical (= commutative) martingale inequalities. In what follows, M will be a von Neumann algebra equipped with a normal faithful ﬁnite normalized trace τ . We begin with some necessary deﬁnitions. Let N ⊂ M be a von Neumann subalgebra. The non-commutative Lp -space associated with (N , τ |N ) is naturally identiﬁed with a subspace of Lp (M). There is a unique normal faithful conditional expectation E : M → N preserving the trace τ , i.e., τ (E(x)) = τ (x) for all x ∈ M. For any 1 p ∞, E is extended to a contractive projection from Lp (M) onto Lp (N ), still denoted by E. Now let (Mn )n0 be an increasing sequence of von Neumann subalgebras of M such that the union of all the Mn ’s is w∗ -dense in M. Let En be the conditional expectation from M onto Mn . Then as usual, we deﬁne a non-commutative martingale (with respect to (Mn )n0 ) as a sequence x = (xn )n0 in L1 (M) such that En (xn+1 ) = xn ,

∀n 0.

If additionally all xn ’s are in Lp (M), x is called an Lp -martingale. Then we set xp = sup xn p . n0

If xp < ∞, x is called a bounded Lp -martingale. The difference sequence of x is deﬁned as dx = (dxn )n0 with dx0 = x0 and dxn = xn − xn−1 for all n 1.

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R EMARK . Let x∞ ∈ Lp (M). Set xn = En (x∞ ) for all n 0. Then x = (xn ) is a bounded Lp -martingale and xp = x∞ p ; moreover, xn converges to x∞ in Lp (M) (relative to the w∗ -topology in the case p = ∞). Conversely, if 1 < p < ∞, every bounded Lp -martingale converges in Lp (M), and so is given by some x∞ ∈ Lp (M) as previously. Thus one can identify the space of all bounded Lp -martingales with Lp (M) itself in the case 1 < p < ∞. The main result of [156] can be stated as follows. Recall that Ap , Bp , . . . , denote constants depending only on p. T HEOREM 7.1. Let M and (Mn )n0 be as above. Let 1 < p < ∞, and let x = (xn )n0 be a ﬁnite Lp -martingale with respect to (Mn )n0 . Then A−1 p Sp (x) xp Bp Sp (x),

(7.1)

where for 2 p < ∞, Sp (x) = dxLp (M;l 2 )∩Lp (M;l 2 ) , C

R

and for 1 < p < 2, Sp (x) = inf dyLp (M;l 2 ) + dzLp (M;l 2 ) , C

R

the inﬁmum being taken over all decompositions x = y + z with Lp -martingales y and z. This is the non-commutative Burkholder–Gundy inequalities. Note that in the commutative case, Sp (x) is the Lp -norm of the usual square function of x (so that the above difference between the cases 2 p < ∞ and 1 < p < 2 disappears). The proof of Theorem 7.1 in [156] is rather tortuous, due to the fact that the usual techniques from classical martingale theory, such as maximal functions, stopping times, etc., are no longer available in the non-commutative setting. See [156] and [19] for applications to non-commutative stochastic integrals. For Clifford martingales, some particular cases of Theorem 7.1 also appear in [34]. R EMARK 7.2. The second inequality in (7.1) holds for p = 1 too. This follows from the duality between H1 and BMO, proved in [156]. Like in the commutative case, Theorem 7.1 implies the unconditionality of noncommutative martingale differences. Let us record this explicitly as follows. C OROLLARY 7.3. With the same assumptions as in Theorem 7.1, we have εn dxn Cp dxn , ∀εn = ±1. n0

p

n0

p

(7.2)

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Some rather particular cases of (7.2) also appear in [65,66]. Note that in the case of 2 p < ∞, (7.2) is equivalent to (7.1), modulo the non-commutative Khintchine inequalities. However, in the case of 1 < p < 2, to prove that (7.2) implies (7.1), one needs a noncommutative version of a classical inequality due to Stein. We refer to [156] for more details. R EMARK . If p is an even integer, the second inequality of (7.1) was extended in [155] to sequences more general than martingale difference sequences (the so-called p-orthogonal sequences); moreover, for these values of p, the method of [155] yields that the order of the constant Bp in (7.1) is O(p) (for even integers p), which is optimal as p → ∞. For the convenience of the reader, we recall the optimal order of the constants Ap and Bp in the commutative case (cf., e.g., [31]): Bp is bounded as p → 1 and O(p) as p → ∞; Ap is O((p − 1)−1 ) as p → 1 and O(p1/2 ) as p → ∞. The constants Ap and Bp in (7.1) obtained in [156] are not satisfactory at all (they are of exponential type as p → ∞). Thus ﬁnding the optimal order of Ap and Bp in Theorem 7.1 seemed a very interesting question. Very recently, major progress on this was achieved by Randrianantoanina [165], as follows. T HEOREM 7.4 ([165]). There is a constant C such that for any ﬁnite non-commutative martingale x in L1 (M) and any sequence (εn ) of signs ε dx C dx n n n . n0

1,∞

n0

1

By interpolation, this implies the optimal order of the constant Cp in (7.2), namely, Cp = O(p) as p → ∞. This, in turn, combined with Theorem 6.1, yields better estimates for Ap , Bp in (7.1), namely Ap is O((p − 1)−2 ) when p → 1 and both Ap and Bp are O(p) when p → ∞ (which for Bp is optimal). It was also shown in [102] that O(p) is the optimal order of Ap as p → ∞. Note that this order is the square of what it is in the commutative case. On the other hand, it was proved in [100] that Bp remains bounded as p → 1. We will now discuss two other inequalities: the Burkholder and Doob inequalities. T HEOREM 7.5. With the same assumptions as in Theorem 7.1, we have A−1 p sp (x) xp Bp sp (x),

(7.3)

where for 2 p < ∞, 1/p

1/2 p 2 , sp (x) = max |dx dxn p , E | n−1 n n0

p

n0

∗ 2 1/2 dx E n−1 n n0

p

Non-commutative Lp -spaces

1493

and for 1 < p < 2, 1/p

1/2 p 2 sp (x) = inf dwn p + En−1 |dun | n0

p

n0

∗ 2 1/2 , + En−1 dvn p

n0

where the inﬁmum runs over all decompositions x = w + u + v with Lp -martingales w, u and v. This theorem comes from [100]. It is the non-commutative analogue of the classical Burkholder inequality. Note that in the commutative case ( En−1 (|dxn |2 ))1/2 is the conditioned square function of x. Like in the commutative case, Theorem 7.5 implies a noncommutative analogue of Rosenthal’s inequality concerning independent mean zero random variables; see [100,101] for more details and some applications. T HEOREM 7.6 ([97]). Let M and (Mn ) be as in Theorem 7.1. Let 1 p < ∞. Let (an ) be a ﬁnite sequence of positive elements in Lp (M). Then E (a ) C an (7.4) n n p . n0

p

n0

p

Note that in the commutative case, (7.4) is the dual reformulation of Doob’s classical maximal inequality. Although it is clearly impossible to deﬁne the maximal function of a non-commutative martingale as in the commutative setting, Junge found in [97] a substitute, consistent with [154], which enables him to formulate a non-commutative analogue of Doob’s inequality itself, which is dual to (7.4). Note that the latter result immediately implies the almost everywhere convergence of bounded non-commutative martingales in Lp (M) for all p > 1. Results of this kind on the almost everywhere convergence of noncommutative martingales go back to Cuculescu [43]. The reader is referred to [44] and [90,91] for more information. R EMARKS . (i) Like the constants in (7.1) the constants in (7.3) and (7.4) obtained in [100, 97] are not satisfactory at all. In fact, they depend on those in (7.1) since the proofs of (7.3) and (7.4) in [97] and [100] use (7.1). The more recent results of [165] imply better estimates for these constants. (ii) It was proved in [102] that the optimal order of the constant Cp in (7.4) is O(p2 ) as p → ∞. This is in strong contrast with the commutative case for, in the commutative case, the optimal order of the corresponding constant is O(p) as p → ∞. The same phenomenon occurs for the optimal order of the best constant in the non-commutative Stein inequality proved in [156], namely, this optimal order is O(p) as p → ∞; again it is the square of what it is in the commutative case. We refer to [102] for more information. (iii) All the preceding results hold in the non-tracial case as well (cf. [100,101,97]).

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In the rest of this section, we brieﬂy discuss the UMD property and the analytic UMD property of non-commutative Lp -spaces, a subject closely related to inequality (7.2). Applying Corollary 7.3 to commutative martingales with values in Lp (M), 1 < p < ∞, we get the unconditionality of commutative martingale differences with values in Lp (M), that is, Lp (M) is a UMD space in Burkholder’s sense (cf. [32] for information on UMD spaces). This is a well-known fact, proved in [21] and [16]. In fact, these authors proved that the Hilbert transform extends to a bounded map on Lp (T; Lp (M)) for any 1 < p < ∞; but this property (called “HT” in short) is equivalent to UMD. We also refer to the next section for discussions on Hilbert type transforms. Together with Theorem 3.6 we obtain the C OROLLARY 7.7. Lp (M) is a UMD space for any 1 < p < ∞ and any von Neumann algebra M. We mention an open problem circulated in the non-commutative world for almost two decades on the UMD property for symmetric operator spaces. P ROBLEM 7.8. Let M be a semiﬁnite von Neumann algebra equipped with an n.s.f. trace τ , and let E be a UMD r.i. space on (0, ∞). Is E(M, τ ) a UMD space? We now turn to the analytic UMD property. Let TN be the inﬁnite torus equipped with the product measure, denoted by dm∞ . Let Ωn be the σ -ﬁeld generated by the coordinates (z0 , . . . , zn ), n 0. Let X be a quasi-Banach space. By a Hardy martingale in Lp (TN ; X) (0 < p ∞), we mean any sequence f = (fn ) satisfying the following: for any n 0, fn ∈ Lp (TN , Ωn ; X) and fn is analytic in the last variable zn , i.e., fn admits an expansion as follows ϕn,k (z0 , . . . , zn−1 )znk , fn (z0 , . . . , zn−1 , zn ) = k1

where ϕn,k ∈ Lp (TN , Ωn−1 ; X) for n 0, k 0. If in addition, ϕn,k = 0 for all k 2, f is called an analytic martingale. Note that if X is a Banach space and 1 p ∞, any Hardy martingale in Lp (TN ; X) is a martingale in the usual sense. D EFINITION . X is called an analytic UMD space if for some 0 < p < ∞ (or equivalently for all 0 < p < ∞) there is a constant C such that all ﬁnite Hardy martingales f in Lp (TN ; X) satisfy ε df C df n n n , ∀εn = ±1. n0

p

n0

p

This notion was introduced in [69]. The apparent weakening obtained by requiring the above inequality be veriﬁed only for analytic martingales, is actually an equivalent deﬁnition of analytic UMD spaces (cf. [69]). Typical examples of Banach spaces which are analytic UMD but not UMD are commutative L1 -spaces. In fact, all commutative

Non-commutative Lp -spaces

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Lp -spaces, 0 < p 1, are analytic UMD spaces. We refer to [69] for more information (see also [32]). However, this no longer holds in the non-commutative setting: P ROPOSITION 7.9. Let M be a von Neumann algebra and 0 < p 1. Then Lp (M) is an analytic UMD space iff M is isomorphic, as Banach space, to a commutative L∞ -space. It was proved in [80] that the trace class S 1 is not an analytic UMD space. The ingredient of the proof there is the unboundedness of the triangular projection on S 1 (cf. [111]). (This projection is in fact a non-commutative Riesz projection in the context of the next section.) The same idea also shows that S p is not an analytic UMD space for 0 < p < 1. Noting that the analytic UMD property is “local”, we then deduce the general case from Theorem 3.5.

8. Non-commutative Hardy spaces A classical theorem of Szegö says that if w is a positive function on the unit circle T such that log w ∈ L1 (T), there is an outer function ϕ such that |ϕ| = w a.e. on T. A lot of effort has been made to extend this theorem to operator valued functions, not only for its intrinsic interest, but also because it is the gateway to many useful applications (cf., e.g., [85,86, 46,179,198]). This problem makes sense in the broader context of subdiagonal algebras, introduced by Arveson in the 60’s in order to unify several frequently used non-selfadjoint algebras such as triangular matrices and bounded analytic operator valued functions. In this section we will present the extension to this general context of some classical results for analytic functions in the unit disc, including Szegö’s theorem, boundedness of the Hilbert transform and the Riesz factorization theorem. Throughout this section, unless explicitly indicated otherwise, M will denote a ﬁnite von Neumann algebra equipped with a normal faithful ﬁnite normalized trace τ . Let D be a von Neumann subalgebra of M. Let E be the (unique) normal faithful conditional expectation of M with respect to D which leaves τ invariant. D EFINITION . A w∗ -closed subalgebra H ∞ (M) of M is called a ﬁnite subdiagonal algebra of M with respect to E (or to D) if (i) {x + y ∗ : x, y ∈ H ∞ (M)} is w∗ -dense in M; (ii) E(xy) = E(x)E(y), ∀x, y ∈ H ∞ (M); (iii) {x: x, x ∗ ∈ H ∞ (M)} = D. D is then called the diagonal of H ∞ (M). This notion can be generalized further (see [12]). However, the theory we will give below is, on one hand, satisfactory only for ﬁnite subdiagonal algebras as above, and on the other, interesting enough to cover many important cases. R EMARKS . (i) If H ∞ (M) is a ﬁnite subdiagonal algebra of M, it is automatically maximal in the sense that it is contained in no proper subdiagonal algebra with respect to E other than itself (see [61]).

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(ii) Consequently, H ∞ (M) admits the following useful characterization (cf. [12]) H ∞ (M) = x ∈ M: τ (xy) = 0, ∀y ∈ H0∞ (M) , where H0∞ (M) = x ∈ H ∞ (M): E(x) = 0 . Here are some examples (see [12] for more). (i) Triangular matrices. Let Mn be the full algebra of all complex n × n matrices equipped with the normalized trace. Let Tn be the algebra of all upper triangular matrices in Mn . Then Tn is a ﬁnite subdiagonal algebra of Mn . In this case, the theory we will give below is partly contained in [73]. (ii) Nest algebras. Let P be a totally ordered family of projections in M containing 0 and 1. Let N (P) = {x ∈ M: xe = exe, ∀e ∈ P}. Then N (P) is a ﬁnite subdiagonal algebra of M. The above example on triangular matrices is a special case of nest algebras. (iii) Analytic operator valued functions. Let (M, τ ) be a ﬁnite von Neumann algebra. Let (L∞ (T), dm) ⊗ (M, τ ) be the von Neumann algebra tensor product (recalling that T is the unit circle equipped with normalized Lebesgue measure dm). Let H ∞ (T, M) be the subalgebra of (L∞ (T), dm) ⊗ (M, τ ) consisting of all functions f such that

τ xf (z) z¯ n dm(z) = 0,

∀x ∈ L1 (M), ∀n ∈ Z, n < 0.

Then H ∞ (T, M) is a ﬁnite subdiagonal algebra of (L∞ (T), dm) ⊗ (M, τ ). This is the algebra of “analytic” functions with values in M. More precisely, each element f in H ∞ (T, M) can be extended, using Poisson integrals, to an M-valued function, analytic and bounded in the unit disc admitting f as its (radial or non-tangential) weak-∗ boundary values. The particularly interesting case H ∞ (T, Mn ) or H ∞ (T, B(l2 )) was extensively studied (cf., e.g., [85,46]). Note that B(l2 ) does not ﬁt into our setting; however, for almost all problems we are concerned with, it can be recovered from Mn by approximation. In the remainder of this section, unless speciﬁed otherwise, H ∞ (M) will denote a ﬁnite subdiagonal algebra of M with diagonal D. For 0 < p < ∞ the corresponding Hardy space H p (M) is deﬁned as the closure of H ∞ (M) in Lp (M). Many results on the classical Hardy spaces in the unit disc have been extended to the present setting. We refer, for instance, to [13,93,127,123,124,126,159,173,175] and [177] for more information and references. We now give some of these extensions. The ﬁrst one is the Szegö type theorem. T HEOREM 8.1. Suppose w ∈ M and w−1 ∈ L2 (M). Then there are a unitary u ∈ M and ϕ ∈ H ∞ (M) with ϕ −1 ∈ H 2 (M) such that w = uϕ.

Non-commutative Lp -spaces

1497

This theorem, proved by Saito [175], improves a previous factorization theorem due to Arveson [12], in which both w and w−1 are supposed to belong to M. Saito’s proof essentially follows the same fashion set out by Arveson, although some extra technical difﬁculties appear. R EMARKS . (i) The above theorem can be still improved as follows: let 0 < p, q ∞ and w ∈ Lp (M) with w−1 ∈ Lq (M). Then there are a unitary u ∈ M and ϕ ∈ H p (M) with ϕ −1 ∈ H q (M) such that w = uϕ. (ii) In the classical case of analytic functions in the unit disc, for a positive function w on T, the condition log w ∈ L1 (T) is necessary and sufﬁcient for the existence of a factorization w = uϕ, with u ∈ L∞ (T) unimodular and ϕ an outer function. It is an open problem to extend this to the non-commutative setting. Some partial results can be found in [46,86] and [198]. The following is an immediate consequence of Theorem 8.1 (and also of the remark (i) above). C OROLLARY 8.2. Let w ∈ L1 (M) such that w 0 and w−1 ∈ Lp (M) for some 0 < p ∞. Then there is ϕ ∈ H 2 (M) such that w = ϕ ∗ ϕ. By a rather standard argument, one can deduce from Theorem 8.1 the following Riesz factorization theorem, which was proved in [124] (see also [177] for the case where p = q = 2). T HEOREM 8.3. Let 1 p, q, r ∞ with 1/r = 1/p + 1/q. Then any x ∈ H r (M) can be factored as x = yz with y ∈ H p (M) and z ∈ H q (M); moreover, xr = inf yp zq : x = yz, y ∈ H p (M), z ∈ H q (M) . R EMARKS . (i) It seems unclear whether the inﬁmum above is attained. (ii) With the notations in Theorem 8.3, one has the following more precise statement: for any ε > 0 there are y ∈ H p (M) and z ∈ H q (M) such that x = yz and

r/p , μt (y) μt (x) + ε

r/q μt (z) μt (x) + ε ,

∀t > 0.

In particular, if x ∈ H ∞ (M), then y, z ∈ H ∞ (M) and yp zq = xr + o(1) as ε → 0. This allows to partially extend Theorem 8.3 to the case of indices less than 1 (at least, for elements x ∈ H ∞ (M) ⊂ H r (M)). However, it is unknown whether Theorem 8.3, in its full generality, still holds for indices less than 1. The reader can ﬁnd applications of Theorem 8.3 to Hankel operators in [179,123], to invariant subspaces of Lp (M) in [175], and to the uniform H-convexity in [201] and [203].

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We will now describe the Hilbert transform and Riesz projection. Let x ∈ {a + b∗ : a, b ∈ It is easy to see that x admits a unique decomposition

H ∞ (M)}.

x = a + d + b∗,

with a, b ∈ H0∞ (M), d ∈ D.

Then we deﬁne the Hilbert transform H by

H x = −i a − b∗ . Clearly, x + iH x ∈ H ∞ (M); moreover if x is self-adjoint, H x is the unique self-adjoint element in {a + b∗ : a, b ∈ H ∞ (M)} such that x + iH x ∈ H ∞ (M) and E(H x) = 0. Note that

⊥ L2 (M) = H02 (M) ⊕ L2 (D) ⊕ H 2 (M) , where H02 (M) = {x ∈ H 2 (M): E(x) = 0}. One easily checks that H02 (M) (resp. (H 2 (M))⊥ ) is the closure of H0∞ (M) (resp. {x ∗ : x ∈ H0∞ (M)}) in L2 (M). This decomposition of L2 (M) shows that H extends to a contraction on L2 (M), still denoted by H . Now let P be the orthogonal projection of L2 (M) onto H 2 (M) (i.e., P is the “Riesz projection”). Like in the classical case, H and P are linked together as follows 1 1 P = (idL2 (M) + H ) + E. 2 2 Thus, as far as boundedness is concerned, it sufﬁces to consider one of them. T HEOREM 8.4. (i) H extends to a bounded map on Lp (M) for any 1 < p < ∞; more precisely, one has H xp Cp xp ,

∀x = a + b∗ , a, b ∈ H ∞ (M),

where Cp Cp2 /(p − 1) with C a universal constant. (ii) H also extends to a bounded map from L1 (M) into L1,∞ (M) (the non-commutative weak L1 -space). This result was proved in [160]. Of course, (i) above (for the case 1 < p < 2) follows by interpolation from (ii) and the L2 -boundedness of H (and by duality for the case 2 < p < ∞). However, (i) admits a much simpler separate proof (see the discussion below). In the case of triangular matrices, (i) above is often referred to as Matsaev’s theorem (cf. [73]). In this case, the corresponding Riesz projection is the usual triangular projection (see [111] for more results on this projection; see also [208] for related results). Let us discuss another particularly interesting case, that of analytic operator valued functions. Then Theorem 8.4(i) is equivalent to the UMD property of Lp (M) that we already

Non-commutative Lp -spaces

1499

saw in the last section. Indeed, considering the ﬁnite subdiagonal algebra H ∞ (T, M) of L∞ (T) ⊗ M, one sees that H = H ⊗ idL2 (M) , where H is the usual Hilbert transform on T. Thus the boundedness of H on Lp (L∞ (T) ⊗ M) is equivalent to the fact that H ⊗ idL2 (M) extends to a bounded map on Lp (T; Lp (M)) (noting that Lp (L∞ (T) ⊗ M) = Lp (T; Lp (M))). In other words, Lp (M) has the “HT property”, which is equivalent to the UMD property, as already mentioned in the last section. The main idea of the proof is an old trick due to Cotlar, which still works in the general setting as in Theorem 8.4. The ingredient is the following formula, whose proof is straightforward. L EMMA 8.5. For any x = a + b∗ with a, b ∈ H0∞ (M)

(H x)∗ H x = x ∗ x + H x ∗ H x + (H x)∗ x . Using Lemma 8.5, we easily check that the boundedness of H on Lp (M) implies that on = 2 and iterating, n we deduce that H is bounded on L2 (M) for all integers n 1. Finally, interpolation and duality yield Theorem 8.4(i). We also point out that this argument gives the optimal order of the constant Cp as stated in Theorem 8.4. L2p (M) (see also the proof of Theorem 5.3 above). Then starting from p

R EMARKS . (i) It was shown in [150] that in the case of triangular matrices or analytic operator valued functions, the non-commutative Hardy spaces form an interpolation scale with respect to the real and complex methods. The same arguments work in the general case as well. Thus for any 0 < p0 , p1 ∞ and 0 < θ < 1 p

H 0 (M), H p1 (M) θ,p = H p0 (M), H p1 (M) θ = H p (M), where 1/p = (1 − θ )/p0 + θ/p1 . (ii) The Hilbert transform H enables us to identify the dual of H 1 (M) with the noncommutative analog of the space BMO (for bounded mean oscillation) as in Fefferman’s classical result, namely the space BMO(M) deﬁned as follows: BMO(M) = x + Hy: x, y ∈ L∞ (M) equipped with the norm z = inf x∞ + y∞ : z = x + Hy, x, y ∈ L∞ (M) . We refer to [124,125] for more information. We end this section by an open problem. A famous theorem due to Bourgain states that the quotient space L1 (T)/H 1 (T) is a GT space of cotype 2 (cf., e.g., [146]). It is not clear at all how to extend this theorem to the non-commutative case. P ROBLEM 8.6. Let H ∞ (M) be a ﬁnite subdiagonal algebra in M. Is L1 (M)/H 1(M) of cotype 2? or merely of ﬁnite cotype?

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In the case of triangular matrices or vector-valued analytic functions, this problem has been circulated in Banach space theory almost since Bourgain’s theorem. By the way, note that any quotient of L1 (M) by a reﬂexive subspace is of cotype 2. This is the non-commutative version of a theorem due to Kisliakov and Pisier (cf. [146]). It follows from [92] and [145].

9. Hankel operators and Schur multipliers In general it is not so easy to compute (up to equivalence) the S p -norm of an operator x in S p , except when x = (xij ) is a column (or row) matrix and when it is a diagonal one, as follows: xi ei1

xi e1i =

xi eii

=

Sp

Sp

=

|xi |2

1/2

and

Sp

|xi |p

1/p

.

In view of their importance and ubiquity in Analysis, it was natural to wonder about the case when x = (xij ) is a Hankel matrix, i.e., there is a (complex) sequence γ in 2 such that xij = γi+j ,

∀i, j 0.

(9.1)

This case was solved in Peller’s remarkable paper [136] as follows. T HEOREM 9.1. Let x = (xij ) be given by (9.1) and let 1 < p < ∞. Let ϕ(z) = be “its symbol” and let Δ0 ϕ(z) = x00,

Δn ϕ(z) =

z j γj ,

j 0 z

jγ

j

∀n 1 (z ∈ T).

2n−1 j <2n

p Then x ∈ S p iff n 2n Δn ϕp < ∞ (here · p denotes the Lp -norm on the unit circle with normalized Lebesgue measure). Moreover, xS p is equivalent to

2

n

p Δn ϕp

1/p .

(9.2)

n0

Actually, Peller also solved the cases p = 1 and 0 < p < 1 but the solution is then a bit more complicated to state (cf. [136] and [138]). The case 0 < p < 1 was obtained independently by Semmes (see [139] for this and for additional references). More generally,

Non-commutative Lp -spaces

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Peller [137] proved an extension of this result for Hankel matrices x = (xij ) with entries xij = γi+j in S p (H ). In that case, Peller proved that xS p (2 (H )) is equivalent to

2

n

p Δn ϕLp (S p (H ))

1/p .

(9.3)

n0

Unfortunately, while (9.2) is very easy to use, in general the norm of Δn ϕ in Lp (S p ) p seems as juntractable as that of x in S (2 (H )). However, when the spectrum of the symbol ϕ = z γj is restricted to be in a suitably “thin” set of integers Λ (meaning that the Fourier coefﬁcients γj ∈ S p are zero when j ∈ / Λ), then (9.3) can effectively be used, as shown in the recent paper [83] (see (9.10) below). To explain this, we will work in a (possibly non-commutative) discrete group Γ but the case of Γ = Z is the most interesting one. D EFINITION 9.2. Let p 2 be an integer and let Λ ⊂ Γ be a subset. Let ε = +1 if p is even and ε = −1 otherwise. (i) Λ is called a B(p)-set if whenever two p-tuples (si ) and (ti ) in Λ satisfy s1 t1−1 s2 t2−1 · · · sp tp−1 = e we have necessarily {s1 , . . . , sp } = {t1 , . . . , tp } with multiplicity (meaning that if an element is repeated, it appears the same number of times in both sets). (ii) For any t in Γ , we denote by Rp (t, Λ) the number of p-tuples t1 , t2 , . . . , tp in Γ with ti = tj for all i = j such that t1−1 t2 t3−1 · · · tpε = t; moreover we let Z(p, Λ) = sup{Rp (t, Λ) | t = e}. We say that Λ is a Z(p)-set if Z(p, Λ) < ∞. The above is inspired by Zygmund’s study of the sets (called here Z(2)-sets) Λ ⊂ Zd such that Z(2, Λ) = supt =0 card{(n, m) ∈ Λ2 | n − m = t} < ∞. As observed by Zygmund, the ﬁnite subsets ΛN = {(n, m) ∈ Z2 | n2 + m2 = N} are uniformly Z(2)-sets, more precisely we have sup Z(2, ΛN ) 2.

(9.4)

N1

√ Actually, the same is true if, replacing Z2 by R2 , we consider the circle of radius N instead of ΛN . A mere look at the picture of such a circle then establishes (9.4). The paper [83] also shows that (generic) random subsets of [1, . . . , N] with cardinality N 1/2 are Z(2)-sets with constants uniformly bounded over N . On the other hand, as pointed out in [83], B(p)-sets are a fortiori Z(p)-sets and this provides examples of a different kind: for instance free sets as well as any subset Λ ⊂ Γ which does not satisfy any non-trivial relation of length 2p. More generally, the generators of the free Abelian groups such as Zd or ZN are B(p)-sets. On the other hand, because of torsion, the Rademacher functions (= coordinates on {−1, 1}N ), identiﬁed to a subset R ⊂ {−1, 1}(N), do not form a B(p)-set, but it is easy to see that they form a Z(p)-set for any p 2 (with Z(p, R) = p!).

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Thus, the following result in [83] can be viewed as an extension of Theorem 6.1 (= the non-commutative Khintchine inequalities). Recall that vN(Γ ) is equipped with its normalized trace τΓ (see Section 1). T HEOREM 9.3. Let Γ be any discrete group. Let p 4 be an even integer and let Λ = {tn | n 0} ⊂ Γ be a Z(p/2)-subset. Then there is a constant C such that for any semiﬁnite M and for any ﬁnite sequence a = (an )n0 in Lp (M) we have aLp (M;l 2 )∩Lp (M;l 2 ) λ(tn ) ⊗ an C R

Lp (vN(Γ )⊗M)

n0

CaLp (M;l 2 )∩Lp (M;l 2 ) . C

(9.5)

R

Moreover, the left side of (9.5) is actually valid for Λ = Γ . R EMARK 9.4. When Γ is commutative and dim(M) = 1, the sets satisfying (9.5) are exactly the Λ(p)-sets in Rudin’s sense (see [171] and [22]). Because of this, the sets satisfying (9.5) when Lp (M) = S p are called Λ(p)cb -sets and are studied in [83]. D EFINITION 9.5. Consider two (commutative) Lp -spaces Lp (μ), Lp (ν) (1 p < ∞) and (closed) subspaces E ⊂ Lp (μ), F ⊂ Lp (ν). A linear mapping u : E → F is called completely bounded (in short c.b.) if there is a constant C such that u(xi )(·) yi

Lp (ν;S p )

C xi (·) yi

Lp (μ;S p )

∀xi ∈ E, ∀yi ∈ S p .

We denote by ucb the smallest C for which this holds. This deﬁnition is coherent with the one used in the theory of operator spaces (cf. [154]). R EMARK 9.6. More generally if Lp (μ), Lp (ν) are non-commutative Lp -spaces associated to semiﬁnite traces μ, ν the preceding deﬁnition still makes sense using Lp (μ ⊗ Tr) and Lp (ν ⊗ Tr) instead of Lp (μ; S p ) and Lp (ν; S p ). C OROLLARY 9.7. Let Λ and p be as in Theorem 9.3 (more generally, what follows is valid for any p > 2 if Λ is assumed Λ(p)cb ). Let (εn ) denote the Rademacher functions on (Ω, F , P ), as in Section 6. Let ER (resp. EΛ ) be the closed subspace of Lp (Ω, F , P ) (resp. Lp (vN(Γ ))) generated by {εn | n 0} (resp. {λ(tn ) | n 0}). Then the linear mappings u and u−1 deﬁned on the linear spans by u(εn ) = λ(tn ) and u−1 (λ(tn )) = εn extend to c.b. maps u : ER → EΛ and u−1 : EΛ → ER with ucb C and u−1 cb Bp . Moreover, the (orthogonal) projection P : Lp (vN(Γ )) → EΛ , deﬁned by P (λ(t)) = λ(t) if t ∈ Λ, and = 0 if t ∈ Λ, is c.b. on Lp (vN(Γ )).

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The preceding results provide non-trivial new examples of c.b. Fourier multipliers on Lp (T). We now turn to Schur multipliers. A linear map T : S p → S p (resp. T : B(2 ) → B(2 )) is called a Schur multiplier if it is of the form

T (x) = ϕ(i, j )xij for some function ϕ : N × N → C. In this case, we write T = Mϕ . The case p = 2 is of course trivial: we have then Mϕ = supi,j |ϕ(i, j )|. In the case p = ∞, it is well known (due essentially to Grothendieck) that bounded Schur multipliers T = Mϕ : B(2 ) → B(2 ) are all of the following special form: there are bounded sequences (xi ) and (yj ) in 2 such that ϕ(i, j ) = xi , yj .

(9.6)

Moreover, we have Mϕ = inf sup xi sup yj , i

j

where the inﬁmum runs over all possible (xi ) and (yj ) satisfying (9.6). This implies in particular (due to Haagerup) that bounded Schur multipliers on B(2 ) (or on S 1 ) are “automatically” c.b. (see [152]). However, the following remains open (we conjecture that the answer is negative): P ROBLEM 9.8. Is every bounded Schur multiplier on S p (1 < p = 2 < ∞) c.b.? Note that it is rather easy to give examples of bounded Fourier multipliers on Lp (G) which are not c.b. when G is any compact inﬁnite commutative group and 1 < p = 2 < ∞ (see [83] or [152], p. 91). P ROBLEM 9.9. Is there a description of c.b. Schur multipliers on S p extending (9.6) to 1 < p = 2 < ∞? It is known ([83], see also [206]) that the space of bounded (or c.b.) Schur multipliers of S p (2 < p < ∞) does not coincide with any interpolation space between the cases p = 2 and p = ∞. D EFINITION 9.10. A subset A ⊂ N × N is called a σ (p)-set (p 2) if {eij | (i, j ) ∈ A} is an unconditional basic sequence in S p . A simple application of Corollary 6.2 shows that this holds iff there is a constant C such that for any ﬁnitely supported function x : A → C we have |||x|||p x(i, j )e ij (i,j )∈A

Sp

C|||x|||p ,

(9.7)

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where |||x|p is as deﬁned in (6.7) above. It is easy to see by interpolation that, if 2 p ∞, we have |||x|p xS p for all x in S p . Hence (9.7) implies that the idempotent Schur multiplier corresponding to the indicator function of A is bounded on S p with norm C. For example, any set A for which either one of the two coordinate projections is one to one when restricted to A, is obviously a σ (p)-set. The following result provides much less trivial examples. P ROPOSITION 9.11 ([83]). Let p 2. Let Λ ⊂ Z be a Z(p/2)-set, or more generally a Λ(p)cb -set. Then the set AΛ = {(i, j ) ∈ N2 | i + j ∈ Λ} is a σ (p)-set. P ROOF. This follows from (6.4), applied to the series ϕ(z) = n∈Λ zn ( i+j =n x(i, j )eij ). Indeed for any z we have ϕ(z) p = x(i, j )e = a ij n , S Sp

(i,j )∈AΛ

Sp

n∈Λ

where an = i+j =n x(i, j )eij . By the Λ(p)cb -property of Λ (see Remark 9.4) there is a constant C such that: 1/2 1 n z an p p max an∗ an an an∗ p, L (S ) S C z n an p p .

1/2

Sp

L (S )

But as we just observed we have

an∗ an =

zn an Lp (S p ) =

x(i, j )2 ejj , j

(9.8)

an S p and

an an∗ =

i

x(i, j )2 eii . i

j

Hence (9.8) implies (9.7) with A = AΛ .

R EMARK . In the situation of Proposition 9.11, the same argument shows that if A = AΛ then for any ﬁnitely supported function x : A → S p we have Q(x) eij ⊗ x(i, j ) CQ(x), (9.9) S p (2 ⊗2 )

(i,j )∈A

where 1/2 p 1/p ∗ Q(x) = max x(i, j ) x(i, j ) , j

i

Sp

1/2 p 1/p ∗ x(i, j )x(i, j ) . p i

j

S

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A set A ⊂ N2 satisfying (9.9) for some constant C is called a σ (p)cb -set. Equivalently, this means, by (6.4), that {eij | (i, j ) ∈ A} is “completely unconditional” (see Remark 9.6), i.e., for any choice of signs εij = ±1, the transformations

x(i, j )eij →

(i,j )∈A

εij x(i, j )eij

(i,j )∈A

are c.b. on the closure in S p of {eij | (i, j ) ∈ A}. Since the left side of (9.9) remains valid for A = N2 , (9.9) implies that the indicator function of A is a c.b. Schur multiplier on S p . In particular, if x = (x(i, j )) is Hankelian, i.e., x(i, j ) = γ (i + j ) for some ﬁnitely supported function γ : Λ → S p , then (9.9) implies q(γ ) γ (n) ⊗ eij i+j =n

n∈Λ

S p (l2 ⊗l2 )

Cq(γ ),

(9.10)

where 1/2 p 1/p ∗ γ (n) γ (n) , q(γ ) = max j

nj

Sp

1/2 p 1/p ∗ . γ (n)γ (n) p j

nj

S

Thus we can “compute” (up to C) the norm of a Hankel operator with “spectrum” in Λ. C OROLLARY 9.12 ([83]). Let p > 2 be an even integer. (i) There are δ > 0 and C such that, for any n, there is a subset An ⊂ [1, . . . , n]2 with |An | δn1+2/p such that {eij | (i, j ) ∈ An } is C-unconditional in S p , i.e., a σ (p)set. (ii) There is an idempotent Schur multiplier T (idempotent means here T 2 = T ) which is bounded on S p but unbounded on S q for any q > p. P ROOF. The proof combines Theorem 9.3 with Rudin’s (combinatorial and number theoretic) construction of a B(p/2) set Λ ⊂ Z such that lim sup sup N −2/p Λ ∩ [a, a + bN] > 0. N→∞ a,b∈N

R EMARKS . (i) It is shown in [84] that, for any p > 2, n1+2/p is the maximal possible order of growth in the ﬁrst part of Corollary 9.12. (ii) The preceding corollary almost surely remains valid when p > 2 is not an even integer, but no proof is known at the time of this writing. (iii) It is proved in [136] (see also [105] for related estimates on the case p = ∞) that the orthogonal projection from S 2 onto the subspace of all Hankel matrices (i.e., the averaging

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projection) is bounded on S p iff 1 < p < ∞, and for p = 1, it is bounded from S 1 to S 1,2 (a fortiori it is of “weak” type (1, 1)). See [2] for more recent results on (Hankel and Toeplitz) Schur multipliers, in particular for the case S p with p < 1.

10. Isomorphism and embedding In this section we discuss isomorphism and embedding of non-commutative Lp -spaces. Unless explicitly stated otherwise, we will assume all Lp -spaces considered in this section are separable and inﬁnite-dimensional, or equivalently, the underlying von Neumann algebras are inﬁnite-dimensional and act on separable Hilbert spaces. Throughout this section, Lp denotes the classical commutative Lp -space on [0, 1]. The isomorphic classiﬁcation of commutative Lp -spaces is extremely simple, for there are only two non-isomorphic commutative Lp -spaces: l p and Lp . However, in the noncommutative setting, the situation is far from simple. In fact, it is impossible to list all non-commutative Lp -spaces up to isomorphism. It even seems very hard to classify them according to the different types of the underlying von Neumann algebras. Despite these difﬁculties, considerable progress has been achieved in the last few years. p Let K p denote the direct sum in the l p -sense of the Sn ’s, i.e., Kp =

/ n1

p

Sn

. p

p p Note that . K is the non-commutative L -space associated with the von Neumann algebra M = n1 Mn , the direct sum of the matrix algebras Mn , n 1. We also recall that if X is a Banach space, Lp (X) stands for the usual Lp -space of Bochner measurable p-integrable functions on [0, 1] with values in X. If X = Lp (M), Lp (X) is just the non-commutative Lp -space associated with L∞ (0, 1) ⊗ M. We should call the reader’s attention to the two different notations for the Schatten classes, equally often used in the literature: S p in our notation is sometimes denoted by C p , and K p by S p ! Recall that R denotes the hyperﬁnite II1 factor.

T HEOREM 10.1. Let M be a hyperﬁnite semiﬁnite von Neumann algebra. Let 1 p < ∞, p = 2. Then Lp (M) is isomorphic to precisely one of the following thirteen spaces:

l p , Lp , K p , S p , Lp ⊕ K p , Lp ⊕ S p , Lp K p , S p ⊕ Lp K p ,

Lp S p , Lp (R), S p ⊕ Lp (R), Lp S p ⊕ Lp (R), Lp R ⊗ B l 2 . R EMARKS . (i) The ﬁrst nine spaces in the above list give precisely all non-commutative Lp -spaces, up to isomorphism, associated with von Neumann algebras of type I. (ii) Theorem 10.1 is proved in [82]. Prior to that, the case of type I was studied in [187]. [82] also contains results on non-commutative Lp -spaces associated with hyperﬁnite factors of type III and free group von Neumann algebras. More precisely, it is shown there

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that the non-commutative Lp -spaces associated with hyperﬁnite factors of type IIIλ for all λ ∈ (0, 1] are isomorphic, and the non-commutative Lp -space associated with a free group von Neumann algebra is independent, up to isomorphism, of the number of generators as soon as this number is not less than 2. We refer the interested reader to [82] for more information. The proof of Theorem 10.1 can be reduced to the non-embedding of one noncommutative Lp -space into another. The main general result on this is the following T HEOREM 10.2. Let 0 < p < ∞, p = 2. Let M be a ﬁnite von Neumann algebra. Then S p does not embed (isomorphically) into Lp (M). We get immediately the following corollary. C OROLLARY 10.3. Let 0 2, in [82] for 1 p < 2, and in [188] for p < 1. Note that in the special case where M = L∞ (0, 1), Theorem 10.2 was established by McCarthy in the pioneering paper [128]. His result was considerably improved in [74]. In particular, Theorem 4.1 implies that K p does not embed into Lp . In the converse direction, it was proved in [11] that Lp does not embed into S p . In the case of 0 < p < 2, we have the following result, much stronger than Theorem 10.2. T HEOREM 10.4. Let 0 < p < 2. Let M be a ﬁnite von Neumann algebra. Let (ui,j )i,j 1 be an inﬁnite matrix of elements in Lp (M) such that supi,j ui,j p < ∞. Suppose that all rows, columns and generalized diagonals of (ui,j )i,j 1 are unconditional. Then one of the following three alternatives holds (i) Some row or column has a subsequence equivalent to the canonical basis of l p ; (ii) There is a constant λ > 0 such that for every integer n some row and some column p contain n elements λ-equivalent to the canonical basis of ln ; (iii) There is a generalized diagonal (uik ,jk )k1 such that n 1 lim 1/p uik ,jk = 0. n→∞ n k=1

p

Here by a generalized diagonal of (ui,j )i,j 1 we mean a sequence (uik ,jk )k1 with i1 < i2 < · · · and j1 < j2 < · · ·. Theorem 10.4 was proved in [82] for 1 p < 2 and in [188] for p < 1. Using Theorem 10.4, we can deduce the following reﬁnement of Theorem 10.1, which comes from [82] for 1 p < 2, and from [188] for p < 1. T HEOREM 10.5. Let M be as in Theorem 10.1, and let 0 < p < 2. If X = Y are listed in the tree in the following ﬁgure, then X embeds into Y iff X can be joined to Y through a descending branch

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Several non-embeddings in Theorem 10.5 are already contained in Corollary 10.3 and the discussion just after it. On the other hand, the non-embedding of Lp (K p ) into Lp ⊕ S p was established in [187], and that of Lp (R) into Lp (S p ) in [157]. The proof for the ﬁrst non-embedding in [187] uses the classical result that Lp contains a subspace isomorphic to l q for all 0 < p < q < 2. This classical result admits a non-commutative version, which is a remarkable result recently obtained by Junge (see Corollary 10.12 below), and which is the main ingredient for the non-embedding of Lp (R) into Lp (S p ). The remaining nonembeddings in Theorem 10.5 can be reduced to the following T HEOREM 10.6. Let 0 < p < 2, and let M and N be ﬁnite von Neumann algebras. Let X ⊂ Lp (M) be a closed subspace which contains no subspace isomorphic to l p , and let Y be a quasi-Banach space which contains no subspace isomorphic to X. Then X ⊗p S p does not embed into Y ⊕ Lp (N ), where X ⊗p S p denotes the closure of the algebraic tensor product X ⊗ S p in Lp (M ⊗ B(l 2 )). Theorem 10.6 was proved in [188]. It extends some results in [82]. Like in [82], its proof heavily relies upon Theorem 10.4. Using this theorem and Corollary 10.12 below (and its commutative counterpart, cited above), we deduce that Lp (S p ) (resp. Lp (R ⊗ B(l 2 ))) does not embed in S p ⊕ Lp (R) (resp. Lp (S p ) ⊕ Lp (R)). Subspaces of Lp (M), which have no copy isomorphic to l p , can be characterized as follows. T HEOREM 10.7. Let 0 < p < ∞, p = 2. Let M be a ﬁnite von Neumann algebra and X ⊂ Lp (M) a closed subspace. Then the following assertions are equivalent:

Non-commutative Lp -spaces

(i) (ii) (iii) (iv)

1509

X contains a subspace isomorphic to l p . For any λ > 1, X contains a subspace λ-isomorphic to l p . p X contains ln ’s uniformly. For any q such that 0 < q < p the norms · q and · p are not equivalent on X.

R EMARK . If one of the preceding assertions holds, then X contains a perturbation of a normalized sequence formed of operators with disjoint support; consequently, if p 1, X contains, for any λ > 1, a subspace λ-isomorphic to l p and λ-complemented in Lp (M). The above theorem is the extension to the non-commutative setting of the classical Kadets–Pełczy´nski results for commutative Lp -spaces (cf. [103,170]). It was proved in [186] for p > 2, in [82] for 1 p < 2, and in [163] and [188] for 0 2, Theorem 10.7 yields the non-commutative analogue of the following striking dichotomy: C OROLLARY 10.8. Let M and X be as in Theorem 10.7 with 2 < p < ∞. Then either X is isomorphic to a Hilbert space or X contains a subspace isomorphic to l p . R EMARKS . (i) The above corollary is easier for subspaces of S p , and there it holds for all 0 < p < ∞ (cf. [68]). (ii) More generally, Theorem 10.7 was extended in [169] to non-commutative Lp -spaces associated with any von Neumann algebra. (iii) [162,164] and [169] contain more results closely related to Theorem 10.7 and Corollary 10.8. There are many open problems on the subject discussed above. Below we give two of them. Let M and N be two von Neumann algebras of type λ and μ, respectively, where λ, μ ∈ {I, II1, II∞ , III}. Combining Corollary 10.3, Theorem 10.5 and Theorem 3.5, we see that if λ < μ and (λ, μ) = (II∞ , III), then Lp (M) and Lp (N ) are not isomorphic for all 0 < p < ∞, p = 2. It is unknown whether this is still valid for (λ, μ) = (II∞ , III). P ROBLEM 10.9. Let M and N be two von Neumann algebras of type II∞ and III, respectively. Are Lp (M) and Lp (N ) isomorphic for p = 2? Theorem 10.5 solves the embedding problem for all spaces listed there in the case of p < 2. On the other hand, Corollary 10.3 provides some partial solutions in the case of p > 2. However, we do not know whether Theorem 10.5 holds in full generality for p > 2. Below we state three of the most important cases left unsolved in Theorem 10.5. P ROBLEM 10.10. Let p > 2, and let (X, Y ) be one of the three couples (Lp (K p ), S p ⊕ Lp ), (Lp (S p ), S p ⊕ Lp (K p )) and (Lp (R), Lp (S p )). Does X embed into Y ? All previous non-embedding results deal with a couple of non-commutative Lp -spaces with the same index p. However, Junge’s theorem already mentioned above says that S q does embed into Lp (R) for p < q < 2. In fact, Junge [96] proved the following striking result, much stronger than the embedding of S q into Lp (R).

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T HEOREM 10.11. Let 0 < p < q < 2. Then Lq (R ⊗ B(l 2 )) embeds isometrically into Lp (R). As an immediate consequence, we get the C OROLLARY 10.12. Let 0 < p < q < 2. Then both S q and Lq (R) embed isometrically into Lp (R). q

In the commutative case, it is well-known that any ln embeds (uniformly over n) into p some lN . Junge also obtained the non-commutative version of this in [95]. T HEOREM 10.13. Let 0 0, n ∈ N. Then there is N = N(p, q, ε, n) such p q that SN contains a subspace (1 + ε)-isomorphic to Sn . Like in the commutative case, Junge’s arguments for the preceding results are probabilistic. They use non-commutative analogues of p-stable or Poisson processes. The reader is referred to [95,96] for more details and more embedding results. We conclude this section by a few words about the local theory of the non-commutative Lp -spaces, very recently developed in [99], in analogy with the classical Lp -space theory. Actually, it is better (and more convenient in some sense) to develop this theory in the operator space framework. Then the corresponding Lp -spaces are called OLp -spaces in [60]. Many classical results concerning Lp -spaces have been transferred to this noncommutative setting. In particular, any separable OLp -space (with an additional assumption) has a basis. It was also proved that Lp (M) (1 < p < ∞) is an OLp -space when M is injective or the von Neumann algebra of a free group (in the former case, p can be equal to 1). Consequently, these non-commutative Lp -spaces have bases. In the case of p = ∞, it was shown that any separable nuclear C ∗ -algebra has a basis. The interested reader is referred to [99] for more information.

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CHAPTER 35

Geometric Measure Theory in Banach Spaces David Preiss Department of Mathematics, University College London, London WC1E 6BT, UK E-mail: [email protected]

Contents 1. Finite-dimensional geometric measure theory in inﬁnite-dimensional situations 1.1. Rectiﬁability and density . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Inﬁnite-dimensional geometric measure theory . . . . . . . . . . . . . . . . . . 2.1. Differentiable measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Surface measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Measures and balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Differentiation theorems for Gaussian measures . . . . . . . . . . . . . . 3. Exceptional sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Lipschitz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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We will consider the question to what extent the classical relations between measure, derivative and geometry carry over to inﬁnite-dimensional Banach spaces. Such relations may be strangely distorted, like the seemingly simple question of recovering the Radon– Nikodým derivative by the limit of ratios of measures of balls, or answers may not be known, even in basic cases such as existence of common points of Fréchet differentiability of ﬁnitely many real-valued Lipschitz functions on a separable Hilbert space. Our goal is to describe some basic notions and results; these notes should be considered as an invitation to the subject and not as a survey of the subject, since many important concepts have necessarily been left out. We ﬁrst visit two themes which are essentially ﬁnite-dimensional even though the surrounding space is inﬁnite-dimensional, the problem of relations between rectiﬁability and density in general metric spaces and the recently developed theory of currents in metric spaces. In both cases, the structure of a Banach space is not essential for the setting, but it may always be assumed and it was needed to prove some of the deep results. For practically all problems of inﬁnite-dimensional geometric measure theory, the most important difference between ﬁnite-dimensional and inﬁnite-dimensional Banach spaces is due to non-existence of a reasonably ﬁnite, translationally invariant measure in the latter case: since any ball B(x, r) contains inﬁnitely many disjoint balls of radius r/3, if μ is a non-zero Borel measure on an inﬁnite-dimensional separable Banach space X and the μ-measure of balls depends only on their radii, then every non-empty open set has inﬁnite measure. In fact, if μ is a σ -ﬁnite measure on X, the shifted measure is singular with respect to μ for many shifts from X. One way to see this is to assume, as we may, that μ is ﬁnite and the norm is square integrable and consider the Cameron–Martin space of μ, H = {x ∈ X; sup{|x ∗ (x)|: x ∗ L2 (μ) } < ∞}. The identity from H to X is compact (in fact, it is much better; for example, if X is Hilbert, it is Hilbert–Schmidt), and it is easy to see that the shift of μ by any vector not belonging to H is singular with respect to μ. Nevertheless, there are measures in inﬁnite-dimensional Banach spaces that have some of the properties normally associated with the Lebesgue measure. In many instances, Gaussian measures have been used as a replacement for the Lebesgue measure. A more general class of such measures is formed by those that are quasi-invariant in a dense set of directions, where a measure is called quasi-invariant in a direction u if its null sets are preserved when shifted by u. We will restrict ourselves to a discussion of the notions of measures differentiable in a direction in 2.1 and to brieﬂy pointing out connections to the possibility of deﬁning surface measures. In 2.3 we visit the amusing results obtained by attempting to understand to what extent exact naïve analogues of ﬁnite-dimensional results fail in inﬁnite dimensions. In the short Section 3 we give several notions of exceptional sets. Some readers may ﬁnd it more convenient to read the deﬁnitions from this section only after encountering their use in Section 4, where we treat the problem of existence of Gâteaux or Fréchet derivative of Lipschitz functions and also brieﬂy consider few more exotic derivatives. Out of the directions that have been omitted one should deﬁnitely mention the study of various notions of generalized derivatives (or subdifferentials) which often parallels the development of classical real analysis. As an example, see [11]. For many other problems that could ﬁt into this text as well as a number of results relevant to it see [5]. Although from time to time we mention the case of Gaussian measures (deﬁned as those measures whose every one-dimensional image by a continuous linear functional is Gaussian or Dirac), much

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of the extensive research devoted to them has not been mentioned even though it often has deep connections to problems of geometric measure theory. To avoid possible misunderstandings, we will consider only separable Banach spaces unless non-separability is speciﬁcally permitted, and only Borel measures. Normally, measures will be positive and ﬁnite. There are, however, important exceptions: signed (i.e., realvalued) measures are appearing, out of necessity, in our considerations of differentiability of measures in 2.1 and the Hausdorff measures are positive but often notoriously inﬁnite.

1. Finite-dimensional geometric measure theory in inﬁnite-dimensional situations Many situations from the classical geometric measure theory (including the notions of fractal geometry such as deﬁnition of fractal sets via iterated function systems) can be easily transferred to the case when the ambient space is inﬁnite-dimensional. For example, the k-dimensional Hausdorff measure ∞ ∞ 1 k lim inf H (A) = diam (Ai ): A ⊂ Ai , diam(Ai ) < δ α(k) δ→0 k

i=1

i=1

as well as a number of several other k-dimensional measures (spherical measures, packing measures, etc.) have been studied in arbitrary metric spaces. (The constant factor α(k) is chosen so that the k-dimensional Hausdorff measure in Rk coincides with the Lebesgue measure; since we will not need this fact here, for the purpose of these notes we may set α(k) = 1.) The natural setting is often that of metric spaces (and, because these can be embedded into Banach spaces, it usually sufﬁces to consider only these), but with a few notable exceptions, the generality does not bring much new, even though it may help to clarify the assumptions or provide natural proofs. Here we will brieﬂy consider two of the situations in which the interaction with geometry of Banach spaces went much farther.

1.1. Rectiﬁability and density Much of the development of classical geometric measure theory was driven by attempts to show, under various geometric assumptions on a subset A of Rn of ﬁnite k-dimensional measure, that A is k-rectiﬁable, i.e., that Hk -almost all of A can be covered by a Lipschitz image of a subset of Rk . (The restriction to the Hausdorff measure and sets of ﬁnite measure is not necessary, but it is convenient and sufﬁces for this presentation.) Perhaps the most useful rectiﬁability criterion, the Besicovitch–Federer projection theorem (see [17, Theorem 3.3.12]), did not seem to have any natural counterpart for inﬁnite-dimensional ambient spaces till one was found in the modern setting of currents (see below). Another useful rectiﬁability criterion (due, in increasing level of generality, to Besicovitch, Marstrand and Mattila) says that (under the above assumptions) A is k-rectiﬁable if and only if its k-dimensional density Θ k (A, x) = limδ→0 Hk (A ∩ B(x, r))/(α(k)r k ) is equal to one at Hk almost every x ∈ A. This could well be true in every metric space, although

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it was shown only in the cases when either k = 1 or the space in question is a subset of a uniformly convex Banach space. (So the simplest unknown case is the two-dimensional measure in ∞ 3 .) The implication saying that rectiﬁable sets have density one almost everywhere was proved by Kirchheim [27] in full generality. The key behind this is the existence of metric derivative (see 4.2.6). One of the corollaries of this work is that the area formula remains valid in the general situation; in the simpler case of injective Lipschitz mappings this says that the k-dimensional measure of the image can be calculated as an integral of a suitable Jacobian over the domain, where the Jacobian depends just on the metric derivative of the mapping. 1.2. Currents The development of the theory of currents was motivated by the difﬁculty to prove existence results for higher-dimensional minimal surfaces by classical methods. The basic ideas behind the (ﬁnite-dimensional) theory of currents, as found, for example, in [17], were similar to those behind the introduction of distributions: a k-dimensional oriented surface gives rise to a linear functional on the space of differential k-forms; among such functionals one chooses (and calls currents) those that still form a (weak∗ ) compact set but already have many features of ‘genuine surfaces’. The main notions may be transfered to Hilbert spaces (or even to those Banach spaces in which suitable results on differentiability of Lipschitz mappings hold) without any change, but deeper results appear to be based on concepts that are not available beyond the ﬁnite-dimensional situation. We will now describe an important recent development which shows that the theory needs no concept of differentiability and that many strong results remain valid even when the ambient space is inﬁnite-dimensional. It is due to Ambrosio and Kirchheim [2], based on an idea of De Giorgi and, incidentally, does not even use the concept of differential forms. Instead of considering integration over a k-dimensional (smooth) oriented surface S ⊂ Rn as a linear functional on k-dimensional differential forms, we will consider it as a (k + 1)-linear functional on the space Dk+1 (Rn ) of (k + 1)-tuples of Lipschitz functions (f, π1 , π2 , . . . , πk ). So, in the simplest case of S = [a, b] being an interval on the line (so b k = n = 1), the associated current is T (f, π) = [a,b] f dπ = a f (x)π (x) dx. Somewhat more generally, if g is a Lebesgue integrable function on Rn (which can be imagined as the multiplicity of the k = n-dimensional oriented surface S = {x: g(x) = 0}), the associated current is T (f, π1 , π2 , . . . , πk ) = fg dπ1 ∧ dπ2 ∧ · · · ∧ dπk S

= f (x)g(x) det π1 (x), π2 (x), . . . , πk (x) dx, and analogous formulas associate such (k + 1)-linear forms to (smooth) k-dimensional surfaces with real multiplicity also in case when k < n. The key observation is that the functionals T in the above examples have the following three properties.

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(a) T is (k + 1)-linear. (b) The restriction of T to the set of (k + 1)-tuples with |f | 1 and Lip(πi ) 1 is continuous in the topology of uniform convergence of the f ’s and pointwise convergence of the πi ’s. (c) T is local in the sense that T (f, π1 , π2 , . . . , πk ) = 0 whenever some πi is constant on {x: f (x) = 0}. For the currents from our examples, (c) expresses the fact that they depend on the derivatives of the πi ’s. The fact that (b) holds (which is not so obvious) has been recognised (often implicitly) as important in a number of other connections. We can now deﬁne a k-dimensional current T in a complete metric space E as a functional on Dk+1 (E) satisfying (a)–(c). The key requirement of locality means that T depends in a weak sense on the derivative of the πi ’s, and it is the main point in showing that currents have the property that transforming the πi ’s by a Lipschitz mapping of Rk multiplies the f by the determinant of the transformation. (In particular, currents are anti-symmetric in the πi ’s.) One may therefore use the more suggestive notation T (f dπ1 ∧ · · · ∧ dπk ) for T (f, π1 , . . . , πk ), although the symbols dπi themselves may have no meaning. Currents behave like measures in the* ﬁrst variable; more precisely there is a ﬁnite measure μ such that T (f, π1 , π2 , . . . , πk ) i Lip(πi ) |f | dμ; the least such measure is called the mass of T and denoted by T . The total mass of T is deﬁned as M(T ) = T (E). Standard operations on currents are deﬁned in a natural way. The existence of mass allows one to extend currents to arbitrary bounded Borel f ; in particular, the restriction of a current to a Borel set B may be deﬁned as (f, π1 , . . . , πk ) → T (f χB , π1 , . . . , πk ), where χB is the indicator function of B. The push-forward of T by a Lipschitz mapping φ : E → F is φ+ T (f, π1 , . . . , πk ) = T (f ◦ φ, π1 ◦ φ, . . . , πk ◦ φ) and the boundary of a (k + 1)-dimensional current S is ∂T (f, π1 , . . . , πk ) = S(1, f, π1 , . . . , πk ). However, boundaries are tricky: ∂T is a functional satisfying (a) and (c), but there is no reason why it should satisfy the continuity requirement (b). Currents for which ∂T satisﬁes (b) are called normal and are the ﬁrst of the basic classes of current. Two other concepts arise naturally from the wish to deﬁne a notion that should represent generalized surfaces (with integer multiplicity): a k-dimensional current T is rectiﬁable if its mass is absolutely continuous with respect to the k-dimensional measure on some k-rectiﬁable set and it is an integer current if the push-forward to Rk of any restriction of T to a Borel set is representable by a Lebesgue integrable integer-valued function. It is not known if every k-dimensional current in Rk (k 3) is rectiﬁable (this problem is close to that of describing sets of non-differentiability of Lipschitz mappings in Rk ) but normal k-dimensional currents in Rk are necessarily rectiﬁable and in fact correspond exactly to functions of bounded variation. It follows that, within normal currents in Rn , the new concepts coincide with the standard ones. These classes of currents admit many natural characterizations similar to those obtained for currents in ﬁnite-dimensional spaces. In particular, strong rectiﬁability criteria which are false for sets are valid for currents. In the presence of suitable results on differentiability of Lipschitz mappings these classes of currents may be deﬁned in a more customary way via exterior algebra and integer currents are those whose density is an integer multiple of the corresponding area factor. (The area factor is related to the Jacobians mentioned above

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as well as to the problem of ﬁnding optimal rectangles enclosing a given convex body, which was studied, for example, in [39].) The Plateau problem may be stated in the full generality of a complete metric space E: given a k-dimensional integer rectiﬁable current S with compact support and zero boundary, ﬁnd a (k + 1)-dimensional integer rectiﬁable current T having the least total mass such that ∂T = S. In general, there may exist no currents T with ∂T = S, but such currents always exist if E is a Banach space. However, to assure the existence of a suitable minimizing sequence (whose limit would give a solution to the Plateau problem under fairly general assumptions, for example, if E is dual of a separable space) one needs that the following isoperimetric inequality holds: for every k-dimensional integer rectiﬁable current S with ∂S = 0 there is a (k + 1)-dimensional integer rectiﬁable current T with ∂T = S and M(T ) c(M(S))(k+1)/ k , where c is a constant depending on E and k only. Whether this holds in every Banach space is an open problem; it has been proved in duals of separable Banach spaces having a weak∗ ﬁnite-dimensional decomposition. 2. Inﬁnite-dimensional geometric measure theory The relation between inﬁnite-dimensional measures and geometry of the inﬁnite-dimensional Banach space E is much weaker than in the ﬁnite-dimensional case. Consider just the problem of describing the image of a non-degenerated Gaussian measure γ on E by a continuous linear transformation T : even for simple transformations, such as T x = 2x, the image is singular with respect to γ and so no analogue of the classical substitution theorem can hold for such transformations. The same situation occurs for most shifts. The most common setting is therefore not only a Banach space E equipped with a measure μ, but also with a vector space H of the set of directions in which μ behaves invariantly; it is also often assumed that H is (a continuous image of) a Hilbert space. (The basic example is, of course, a Gaussian measure γ in E with H being its Cameron–Martin space.) The (possibly non-linear) transformations of the form x → x + h(x), where h : E → H are the natural candidates for which the substitution theorem may be valid. The role of geometry of E has nearly disappeared and in fact E is usually just assumed to be a locally convex space. Below we comment on the background of the basic concepts of derivative of measures in Banach spaces and brieﬂy indicate some directions of research. Then we discuss results showing that not only covering theorems but even some of their natural corollaries often fail in inﬁnite-dimensional situation even for Gaussian measures. 2.1. Differentiable measures In much of modern analysis in ﬁnite-dimensional spaces, the role of pointwise derivative has been completely overshadowed by that of derivative in the sense of distributions. If f : R → R is Lebesgue integrable, its distributional derivative may be deﬁned as a Lebesgue integrable function g : R → R such that the formula for integration by parts

(1) φ (t)f (t) dt = − φ(t)g(t) dt

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holds for every smooth φ : R → R with bounded derivative. However, observing that in (1) the functions f and g are only used as acting on functions by integration, i.e., as measures, we may consider it as deﬁning that the distributional derivative of a signed measure μ is a signed measure ν such that that φ (t) dμ(t) = − φ(t) dν(t) for every smooth bounded φ : R → R with bounded derivative. On the real line, this generality is partly spurious, since it is easy to see that a measure μ on R has this derivative if and only if it is a function of bounded variation. (Somewhat loosely, one says that the measure μ(E) = E f (t) dt is a function, namely, the function f .) However, the derivative may well be a measure which is not a function, for example, the derivative of the function f (t) = signum(t) is the Dirac measure. A similar approach is used in Rn to deﬁne distributional partial derivatives; and again their existence means that the measure is a function. In fact, it is again a function of bounded variation, usually by deﬁnition (see, for example, [61]). The deﬁnition of distributional derivatives of measures admits a direct generalization to Banach spaces (where we have no notion of a measure being a function): the derivative of a (ﬁnite Borel) signed measure μ in direction w is a signed measure Dw μ such that

Dw φ(x) dμ(x) = −

φ(x) dDw μ(x)

(2)

for every bounded continuously differentiable φ : X → R with bounded derivative. The deﬁnition immediately implies that the set of directions of differentiability of μ is a linear space, the mapping w → Dw μ is linear and that differentiation commutes with convolution, i.e., Dw (ν ∗ μ) = ν ∗ Dw μ provided that Dw μ exists. Directional derivatives of measures may be equivalently deﬁned by more direct formulae: derivative of μ in the direction w in Skorochod’s sense is deﬁned by (see [52, §21] for details)

φ dDw μ = − lim

r→0

φ(x + rw) − φ(x) dμ(x) r

(3)

provided that the limit exists for every bounded continuous φ : X → R; the functional deﬁned by the limit is necessarily an integral with respect to a measure. Another approach which was developed in ﬁnite-dimensional spaces by Tonneli needs essentially no modiﬁcation in inﬁnite-dimensional spaces: we require that μ has a disintegration

φ dμ =

Y R

φ(y + tw)ψy (t) dt dν(y),

where Y is a complement of span{w}, ν is a probability measure on Y and ψy are (right continuous) functions of bounded variation; under these conditions we deﬁne

φ dDw μ =

Y R

φ(y + tw) dψy (t) dν(y).

(4)

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It is easy to see that the derivatives of measures deﬁned by (3) or (4) satisfy (2). If (2) holds, we obtain (3) by denoting νr (E) = λ{t ∈ [0, r]: tw ∈ E}/r, inferring from the formula for differentiation of convolution that φ(x + rw) − φ(x) dμ(x) φ dνr ∗ Dw μ = − r ﬁrst for every bounded continuously differentiable φ : X → R with bounded derivative and then, by approximation, for every bounded continuous φ : X → R, and by letting r → 0. Finally, to obtain (4) from (2), we disintegrate

h dDw μ =

Y R

h(y + tw) dσy (t) dν(y)

and let ψy (t) = σy (−∞, t]. By approximation, it sufﬁces to show that (4) holds for every continuously differentiable function φ : X → R with bounded derivative and with {t t ∈ R: φ(y + tw) = 0 for some y ∈ Y } bounded. For any τ > 0 denote gτ (y + tw) = −∞ φ(y + sw) − φ(y + (s + τ )w) ds and use (2) and integration by parts to infer that

Dw gτ (x) dμ(x) = −

gτ (x) dDw μ(x) = −

=

Y R

Y R

gτ (y + tw) dσy (t) dν(y)

Dw gτ (y + tw)ψy (t) dt dν(y).

Since Dw gτ (x) = φ(x) − φ(x + τ w), (2) follows by letting τ → ∞. Currently the most useful notion of derivative of a measure μ (often called differentiability in the sense of Fomin) is obtained by requiring additionally that Dw μ be absolutely continuous with respect to μ. This is equivalent to validity of (3) for every bounded Borel measurable function or to differentiability at t = 0 of the function assigning to t ∈ R the measure μ shifted by tw when the space of measures is equipped with the usual norm. The Radon–Nikodým derivative of Dw μ with respect to μ is called the logarithmic derivative of μ in direction w; one readily sees that this term is justiﬁed in the ﬁnite-dimensional situation. All these notions have been treated as a special case of differentiability of mappings of the real line into the space of signed measures equipped with various topologies in [50]; another particular case of this treatment is the notion of differentiability of measures along vector-ﬁelds. (Of course, in this generality some of the equivalences mentioned above may fail.) Under very mild assumptions, these authors also prove the key forb mula dμa / dμb = exp( a -t (x) dt), where -t is the logarithmic derivative of t ∈ R → μt . (See [50] for the history of this formula and its applications.) In the setting when H is a subspace of E consisting only of directions of logarithmic differentiability and h : X → H , one can, under appropriate assumptions, compute the logarithmic derivative of t → (id + th)+ μ and the Radon–Nikodým derivative d(id + th)+ μ/ dμ from the derivative of h and directional logarithmic derivatives of μ – the latter gives a substitution theorem mentioned above (see [51]). The assumptions alluded to here are, of necessity, much stronger than those mentioned so far: since the formulas involve either the trace of the derivative of h in

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the direction of H or the determinant of id + th (x). (This also explains why H is supposed to carry a Hilbert space structure.) A Gaussian measure is logarithmically differentiable exactly in the directions of its Cameron–Martin space and the derivatives may be found explicitly. For these measures, the above results form just a beginning of the story; see, for example, [8] for much more. The natural problem of unique determination of a measure by its logarithmic derivative has been answered negatively in [38]. (Prior to it, several authors noted that a positive answer would not only mean that some correspondence between functions and measures survives to the inﬁnite-dimensional situation but would also have interesting applications.)

2.2. Surface measures Several approaches have been suggested to the deﬁnition of the surface measure induced by a given measure μ on E. A natural way is to assume that the surface is deﬁned as {x: ϕ(x) = 0} where ϕ : E → R is such that for sufﬁciently many functions g on E the measures ϕ+ (gμ) have continuous density kg with respect to the Lebesgue measure; the value of the surface integral of such g is then kg (0)/k1 (0). To prove the assumption of continuity of kg , one may use differentiability of μ together with the Malliavin method. (More details may be found in [8].) Uglanov’s method [56] is based on the idea that, if μ has logarithmic derivative in direction w and G is the graph of a smooth function from a complement of Rw to Rw then Dw μ{a + tw: a ∈ A, t 0} should be a measure on G which is absolutely continuous with respect to the corresponding surface measure on G with known Radon–Nikodým derivative. This can be used to deﬁne the surface measure of subsets of G provided that it does not depend on the choice of w. This independence is shown under suitable assumptions, which appear less stringent than in other methods. It is natural to imagine that the theory of surface measures (or, more generally, measures on surfaces of ﬁnite co-dimension), could be understood also as theory of integration of differential forms and/or currents of ﬁnite co-degree. Such possibility has been explored in a series of papers starting from [49].

2.3. Measures and balls Only little seems to be known about the interplay between geometry of an inﬁnitedimensional Banach space X and behaviour of measures on it. A reasonably clear picture showing that the situation is rather complicated has been obtained concerning the questions that developed from the attempts to ﬁnd valid inﬁnite-dimensional counterparts to the differentiation theorem for measures according to which in ﬁnite-dimensional Banach spaces the Radon–Nikodým derivative g of a Borel measure μ with respect to a Borel measure ν is, at almost every x, obtained by the limit of the ratio of their averages on balls around x: g(x) = limr→0 μ(B(x, r))/ν(B(x, r)). It turned out that the differentiation theorem for measures holds for all Borel measures in a Banach space if and only if it is ﬁnite-dimensional; a similar statement holds even for complete metric spaces (and for

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other families than balls) with a suitable combinatorial deﬁnition of ﬁnite dimensionality. (Like most results mentioned in this subsection, this may be found in [36].) Moreover, in a Hilbert space even much weaker statements are false as is shown by the following rather involved example from [43]. (Some numerical constants in the construction in [43] should not be taken too literally; a more accessible construction should eventually appear in [36].) There are a Gaussian measure ν in 2 and a positive function f ∈ L1 (ν) such that B(x,r) f dν/ν(B(x, r)) tends to inﬁnity as r tends to zero not only for every x but even uniformly on 2 , in other words lim inf

r→0 x∈2 B(x,r)

f dν/ν B(x, r) = ∞.

(5)

Among the possible corollaries of the differentiation theorem for measures that are not negated by the above example the most natural one is the determination of measures by balls: the differentiation theorem applied to μ and μ + ν gives that μ = 12 (μ + ν), hence μ = ν whenever μ and ν coincide on all balls. The hope that at least this statement holds in general metric spaces was dashed by Davies in [14]: there is a compact metric space on which two different Borel probability measures coincide on all balls. The basic idea of this beautiful example is the construction, for any given α, β > 0, of two measures μ, ν on a ﬁnite metric space M in which the only distances are one and two (so M is easy to imagine as the vertex set of a graph; points joined by an edge have distance one, remaining points have distance two) which coincide on all closed balls with radius one and satisfy μ(M) = α and ν(M) = β. Such a space is obtained as a graph on n + n2 vertices consisting of a complete graph on n ‘inner’ vertices, to each of which n different ‘outer’ vertices are joined. The main observation is that each ball of radius one consists either of one inner and one outer vertex, or of n inner and n outer vertices; then a straightforward calculation gives μ and ν provided that n is large enough. (For example, all inner vertices may have μ measure α/2n and ν measure α/2n − (β − α)/(n2 − n), and all outer vertices may have μ measure α/2n2 and ν measure α/2n2 + (β − α)/(n2 − n).) Replacing recursively points by rescaled copies of such spaces, one ﬁnds a compact metric space M0 of diameter one on which two different Borel measures μ0 , ν0 coincide on all balls of radius less than one. The ﬁnal space and measures are obtained as M0 ∪ M1 , μ0 + ν1 and ν0 + μ1 , where M1 is another copy of M0 (with corresponding measures μ1 , ν1 ) and the distance between points of M0 and M1 is deﬁned as one. The above construction showing that even the determination of measures by balls may be false in general metric spaces clearly leads only to highly non-homogeneous spaces and it cannot produce, for example, a Banach space. Indeed, for Banach spaces such an example does not exist ([45]): two ﬁnite Borel measures μ and ν coinciding on all balls in a separable Banach space X necessarily coincide on all Borel subsets of X. The argument blows suitably placed balls to show that μ and ν coincide on many convex cones that factor through a ﬁnite-dimensional subspace; a simple consideration of ﬁnite-dimensional projections of μ and ν then gives that they coincide on all half-spaces given by linear functionals belonging to a weak∗ dense subset of the dual unit ball. Hence μ and ν have the same Fourier transform, and the statement follows.

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It is an open problem if the determination of measures by balls holds under relatively weak homogeneity assumptions, e.g., in complete separable metric Abelian groups. In particular, except for a few special cases it is not known whether two ﬁnite Borel measures in a Banach space coincide provided that they coincide on all balls with radius at most one. (Note that by the above proof this holds for balls of radius at least one.) This motivates the attempts to prove the Banach space result without the use of Fourier transform. One such attempt noted that it would be enough to show that the family of Borel sets in a separable Banach space X is the smallest family of subsets of X containing balls and closed under complements and countable disjoint unions; the latter family is necessarily closed also under countable monotone unions and intersections. It has been recently shown that this statement holds in ﬁnite-dimensional spaces (it is not easy; both existing proofs ([20] and [60]) use Besicovitch’s covering theorem), but not in an inﬁnite-dimensional Hilbert space (any non-trivial intersection of two balls supplies a counterexample [25]). The extent of the difference between ﬁnite- and inﬁnite-dimensional situation is apparent from the following two amusing statements concerning separable Hilbert spaces H : (A) The statement “whenever μ(B) ν(B) for all balls B with radius 1, then μ ν” holds if and only if H is ﬁnite-dimensional. (B) The statement “whenever μ(B) ν(B) for all balls B with radius 1, then μ ν” holds if and only if H is inﬁnite-dimensional. The statements considered in (A) and (B) are sometimes called positivity principles for small and large balls, respectively. The positivity principle for small balls follows from the differentiation theorem for measures, and so it holds in all ﬁnite-dimensional Banach spaces. The example behind the other implication of (A) uses the Gaussian measure ν in 2 and positive function f ∈ L1 (ν) such that (5) holds: a simple modiﬁcation achieves f dν < 1 and B(x,r) f dν > ν(B(x, r)) for r 1; and the example needed for (A) is ob tained with μ(A) = A f dν. This example also answers negatively the question of validity of positivity principle for all balls in general Banach spaces: In the space 2 ⊕∞ R consider ν1 + ν−1 + μ0 and ν0 , where the index r indicates the image measure under the mapping x ∈ 2 → x ⊕ r. The statement (B) is much easier: in the n-dimensional case one may consider for μ the Lebesgue measure on a ball and for ν a small multiple of the Dirac measure in its centre. In the inﬁnite-dimensional case one shows that the characteristic function of each cylinder (set of the form π −1 (B), where π is an orthogonal projection with a ﬁnite-dimensional range and B is any ball in the range), can be obtained as a limit of functions from the convex cone generated by characteristic functions of balls with radius 1. This reduces the proof to the ﬁnite-dimensional case of small balls. The negative results mentioned above are remarkably unstable. Riss [48] shows that every Banach space may be renormed so that the positivity principle holds for large balls. Only future investigations may reveal the fate of this little corner of geometric measure theory. It is possible that useful connections exist to probability theory on Banach spaces, especially to problems of large deviations; a small indication of this may be given by the use of Chernoff’s theorem in [15] to prove the determination of measures by balls for measures with ﬁnite Laplace transform, or by the fact that more information may be sometimes obtained by using large balls instead of small balls.

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2.4. Differentiation theorems for Gaussian measures The importance of Gaussian measures justiﬁes their separate treatment, although from the point of view of the problems considered here the results are rather sparse. It is not difﬁcult to see that Gaussian measures on Banach spaces are determined by their values on balls of radii at most one [9]. Nevertheless, in 2.3 we noted that even for a Gaussian measure ν in a separable Hilbert space the differentiation theorem for L1 functions may fail not only almost everywhere, but even uniformly (see Eq. (5)). The density theorem may fail as well [42]: there are a Gaussian measure ν in 2 and a Borel set E ⊂ 2 of positive ν measure such that limr→0 ν(E ∩ B(x, r))/ν(B(x, r)) = 0 for ν almost every x, although here it is not known if the failure may be uniform. On the other hand, by [55] there are inﬁnite-dimensional Gaussian measures in separable Hilbert spaces for which the differentiation theorem holds for all functions from Lp for p > 1; a sufﬁcient condition is that the eigenvalues σk of their covariance satisfy σj +1 cσj /j α for some α > 5/4. The proof is rather technical and its important ingredient is the dimension-independent estimate of the Hardy–Littlewood maximal operator from [54].

3. Exceptional sets Here we discuss several notions of “null” or “negligible” sets in a Banach space that appear as exceptional sets in various questions of behaviour of mappings between inﬁnitedimensional spaces. With the notable exception of the topological notion of the sets of the ﬁrst category, these involve metric or linear conditions and sometimes behave in an unexpected way. First category sets are, unfortunately, rarely useful in problems where main point is to capture some of the roles played in ﬁnite-dimensional spaces by the Lebesgue null sets. When convenient, we will give the deﬁnition for Borel sets only with the understanding that a possibly non-Borel set is null if it is contained in a Borel null set. (However, it should be pointed out that Borel measurability is not a point of pedantry; leaving it out may easily not only change the deﬁnitions but render them meaningless.) The most appealing replacement for Lebesgue null sets in inﬁnite-dimensional Banach spaces or even in complete separable metric Abelian groups is due to Christensen [12]. Let G be an Abelian topological group whose topology is metrizable by a complete separable metric. A Borel set E ⊂ G is Haar null if there is a Borel probability measure μ on G such that every translate of G has measure zero. (Sometimes these sets are referred to as Christensen null. They have also been rediscovered under the name of shy sets.) Haar null sets form a σ -ideal since, if En are Haar null and μn are the corresponding measures, the measure obtained as an inﬁnite convolution of suitable portions of μn witnesses that ∞ n=1 En is Haar null. If the group is locally compact, these null sets coincide with those of Haar measure zero. If, however, the group is not locally compact, then every compact set is Haar null; this follows from the following important generalization of Steinhaus’s theorem: if a Borel set A is not Haar null then A − A contains a neighbourhood of the identity in G. Haar null sets have been, deservedly, investigated in their own right. Since measures on G are inner regular with respect to compact sets, any subset of G containing a translate of

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every compact set is not Haar null. This has been used in [53] to show that if G is not locally compact, then it contains an uncountable collection of disjoint closed subsets which are not Haar null. Reﬂexive Banach spaces have been characterized as those in which every closed convex set with empty interior is Haar null, see [35] and [34]. Relatively simple examples show that a Fubini-type result is false for Haar null sets. A property useful in application to differentiability is that a Haar null set is Lebesgue null on lines parallel and arbitrarily close to any given line. Although the knowledge that an exceptional set is Haar null sufﬁces for many applications (in particular, if we just need to have one non-exceptional point), there are situations in which this is not the case. Several seemingly different stronger notions of negligible sets in a Banach space X were deﬁned (implicitly or explicitly) to improve, in particular, the results on Gâteaux differentiability of Lipschitz functions. Recall that a cube measure in X is N under the mapping t → x + tk xk any image of the product Lebesgue measure on [0, 1] provided that xk < ∞. A cube or Gaussian measure is non-degenerate if every closed hyperplane gets measure zero. A Borel set E is a separable Banach space X is called (a) cube null if it is null for every non-degenerate cube measure on X [32], (b) Gauss null if it is null for every non-degenerate Gaussian measure on X [40], and (c) Aronszajn null if for every sequence un ∈ X with dense span, E can be written as a union of Borel sets En such that the intersection of En with any line in the direction un is of one-dimensional Lebesgue measure zero [3]. Clearly, Aronszajn null sets are Gauss as well as cube null. It is in fact not difﬁcult to see that Gauss and cube null sets coincide, and may be equivalently deﬁned as those Borel sets that are null for every measure with a dense set of directions of differentiability (cf. [6]). A remarkable result of Csörnyei [13] shows that every cube null set is Aronszajn null, and so all these notions coincide. The main difﬁculty of the proof stems from the requirement that the sets En be Borel; note however that without assuming this the deﬁnition (c) would become meaningless since by an observation from [6] the whole space would be Aronszajn null. Clearly, (a) or (b) immediately show that every Aronszajn null set is Haar null. The converse is false because there are compact Aronszajn non-null sets (e.g., cubes) while every compact set K is Haar null. This example also shows that the analogue of Steinhaus’s theorem is no longer valid for Aronszajn null sets. As one would expect, in ﬁnite-dimensional spaces Aronszajn null sets coincide with Lebesgue null sets. By an example from [7], Aronszajn null sets are not invariant under C ∞ Lipschitz isomorphisms. It follows that these sets cannot provide full characterization of exceptional sets that are invariant under such mappings (such as sets of non-differentiability). A possible remedy, suggested by Bogachev [7], is to consider sets which are null for all measures differentiable with respect to a spanning sequence of vector-ﬁelds un (i.e., such that for every x the vectors un (x) span X). Another possible remedy, from [47], is to consider sets which can be written as a union of Borel sets En such that, for some un ∈ X and εn > 0, the intersection of En with any curve γ : R → X with Lip(t → γ (t) − tun ) < εn is of onedimensional measure zero. Many questions concerning these sets are open. For example, it is not known if a formally stronger notion in the spirit of (c) is equivalent to the one described here or if an analogue of Csörnyei’s result holds in this situation. In fact, it is

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not even known if in ﬁnite-dimensional spaces (of dimension at least three) these null sets coincide with Lebesgue null sets. Intriguing invariance problems for the null sets deﬁned above remain open and deserve to be mentioned because of their possible application to the problem of classifying Banach spaces up to Lipschitz isomorphisms. We have already noted that Aronszajn null sets are not invariant under Lipschitz isomorphisms. The same holds for Haar null sets; moreover, in [28] there is an example of a Haar non-null set in 2 which can be transformed by a Lipschitz isomorphism into a hyperplane; note that hyperplanes are not only Aronszajn null, but have to be null for any notion for which we wish to have any reasonable statement on differentiability of Lipschitz functions almost everywhere. For the intended application to the Lipschitz isomorphism problem it would be enough to know that a null set cannot be transformed into a set whose complement is null; as the title of [28] indicates, even this may happen for Haar null sets, but it is unknown for Aronszajn null sets. No pertinent examples are known for the non-linear notions of null sets. We now brieﬂy describe another appealing notion of exceptional sets which is a metric strengthening of the notion of ﬁrst category sets; unfortunately, for our purposes, it has the disadvantage that it cannot describe non-differentiability sets of Lipschitz functions since these may well be of second category. These sets, however, play an important role in studying more exceptional behaviour (for example, non-differentiability of continuous convex functions). Out of the huge number of non-equivalent notions (cf. [58]), the two most natural ones (in our context) are σ -porous sets and σ -directionally porous sets. It sufﬁces to deﬁne the notions of porous and directionally porous sets only, since the preﬁx ‘σ ’ means ‘the union of countably many of.’ A set E ⊂ X is porous if there is 0 < λ < 1 such that for every x ∈ E there are xn ∈ X converging to x and rn > λxn − x such that the balls B(xn , rn ) are disjoint from E. It is porous in direction u if the xn may be found on the line through x in direction u, and it is directionally porous if there is u such that it is porous in direction u. The same notion of σ -porosity (though a different notion of porosity) is obtained if we deﬁne porosity with λ independent of x; for σ -directional porosity we could even allow dependence of u on x. The porosity σ -ideals are Borel: every σ -(directionally) porous sets is contained in a Borel σ -(directionally) porous set. Clearly, σ -directionally porous sets are σ -porous, and σ -porous sets are ﬁrst category. Every σ -directionally porous set is Aronszajn null: if E is porous in direction u, it is porous in every direction sufﬁciently close to u; so, if μ has a dense set of directions of differentiability, then E is porous in a direction v of differentiability of μ and the disintegration along this direction shows that μ(E) = 0. In ﬁnite-dimensional spaces the notions of σ -porous and σ -directionally porous sets coincide, otherwise they differ: by [46], every inﬁnite-dimensional space is a union of a σ -porous set and of an Aronszajn null set. For super-reﬂexive spaces a much stronger decomposition statement (using sets that are sometimes called strongly very porous) can be found in [33]. (Both these statements have been used to give examples concerning nondifferentiability, and so we will meet them again.) These constructions are based on the trivial observation that a Borel set which meets every k-dimensional afﬁne subspace (where k is ﬁxed) in a Lebesgue null set is necessarily Aronszajn null. Our next classes of exceptional sets consist of very small sets indeed (at least compared to the previous classes). Their importance stems from the fact that they have been used in the only results in differentiability in which we have complete characterizations (see

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Section 4.1.1). They are the σ -ideals of sets that can be covered by countably many k codimensional Lipschitz, respectively δ-convex, hypersurfaces. The k codimensional Lipschitz, respectively δ-convex, hypersurfaces are deﬁned as sets of the form {w + φ(w): w ∈ W }, where X is a direct sum X = W ⊕ U and φ : W → U is Lipschitz, respectively δ-convex. Recall that a Lipschitz mapping φ : X → Y is δ-convex if for every y ∗ ∈ Y ∗ the composition y ∗ ◦ φ may be expressed as a difference of two continuous convex functions. These σ -ideals are Borel and are properly contained in all the previous σ -ideals; they are becoming smaller with k increasing, and those deﬁned using δ-convex hypersurfaces are properly contained in the σ -ideals deﬁned via Lipschitz hypersurfaces. In case k = dim(X), both notions give just countable sets. The preﬁx ‘k-codimensional’ will be omitted if k = 1. Finally, we meet a σ -ideal combining measure and category in a useful (and non-trivial way). It has been used in [31] to obtain ﬁrst (and so far only) inﬁnite-dimensional results on Fréchet differentiability of Lipschitz functions almost everywhere. The basic idea is to consider a suitable completely metrizable space of measures differentiable in direction of a spanning sequence of vector-ﬁelds (which may depend on the measure) and deﬁne that a set is null if it is null for residually many of these measures. This, however, appears to be technically complicated, and so the deﬁnition uses a parametric approach (in which the condition of differentiability of measures is not so apparent). Let Σ = [0, 1]N be endowed with the product topology and the product Lebesgue measure μ, and let Γ (X) denote the space of continuous maps γ : Σ → X having continuous partial derivatives. We equip Γ (X) with the topology of uniform convergence of the maps and their partial derivatives and deﬁne a Borel set E ⊂ X to be Γ -null if μ t ∈ Σ: γ (t) ∈ E = 0 for residually many γ ∈ Γ (X). Note that, since Γ (X) is completely metrizable, the family of null sets forms a proper σ -ideal of subsets of X. A simpler but less useful variant of the notion of Γ -null sets may be obtained by replacing Σ by [0, 1]k . These notions of Γ -null sets can be viewed as a special case of a general scheme deﬁning negligibility of a set A in the space by requiring that it is negligible inside all except negligibly many elements of the hyperspace (space of subsets, space of measures). For example, the set P(X) of Borel probability measures on a complete separable metric space X considered as a subset of the dual to the space of bounded continuous functions with the weak∗ topology is completely metrizable, so one can try to consider as negligible those sets A ⊂ X that satisfy μ(A) = 0 for residually many μ ∈ P(X). Another, purely topological, example can be obtained by using the space K(X) of non-empty compact subsets of X equipped with the Hausdorff metric and deﬁning negligibility of a set A ⊂ X by requiring that C ∩ A is of the ﬁrst category in C for residually many C ∈ K(X). The usefulness of the notions introduced in these two examples is somewhat diminished by the easily seen fact that for Borel sets they are both equivalent to the notion of the sets of the ﬁrst category. (In this connection, one may note that, without any condition on a set A ⊂ X, A is of the ﬁrst category if and only if C ∩ A = ∅ for residually many C ∈ K(X). This follows immediately by noting that the union of the compacts belonging to a Gδ subset of K(X) is a Suslin set.) Nevertheless, the negligibility notion from the second example (in possibly non-separable spaces) and its variants have been successfully used in [37] to

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show existence of points of Gâteaux differentiability of continuous convex functions on certain non-separable Banach spaces (where the set of points of differentiability need not be Borel).

4. Differentiability We will consider the question of existence of points of Fréchet and Gâteaux differentiability of continuous convex functions and of Lipschitz mappings between Banach spaces. For deﬁnitions, basic information and, in particular, for the concept of Radon–Nikodým property of a Banach space X and the result that all Lipschitz mappings of ﬁnite-dimensional spaces into X are differentiable (at least at one point or, equivalently, almost everywhere) see Section 7 of [21].

4.1. Convex functions The study of differentiability problems for continuous convex functions is greatly simpliﬁed by several facts (cf. [21]): the sets of points of Gâteaux as well as of Fréchet differentiability are Gδ (the latter even in non-separable spaces), if one term of a sum of such functions is (Gâteaux or Fréchet) non-differentiable at x, then the sum is non-differentiable at x, at every point the one-sided directional derivatives exist and form a convex continuous and positively 1-homogeneous function of the directions, hence Gâteaux differentiability at x is equivalent to the requirement that f (x + th) + f (x − th) − 2f (x) = o(t) as t → 0 and Fréchet differentiability is equivalent to the requirement that f (x + h) + f (x − h) − 2f (x) = o(h) as h → 0, the subdifferential ∂f (x) = {x ∗ ∈ X∗ : x ∗ (u) f (x + u) − f (x) for all u ∈ X} is non-empty and differentiability has a simple description as a property of the subdifferential: f is Gâteaux differentiable at x if and only if its subdifferential at x is a singleton and f is Fréchet differentiable at x if and only if the multi-valued mapping y → ∂f (y) is single valued and norm continuous at x, i.e., for every ε > 0 there is δ > 0 such that ∂f (y) ⊂ B(x ∗ , ε) for all y ∈ B(x, δ) and x ∗ ∈ ∂f (x). A number of results on convex functions has been generalized to statements about monotone operators: a mapping T of a set E ⊂ X to the family of non-empty subsets of X∗ is called monotone if (y ∗ − x ∗ )(y − x) 0 for all x, y ∈ E, x ∗ ∈ T (x), and y ∗ ∈ T (y). (Note that some authors require E = X but allow T (x) to be empty.) Basic properties and references to situations in which they play a signiﬁcant role may be found in [41]. Important examples of monotone operators are provided by subdifferentials of continuous convex functions. Standard results on convex functions have their counterpart in the theory of monotone operators. For example, the simple but useful fact that continuous convex functions are locally Lipschitz may be obtained as a corollary of the fact that monotone operators on open sets are locally bounded. Another direction in which the subdifferential approach may be understood is via selection theorems. It is easy to see that the mapping T (x) = ∂f (x) is weak∗ upper semicontinuous (i.e., the set {x: T (x) ⊂ G} are open for every weak∗ open G ⊂ X∗ ) and has non-empty weak∗ compact values (we abbreviate both these properties of T by saying

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that T is weak∗ usco). If ξ : X → X∗ is a selection for T (i.e., ξ(x) ∈ T (x) for all x) which is norm-to-weak∗ continuous at x, then f is Gâteaux differentiable at x; if it is even norm-to-norm continuous, then f is Fréchet differentiable at x. Since the T given by the subdifferential is locally bounded, one may obtain differentiability results from purely topological statements on the existence of selections of weak∗ usco mappings of, say, topological spaces having the Baire property into the unit ball of X∗ . This approach gives also results for monotone operators, since maximal monotone operators an open sets are (locally bounded and) weak∗ usco. The price paid for the higher generality of the approach is weaker information about the size of the set of points of differentiability; in the selection approach one may hardly expect stronger exceptional sets than those of the ﬁrst category. 4.1.1. Gâteaux differentiability of convex functions The remarkable results of [59] give a complete description of the size of sets of points of Gâteaux non-differentiability of continuous convex functions. They considerably strengthen a series of previous inﬁnitedimensional results starting with Mazur as well as more detailed previous results in the ﬁnite-dimensional case. The set of points of Gâteaux non-differentiability of an arbitrary continuous convex function on a separable Banach space X can be covered by countably many δ-convex hypersurfaces. Conversely, for every set E contained in countably many δ-convex hypersurfaces there is a continuous convex function on X which is Gâteaux nondifferentiable at every point of E. Note that in case dim(X) = 1 we recover the classical statement that the sets of points of non-differentiability of convex functions on R are exactly countable sets. The natural generalization of the question answered in the previous paragraph is the study of those points at which the subdifferential is large. Again, [59] gives a complete answer, which reduces to the previous statement if k = 1: for an arbitrary continuous convex function on a separable Banach space the set of those x at which the dimension of the afﬁne span of the subdifferential is at least k can be covered by countably many k-codimensional δ-convex hypersurfaces. Conversely, for every set E contained in countably many k-codimensional δ-convex hypersurfaces there is a continuous convex function on X such that for every x ∈ E the dimension of the afﬁne span of ∂f (x) is at least k. To indicate the way in which this is proved, let N denote the set of points x for which the dimension of the afﬁne span of ∂f (x) is at least k. Given a k-dimensional subspace U of X, u∗ ∈ U ∗ , and ε > 0, the set B of those x ∈ X for which there is x ∗ ∈ ∂f (x) extending u∗ such that x ∗ < 1/ε and f (x + h) − f (x) x ∗ (h) + εh for all h ∈ U is a subset of N ; moreover, by separability, N is a countable union of such sets B. It is therefore enough to consider one such set B and show that it is covered by a k-codimensional δ-convex hypersurface (which still needs work). To show the converse it sufﬁces to consider the case of one hypersurface; then one can deﬁne the required convex function using the convex functions describing the hypersurface. The above discussion of Gâteaux differentiability may be modiﬁed to show that for any monotone operator T on an open subset of a separable Banach space the set of points at which T (x) contains at least k afﬁnely independent elements can be covered by countably many k-codimensional Lipschitz hypersurfaces. Applying this fact to the subdifferential of a convex function gives, however, only a weaker version of the above results.

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Let us brieﬂy mention some points of the non-separable theory; for more information see [16,18] or [41]. X is said to be a weak Asplund space if every continuous convex function on X is Gâteaux differentiable on a residual set and a Gâteaux differentiability space, if every continuous convex function on X is Gâteaux differentiable at least at one point (or, equivalently, on a dense set). An interesting class of weak Asplund spaces which includes all separable spaces as well as spaces admitting a Gâteaux smooth norm (or just a Gâteaux differentiable Lipschitz bump function) is formed by those Banach spaces whose dual unit ball B ∗ satisﬁes the condition (of nature similar to topological descriptions of Radon–Nikodým property of duals as in 4.1.2(e)) that inthe following ‘fragmentability’ game the second player has a strategy guaranteeing that ∞ k=1 Fk ∩ Gk is a singleton: the ﬁrst player starts by choosing a non-empty weak∗ closed subset F1 of B ∗ , then the second player chooses a weak∗ open set G1 such that F1 ∩ G1 = ∅, then the ﬁrst player chooses a weak∗ closed subset F2 of F1 such that F2 ∩ G1 = ∅, then the second player chooses a weak∗ open set G2 such that F2 ∩ G2 = ∅, etc. This implies that B ∗ belongs to the so called Stegall’s class, i.e., has the property that any weak∗ usco mapping of a Hausdorff topological space Z to B ∗ has a selection which is weak∗ continuous on a residual set. With our deﬁnition, it can be shown by considering a minimal usco mapping of Z to B ∗ and showing that it is single-valued on a residual set with the help of the Banach–Mazur game in Z. (No use of the fact that B ∗ is a ball has been made; the argument works in any compact Hausdorff space satisfying the fragmentability condition.) Recent papers of Kalenda [23,24] and of Kenderov [26] show that these classes are different and do not coincide with weak Asplund spaces. Very recently Moors and Somasundaram [37] used the hyperspace based notions of negligibility mentioned at the end of Section 3 to answer the key open problem of the theory by producing a Gâteaux differentiability space that is not weak Asplund. 4.1.2. Fréchet differentiability of convex functions We have seen in [21] which separable spaces have the property that every continuous convex function on them is Fréchet differentiable on a dense Gδ set; they are precisely those whose dual is separable. Other characterizations follow from this, and similarly satisfactory results hold in non-separable setting. Banach spaces in which continuous convex functions have points of Fréchet differentiability are called Asplund spaces. They are characterized by any of the following equivalent properties: (a) Every continuous convex function on X has a point of Fréchet differentiability. (b) Every continuous convex function on X is Fréchet differentiable on a dense Gδ set. (c) The dual of every separable subspace of X is separable. (d) X∗ has the RNP. (e) For every non-empty bounded set E ⊂ X∗ and every ε > 0 there is a weak∗ open set S meeting E such that S ∩ E has diameter less than ε. (f) For every non-empty weak∗ compact convex E ⊂ X∗ and every ε > 0 there are u ∈ X and δ > 0 such that the weak∗ slice S(E, u, δ) = {x ∗ ∈ E: x ∗ (u) supy ∗ ∈E y ∗ (u) − δ} has diameter less than ε. The equivalence of (a) and (b) follows from the set of points of Fréchet differentiability being Gδ ; see [21]. From the negation of one of the statements (c)–(e) one may prove the negation of (f); if then E is a non-empty weak∗ compact convex set without small weak∗ slices, the function f (x) = supx ∗ ∈E x ∗ (x) is nowhere Fréchet differentiable, and so is the sum of the original norm with f (x) and f (−x). In this way, we even see that every non-Asplund space admits an equivalent norm satisfying lim suph→0 (x + h + x − h − 2x)/h > ε for

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some ε > 0; such norms are sometimes called ε-rough. To prove, say, (e) ⇒ (b), one may use (e) to deduce that every weak∗ usco mapping of a Hausdorff space Z to B ∗ has a selection norm continuous on a residual set; this is a purely topological statement whose proof and use is similar to what was described in 4.1.1 for Gâteaux differentiability. Another possibility is to prove the statement in the separable case only (see later), and use the method of separable reduction. (For a general approach to this method see [10].) The question of describing the size of the set of points of Fréchet differentiability of continuous convex functions is not yet fully understood even in the separable case. However, we have: the set of points of Fréchet non-differentiability of any continuous convex functions on a separable Asplund space is σ -porous. To see this, let xk∗ be dense in the dual and let Fk,l be the set of those x in whose neighbourhood the Lipschitz constant of f does not exceed l and for which there is x ∗ ∈ ∂f (x) such that xk∗ − x ∗ < 1/ l and there are arbitrarily small h such that f (x + h) − f (x) − x ∗ (x) > 4h/ l. Then for any y ∈ B(x + h, h/ l 2 ) and any y ∗ ∈ ∂f (y) we have y ∗ (h) f (y) − f (y − h) / Fk,l . Hence f (x + h) − f (x) − 2h/ l xk∗ (h) + h/ l; so xk∗ − y ∗ 1/ l and y ∈ B(x + h, h/ l 2 ) ∩ Fk,l = ∅, which shows that Fk,l is porous. The proof is ﬁnished by observing that the set of points of Fréchet non-differentiability is covered by the union of Fk,l . The above argument may be modiﬁed to show that for every monotone operator on a Banach space with a separable dual there is a porous set outside of which the operator is single-valued and norm-to-norm upper semi-continuous. In both these results the porosity may be strengthened (to so-called σ -cone porosity, in which the holes, instead of balls, may be cones); for details see [41]. Although these notions are still some way from a description of the size of sets of Fréchet non-differentiability of convex functions, there is a (very strong) porosity condition which enables the construction of badly differentiable functions: if X is uniformly convex, rn 5 0 and E ⊂ X is such that for every x ∈ E and every λ > 1 there are zn such that zn − x < λrn and B(zn , rn /λ) ∩ E = ∅, then there is a continuous convex function on X which is Fréchet non-differentiable at every point of E. The basic idea in construction of such a function is (assuming that E ⊂ B(0, 1)) to consider the supremum of all afﬁne function majorized by the restriction of a uniformly convex function to E ∪ (X \ B(0, 2)). A similar construction (together with adding the functions constructed for a sequence of such sets E) was used by Matoušková [33] to show that on every super-reﬂexive separable space there is an equivalent norm whose set of points of Fréchet differentiability is Aronszajn null. This example, in particular, shows that the results mentioned so far are not strong enough to produce, given a convex continuous function g on a separable Asplund space X and a Lipschitz mapping f of X to an RNP space Y , one point x ∈ X at which g would be Fréchet differentiable and f Gâteaux differentiable. This question was answered by [31]: the set of points of Fréchet non-differentiability of any convex continuous function on a separable Asplund space is Γ -null. (See 4.2.1 for results on Gâteaux differentiability implicitly alluded to here, and 4.2.2 for a more general version of this statement.) Note that the incompatibility of Γ -null sets and σ -porous sets (which probably carries over to σ -cone porous sets) makes it is difﬁcult to conjecture how a characterization of sets of Fréchet non-differentiability may look like.

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4.2. Lipschitz functions One of the problems guiding the development of differentiability results for Lipschitz mappings is the Lipschitz isomorphism problem: if X, Y are Lipschitz isomorphic (i.e., if there is a bijection f : X → Y such that both f and f −1 are Lipschitz) are they linearly isomorphic? Much of the theory described below can be, and has been, successfully used to obtain partial answers. The basic idea is that the derivative of f should provide such an isomorphism. For this to work, one has to assume additional properties of X, Y such as the RNP (or just reﬂexivity, or even just super-reﬂexivity), since otherwise f may be nowhere differentiable. However, at the present time any direct use of this program comes to the obstacle caused by the open problem whether the Gâteaux derivative of a Lipschitz isomorphism of 2 onto itself is surjective at least at one point. Another approach to the Lipschitz isomorphism problem is based on the observation that a Gâteaux derivative of a Lipschitz isomorphism f of X to Y is surjective provided that all compositions y ∗ ◦ f are Fréchet differentiable for all y ∗ ∈ Y ∗ . In fact, a dense set of y ∗ sufﬁces, which gives a good reason why we care so much about the problem of ﬁnding a common point of Gâteaux differentiability of a Lipschitz mapping and of Fréchet differentiability of countably many real-valued Lipschitz functions. In these arguments Fréchet derivative may be replaced by almost Fréchet derivative (see 4.2.4). The weakening of the concept of Fréchet derivative can be pushed even further, to the so called afﬁne approximation property of [4]; we will not consider these results here, but mention that in the super-reﬂexive case most applications of almost Fréchet differentiability (see 4.2.4) to the Lipschitz isomorphism problem may be also obtained with the help of the uniform version of this property. We should recall that differentiability results for general Lipschitz functions cannot be obtained using sets of the ﬁrst category, as there are Lipschitz functions f : R → R which are differentiable only on ﬁrst category sets. In fact, on the real line these results cannot be obtained by any means weaker than the Lebesgue measure, since for every set N ⊂ R of Lebesgue measure zero there is a Lipschitz f : R → R which in non-differentiable at all points of N ; moreover, by [57] the sets of non-differentiability of Lipschitz functions R → R are characterized as Gδσ sets of Lebesgue measure zero. 4.2.1. Gâteaux differentiability of Lipschitz functions The question how small are the sets of points of Gâteaux non-differentiability of Lipschitz functions does not have a complete answer yet; it is not even known if in Rn (n 3) the σ -ideal generated by the sets of non-differentiability of real-valued Lipschitz functions coincides with the Lebesgue null sets. Nevertheless, the fact that locally Lipschitz mappings of separable spaces into RNP spaces are Gâteaux differentiable outside Haar null sets mentioned in [21] or the following stronger result are sufﬁcient for a number of purposes: every locally Lipschitz mapping of a separable space into an RNP space is Gâteaux differentiable outside an Aronszajn null set. There are several remarkably simple proofs of this statement: the basic idea is that, if f : X → Y and un ∈ X have a dense span, the sets En of those x ∈ X for which the directional derivative f (x, un ) = limt →0 (f (x + tun ) − f (x))/t does not exist are Borel and, by the RNP of Y , the intersection of En with any line in the direction un is of onedimensional Lebesgue measure zero. It remains to show that the set of points of Gâteaux

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non-differentiability at which directional derivatives exist in a spanning set of directions is Aronszajn null; this set is reasonably well understood, since: if f is a locally Lipschitz mapping of a separable space X to Y , then the set of those x ∈ X at which f has the directional derivative in a spanning set of directions but is not Gâteaux differentiable is σ -directionally porous. To see this, one shows, for every u, v ∈ X, y, z ∈ Y and ε, δ > 0, the directional porosity of the set E of those x0 ∈ X such that Lip(f, B(x0 , δ)) < 1/ε, f (x0 + tu) − f (x0 ) − ty + f (x0 + tv) − f (x0 ) − tz ε|t| for |t| < δ and there are arbitrarily small |s| such that f (x0 + s(u + v)) − f (x0 ) − s(y + z) > 4ε|s|: if |s| is small and x − (x0 + su) < ε2 |s|, then f (x + sv) − f (x) − sz f (x0 + s(u + v)) − f (x0 ) − s(y + z) − f (x + sv) − f (x0 + s(u + v)) − f (x0 + su) − f (x0 ) − sy − / E. The set in question is covered by countably many f (x0 + su) − f (x) > ε|s|, so x ∈ such sets E since it sufﬁces to consider u, v from a dense countable subset of X, y, z from a dense countable subset of the span of f (X) and rational ε, δ. Intriguing questions are obtained when one attempts to use these results to answer the Lipschitz isomorphism problem. The Gâteaux derivative of a Lipschitz isomorphism f of a separable Banach space X onto an RNP space Y , whenever it exists, is a linear isomorphism onto a closed subspace of Y (so X has the RNP as well). This subspace is complemented if, e.g., Y is reﬂexive (see [5]). Nevertheless, the following key problem is still open: if f is a Lipschitz isomorphism of 2 onto itself, is there a point at which its Gâteaux derivative is a linear isomorphism of 2 onto itself? One may hope that the Gâteaux derivative of any Lipschitz isomorphism between RNP spaces is, at least at one point, a linear isomorphism between them; this more general version of the problem is open as well. Since it is easy to see that if a Lipschitz isomorphism f of X onto Y is Gâteaux differentiable at x and f −1 at f (x), then Df (x) is a linear isomorphism of X onto Y , a positive answer would be obtained if the null sets with respect to which one has the differentiability theorem were invariant under Lipschitz isomorphisms. We have, however, pointed out in Section 3 that this is not the case for Haar null nor for Aronszajn null sets. Other problems on invariance of null sets under Lipschitz isomorphisms treated in Section 3 have been also motivated by the Lipschitz isomorphism problem. For any given notion of null sets, the worst examples would be of the situation when a complement of a null set is mapped onto a set of Gâteaux non-differentiability of some Lipschitz function; such examples are not known even for Haar null sets (and so also not for Aronszajn null sets). Curiously enough, if the Lipschitz isomorphism f : X → Y has all one-sided directional derivatives, then the image of the set at which g : X → Z (with RNP Z) is not Gâteaux differentiable is even Aronszajn null: the image of the set of points at which g is non-differentiable at some direction is contained in the set of non-differentiability of g ◦ f −1 , and the remaining part of the set of non-differentiability points of g is σ -directionally porous, so its f image is also σ -directionally porous, since Lipschitz isomorphisms having one-sided directional derivatives map σ -directionally porous sets to σ -directionally porous sets. The Lipschitz isomorphism problem may well require a strengthening of the above results on Gâteaux differentiability. This motivates the quest for ﬁnding smaller σ -ideals of sets for which the differentiability statement still holds (and is genuinely stronger than the use of Aronszajn null sets). The non-linear concepts of Aronszajn null sets brieﬂy discussed in Section 3 provide such σ -ideals. From these results (or directly) is is also easy to see that every Lipschitz function f from a separable Banach space X to an RNP space

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Y is Gâteaux differentiable Γ -almost everywhere. Note again that for these σ -ideals the problem of invariance under Lipschitz isomorphisms is open. 4.2.2. Fréchet differentiability of Lipschitz functions Until recently, the only general result on Fréchet differentiability of Lipschitz mappings, except the case of ﬁnitedimensional domain where the concepts of Gâteaux and Fréchet differentiability coincide, was that every Lipschitz mapping f of an Asplund space X to R has points of Fréchet differentiability; a small generalization (for X separable) assumes only the weak∗ closure of the set of of Gâteaux derivatives of f norm separable. It is immediate to deduce from this that the set of points of Fréchet differentiability must be uncountable in every non-empty open set, and stronger information on the size of this set can be obtained by use of the mean value estimate (see 4.2.3). The original proof of the Fréchet differentiability result is rather involved [44]; a simpler (but not simple) proof from [30] is based on the following ideas (we assume X separable): denote by Df (E) the set of all Gâteaux derivatives attained at points of E. Let E1 be a ball of radius one, W1 = Df (E1 ) and let u1 ∈ X be such that the slice S(W1 , u1 , δ1 ) has a small diameter. One can show that there is η1 > 0 such that whenever Df (x) ∈ S(W1 , u1 , η1 ), then lim suph→0 |f (x + h) − f (x) − Df (x)(h)|/h is small. Then one deﬁnes E2 as the set of those x from the intersection of E1 and a ball with radius 1/2 at which Df (x)(u1 ) is large and the increments in the direction u1 are uniformly controlled (the real difﬁculty comes at this point; keeping this control is enabled by an involved estimate of behaviour of derivatives in the plane), and we continue in a similar way requiring now that u2 is close to u1 , etc. The limit of the sequence xn ∈ En is the required point. Even from this rough description it should be clear that this approach shows that every slice S(Df (X), u, δ) (u ∈ X) of the set of Gâteaux derivatives of f contains a Fréchet derivative. One of the main difﬁculties in proving Fréchet differentiability results, say, for mappings of 2 to ﬁnite-dimensional spaces is that the analogous slicing statement is false: by a (complicated) example of [46] there is a Lipschitz mapping f = (f1 , f2 , f3 ) : 2 → R3 such that Df1 (e1 ) + Df2 (e2 ) + Df3 (e3 ) = 0 at every point of Fréchet differentiability of f , but not at every point of Gâteaux differentiability. (Except for understandable misprints, this example, by a computer quirk, uses the meaningless symbol “ ” for (π0 z/rm ).) Any attempt to prove Fréchet differentiability almost everywhere (or even existence of a common point of differentiability of ﬁnitely many real-valued functions) is greatly hampered by the fact that there may exist slices of the set of Gâteaux derivatives of f containing no Fréchet derivative. This, however, cannot happen for convex functions. The reason behind this is that they are regular in the following sense: a mapping f : X → Y is called regular at a point x if for every v ∈ X for which the directional derivative f (x, v) exists, (x+t u) limt →0 f (x+t (u+v))−f = f (x, v) uniformly in u with u 1. The key statement on t Fréchet differentiability of Lipschitz mappings with respect to Γ -null sets says (see [31]): if L is a norm separable subspace of the space of linear operators between separable Banach spaces X and Y , then every Lipschitz mapping f : X → Y is Fréchet differentiable at Γ -almost every point of the set at which it is regular, Gâteaux differentiable and its Gâteaux derivative belongs to L. The proof is hard and draws on much of what has been done before. The basic new ingredient comes from the classical descriptive set theory: assuming, for simplicity, that f is Gâteaux differentiable Γ almost everywhere, we observe

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that the mapping γ → Df ◦ γ , being a Borel measurable mapping between complete separable metric spaces Γ (X) and L1 (Σ, L), becomes continuous when restricted to a suitable residual set. To return to general Lipschitz mappings, we observe that the sets of points of their irregularity are σ -porous. Hence every Lipschitz mapping of f : X → R is Fréchet differentiable Γ -almost everywhere if and only if every σ -porous set is Γ -null. The condition of σ -porous sets being null does not hold in 2 (as we will see in 4.2.3, not all Lipschitz f : 2 → R are Fréchet differentiable Γ -almost everywhere) but it can be proved in spaces whose structure is similar to c0 (for example, for spaces containing an asymptotically c0 sequence of ﬁnite co-dimensional subspaces). The basic method of avoiding porous sets is to modify a given γ ∈ Γ (X) close to a point at which it belongs to a given porous set so that it passes through a hole. Unfortunately, the resulting sequence of so modiﬁed γn ∈ Γ (X) may not converge (in the space Γ (X)). However, in the presence of a c0 structure we can make the modiﬁcation on disjoint sets of coordinates and so achieve the convergence. These argument then give that if X is a subspace of c0 , or a space C(K) with K countable compact, or the Tsirelson space, then all the σ -porous subsets of X are Γ -null; hence all real-valued Lipschitz functions on these spaces are Fréchet differentiable Γ -almost everywhere. In fact, if X is a subspace of c0 , or C(K) with K countable compact, then the space of bounded linear operators from X to any RNP space Y is separable, and so every Lipschitz mapping between such spaces is Fréchet differentiable Γ -almost everywhere. 4.2.3. Mean value estimates One of the important applications of derivatives or their generalizations is their use to estimate the increment of a function. The model statement is Lebesgue’s variant of the fundamental theorem of calculus saying that for a real-valued b Lipschitz function f of one real variable f (b) − f (a) = a f (t) dt and its corollary, the mean value estimate, that for every ε > 0 there is t ∈ [a, b] such that f (t)(b − a) > f (b) − f (a) − ε. For real-valued Lipschitz functions on a Banach space X one cannot expect that a point of differentiability can be found on the segment [a, b], and so the mean value estimate either uses a point of differentiability close to [a, b] or replaces the derivative by its generalization (this approach will not be used here). The mean value estimate for Gâteaux derivatives follows immediately from the fact that every Haar null set is null with respect to linear measure on a dense set of lines. In fact this gives a stronger statement: if X is separable, G ⊂ X is open, N ⊂ X is Haar null, and f : G → R is Lipschitz, then for every segment [a, b] ⊂ G and every ε > 0 there is x ∈ G \ N such that Df (x)(b − a) > f (b) − f (a) − ε. Since no almost everywhere result is known for Fréchet derivatives, the mean value estimate for them is proved by following more carefully the construction of points of differentiability: if X is an Asplund space, G ⊂ X is open, and f : G → R is Lipschitz, then for every segment [a, b] ⊂ G and every ε > 0 there is x ∈ G at which f is Fréchet differentiable such that Df (x)(b − a) > f (b) − f (a) − ε. (As in the existence result, it sufﬁces to assume that the weak∗ closure of the set of Gâteaux derivatives of f is norm separable.) The mean value estimate may be used to show that the set of points of Fréchet differentiability of these mappings cannot be too small: if any of its projections on R were not of full outer Lebesgue measure, we would ﬁnd a non-constant everywhere differentiable Lipschitz function ϕ on R having derivative zero at the projection of every point of Fréchet differentiability of f ; adding to

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f a large multiple of the composition of ϕ and the projection would produce a function violating the mean value estimate. If X is separable, this shows that any one-dimensional projection of the set of points of Fréchet differentiability is of full Lebesgue measure (since it is measurable). It is not known if an analogous statement holds also for two-dimensional projections. A higher-dimensional version of the mean value statement for Fréchet derivative of a mapping f of X to a ﬁnite-dimensional space may be understood as the statement that every slice of the set of Gâteaux derivatives of f contains a Fréchet derivative. This holds for mappings which are Fréchet differentiable Γ -almost everywhere. (Basically, one considers a γ representing a small ﬁnite-dimensional parallelepiped on which the mapping is well approximated by a linear mapping belonging to the slice; for a slight modiﬁcation of γ one gets Fréchet differentiability μ-almost everywhere and, by the divergence theorem, the mean of the derivative changes only as little as we wish.) Because of this and of the example of [46] (which was already mentioned above), we see that Fréchet differentiability Γ -almost everywhere is false for real-valued Lipschitz functions on 2 . 4.2.4. Almost Fréchet derivative It has been already mentioned that for the Lipschitz isomorphism problem notions of derivatives weaker than Fréchet derivative may be pertinent. A function f : X → Y is called almost Fréchet differentiable is for every ε > 0 there are x ∈ X and a bounded linear operator T (both x and T may depend on ε) such that lim supu→0 f (x + u) − f (x) − T (u)/u < ε. In [29] these derivatives were shown to exist for mappings of supper-reﬂexive spaces to ﬁnite-dimensional spaces (by a rather involved proof). This result was extended in [22], with a more transparent proof, to the case of asymptotically uniformly smooth spaces; this paper should be consulted for details and applications. 4.2.5. Weak∗ derivative For a Lipschitz mapping f of a separable Banach space X to the dual of a separable space Y one deﬁnes the weak∗ directional derivative of f at x in (x) direction u as the weak∗ limit, as t → 0, of f (x+t u)−f . The weak∗ Gâteaux different tiability of f at x is deﬁned by requiring that this mapping be linear in u. The existence results for Gâteaux derivatives of Lipschitz functions hold also in this setting (and do not need any RNP requirement). Of course, some of the properties of Gâteaux derivatives are lost; in particular, the weak∗ Gâteaux derivative of a Lipschitz isomorphism may well be zero at some points. However, mean value estimates still hold, so these derivatives are not trivial; and, starting from [32] and [19] have been successfully used to study the Lipschitz isomorphism problems for spaces without Radon–Nikodým property. 4.2.6. Metric derivative The standard example of a nowhere differentiable Lipschitz mapping of (0, 1) to L1 (0, 1), given by f : x → χ(0,x) where χE denotes the indicator function of the set E, is an isometry. This is not just a chance, since the one-dimensional case of the following result due to Kirchheim [27] says that every Lipschitz mapping of (0, 1) to a metric space locally (near to a.e. point) multiplies the distance by a constant as if it were differentiable (no RNP type condition on of the range is needed). If f is a Lipschitz mapping of an open subset of Rn to a metric space, then for a.e. x ∈ Rn there is a seminorm · x on Rn such that limt →0 -(x + tu, x + tv)/t = u − vx for all u, v ∈ Rn . (For an

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application see 1.1.) A new proof of this statement, which relates it to differentiability of Banach space valued mappings has been found in [1]. We may assume that the target is the dual of a separable Banach space. Then f is weak∗ differentiable almost everywhere, and it is natural to assume that ux = Df (x)(u) is the required seminorm; this can in fact be shown by decomposing Rk into countably many sets in which the weak∗ derivative does not oscillate much (the oscillation is measured in a metric metrizing the weak∗ topology of a ball in E) and using the density theorem together with mean value estimates.

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CHAPTER 36

The Banach Spaces C(K) Haskell P. Rosenthal∗ Department of Mathematics, The University of Texas at Austin, Austin, TX 78712-1082, USA E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. The isomorphic classiﬁcation of separable C(K)-spaces 3. Some Banach space properties of separable C(K)-spaces 4. Operators on C(K)-spaces . . . . . . . . . . . . . . . . . 5. The complemented subspace problem . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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∗ Research partially supported by NSF Grant DMS-0070547.

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1. Introduction A C(K)-space is just the space of scalar-valued continuous functions on a compact Hausdorff space K. We focus here mainly on the case where K is metrizable, i.e., the case of separable C(K)-spaces. Our main aim is to present the most striking discoveries about the Banach space structure of C(K)-spaces, and at the same time to describe the beautiful, deep intuitions which underlie these discoveries. At times, we go to some length to describe the form and picture of an argument, without giving the full technical discussion. We have also chosen to present proofs which seem the most illuminating, in favor of more advanced and sophisticated but (to us) less intuitive arguments. The following is a summary of our exposition. Section 2 deals with the by now classical isomorphic classiﬁcation of the separable C(K)-spaces, dating from the 50’s and 60’s. It begins with Milutin’s remarkable discovery: C(K) is isomorphic to C([0, 1]) if K is an uncountable compact metric space. We give a fully detailed proof, modulo some standard basic facts (summarized in Lemma 2.5), which follows an argument due to Ditor. This yields that every separable C(K)-space is isometric to a contractively complemented subspace of C(D), D the Cantor discontinuum (Theorem 2.4), through a natural inverse limit argument, given in Lemma 2.11 below. The way inverse limits work (in the metrizable setting) is given in Lemma 2.12, and Theorem 2.4 is deduced after this. The isomorphic classiﬁcation of the C(K)-spaces with separable duals, due to Bessaga and Pełczy´nski, occupies the balance of this section. Their remarkα able result: the spaces C(ωω +) form a complete set of representatives of the isomorphism classes, over all countable ordinals (Theorem 2.14). We do give a detailed proof that C(K) is isomorphic to one of these spaces, for all countable compact K (of course, we deal only with inﬁnite-dimensional C(K)-spaces here). This is achieved through Theorem 2.24 and Lemma 2.26. We do not give the full proof that these spaces are all isomorphically distinct, although we spend considerable time discussing the fundamental invariant which accomα plishes this, the Szlenk index, and the remarkable result of Samuel: Sz(C(ωω +)) = ωα+1 for all countable ordinals α (Theorem 2.15). We give a variation of Szlenk’s original formulation following 2.15, and show it is essentially the same as his in Proposition 2.17. We then summarize the invariant properties of this index in Proposition 2.18, and give the relα atively easy proof that Sz(C(ωω +)) ωα+1 in Corollary 2.21. We also show in Section 2 how the entire family of spaces C(α+) (up to algebraic isometry) arises from a natural Banach space construction: simply start with c0 , then take the smallest family of commutative C ∗ -algebras containing this, and closed under unitizations and c0 -sums. (This is the family (Yα )1α<ω1 , given at the beginning of part B of Section 1.) The isomorphic description, however, is achieved through taking tensor products at successive ordinals and c0 -sums and unitizations at limit ordinals. (This is the transﬁnite family (Xα )α<ω1 given following Deﬁnition 2.22.) Section 3 deals with three unrelated structural properties. The ﬁrst, due to Pełczy´nski, is that every separable C(K)-space X is weakly injective, that is, any isomorph of X in a separable Banach space Y , contains a subspace isomorphic to X and complemented in Y (Theorem 3.1). The second one, due to Bessaga and Pełczy´nski, is that every C(K)space with separable dual is c0 -saturated (Proposition 3.6). This follows quickly from our ﬁrst transﬁnite description of these spaces mentioned above. It is not a difﬁcult result, but

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is certainly fundamental for the structure of these spaces. We also brieﬂy note the rather long standing open problem: is every subspace of a quotient of C(α+) c0 -saturated, for countable ordinals α? We note also: it is unknown if 2 is isomorphic to a subspace of a quotient of C(ωω +). The third result in Section 3 deals with Amir’s theorem: C(ωω +) fails to be separably injective. We also give a fully detailed proof of Milutin’s classical discovery; the Cantor map of {0, 1}N onto [0, 1] induces an uncomplemented isometric embedding of C([0, 1]) in C(D). We give a uniﬁed account of both of these results through the space rcl([0, 1]) of functions on [0, 1] which are right continuous with left limits (also called cadlag by French probabilists). For any countable compact subset K of [0, 1], we let rcl(K) be the analogous function space, just deﬁned on K. If K has enough two-sided cluster points, then C(K) is uncomplemented in rcl(K). Similarly, if D is any countable dense subset of [0, 1], then C([0, 1]) is uncomplemented in rcl([0, 1], D). (The latter is simply the space of all functions on [0, 1] continuous at all x ∈ / D, right continuous with left limits at all x ∈ D.) These results are proved in Theorem 3.14. Section 3 concludes with the proof that if D is the dyadic rationals, then the embedding of C([0, 1]) into C(D) via the Cantor map is essentially just the identity injection of C([0, 1]) in rcl([0, 1], D), and so is uncomplemented (Proposition 3.18). Section 4 deals mainly with several deep ﬁxing results for operators on C(K)-spaces, all of which heavily bear on the famous long standing problem discussed in the ﬁnal section of this article. An operator T between Banach spaces is said to ﬁx a Banach space Z if there is an isometric copy Z of Z in the domain with T |Z an isomorphism. In the present context, it turns out there are isometric copies Z of Z which are ﬁxed. The ﬁrst of these is Pełczy´nski’s theorem that non-weakly compact operators on C(K)-spaces ﬁx c0 , Theorem 4.5. We show this follows quite naturally from Grothendieck’s classical description of weakly compact sets in C(K)∗ (Theorem 4.29), and a relative disjointness result on families of measures, due to the author (Proposition 4.30). Next, we take up characterizations of operators ﬁxing C(ωω +). Our main aim is to give an intuitive picture of the isometric copy of C0 (ωω ) which is actually ﬁxed. Theorem 4.25 itself states Alspach’s remarkable equivalences, which in particular yield that an operator on a separable C(K)-space ﬁxes C(ωω +) if and only if its ε-Szlenk index is at least ω for all ε > 0. We follow Bourgain’s approach here, stating his deep extension of this result to arbitrary countable ordinals in Theorem 4.17. Bourgain achieves his results on totally disconnected spaces K, obtaining the ﬁxed copy as the span of the characteristic functions of a regular family of clopen sets (Deﬁnition 4.15). In turn, the direct Banach space description of C(α+)-spaces is given by Bourgain’s formulation in terms of trees (Deﬁnition 4.13); these yield an intuitive direct description of monotone bases for such spaces, which are actually the clopen sets mentioned above, in the needed concrete realization of these spaces (formulated in Proposition 4.15). We discuss in considerable detail Bourgain’s remarkable result (which rests on 4.25): an operator on a C(K)-space ﬁxes C(ωω +) if and only if it is a non-Banach–Saks operator (see Deﬁnition 4.8). We ﬁrst give Schreier’s proof that C(ωω +) fails the Banach–Saks property, in Propositions 3.8 and 3.9. Next we recall the author’s dichotomy: a weakly null sequence in an arbitrary Banach space either has a subsequence whose arithmetic averages converge to zero in norm, or a subsequence which generates a spreading model

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isomorphic to 1 (Theorem 4.23). We then use this to deduce Bourgain’s non-Banach–Saks characterization (Theorem 4.22). The ﬁnal result discussed in Section 4 is the author’s result: an operator on a separable C(K)-space ﬁxes C([0, 1]) if its adjoint has non-separable range. We formulate three basic steps in the proof, Lemma 4.25, Lemma 4.29, and Proposition 4.30. These are then put together to outline the proof, and ﬁnally the “almost isometric” Lemma 4.25 is explained somewhat, via Lemma 4.31, to give a picture of the actual isometric copy of C(D) which is ﬁnally ﬁxed. Section 5 is purely expository; only obvious deductions are given. The remarkable partial progress on the Complemented Subspace Problem (CSP) illustrates the deep penetration into the structure of C(K)-spaces that has been achieved. The problem itself and especially certain unresolved special cases show there is still much to be understood about their structure (see Problems 1–4 in Section 5). L∞ spaces and L1 (μ) preduals are brieﬂy discussed. Zippin’s fundamental lemma is presented in the context of the CSP as Lemma 5.11. Possibly the most striking of the known results on the CSP, due to Benyamini, rest on Lemma 5.11. These assert that every complemented subspace of a separable C(K)-space is either isomorphic to c0 or contains a subspace isomorphic to C(ωω +). Moreover every complemented subspace X of a separable C(K)-space with X∗ separable is isomorphic to a quotient space of C(α+) for some countable ordinal α (Theorems 5.9 and 5.15). To prove this, Benyamini also establishes an extension result for general separable Banach spaces which actually yields a new proof of Milutin’s theorem (Theorem 5.12). Section 5 concludes with a brief discussion of the positive solution to the CSP in the isometric setting: every contractively complemented subspace of a separable C(K)-space is isomorphic to a C(K)-space. An exciting new research development deals with many of the issues discussed here in the context of C ∗ -algebras. Neither time nor space was available to discuss this development here. We shall only brieﬂy allude to two discoveries. The ﬁrst is Kirchberg’s non-commutative analogue of Milutin’s theorem: every separable non-type I nuclear C ∗ algebra is completely isomorphic to the CAR algebra [38]. The second concerns quantized formulations of the separable extension property, due to the author [61], and the joint theorem of Oikhberg and the author: the space of compact operators on separable Hilbert space has the Complete Separable Complementation Property [50]. For a recent survey and perspective on these developments, see [62].

2. The isomorphic classiﬁcation of separable C(K)-spaces A. Milutin’s theorem Our ﬁrst main objective is the following remarkable result due to Milutin [47]. T HEOREM 2.1. Let K be an uncountable compact metric space. Then C(K) is isomorphic to C([0, 1]). Although this is not an isometric result, its proof is based on isometric considerations. Let us introduce the following deﬁnitions and notation.

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D EFINITION 2.2. Let X and Y be given Banach spaces. (1) X → Y means that X is isomorphic to a subspace of Y . c (2) X → Y means that X is isomorphic to a complemented subspace of Y . cc (3) X → Y means that X is isometric to a contractively complemented subspace of Y . c c c (4) X ∼ Y means that X → Y and Y → X. We say X is complementably equivalent to Y . cc cc cc (5) X ∼ Y means that X → Y and Y → X. We say X is contractively complementably equivalent to Y . (6) X ∼ Y means that X is isomorphic to Y . Of course one has that the ﬁrst three relations are a kind of partial order on Banach spaces; e.g., one easily has that cc

X → Y

cc

and Y → Z

cc

implies X → Z.

(2.1)

c

(The relation → was crystallized by Alspach in some unpublished work.) The relation c → is implicit in the decomposition method given on page 14 of [33], which was developed by Pełczy´nski [51]. The proof in [33] (as well as that in [51]) yields the following result. c

P ROPOSITION 2.3. Let X, Y be Banach spaces. Then X ∼ (X ⊕ X ⊕ · · ·)c0 and X ∼ Y implies X ∼ Y . Milutin’s theorem now easily reduces to the following fundamental result (known as Milutin’s lemma). cc

T HEOREM 2.4. Let K be a compact metric space. Then C(K) → C(D) where D denotes the Cantor discontinuum. We give a proof due to Ditor [23]. We ﬁrst summarize some standard needed results. L EMMA 2.5. Let K be a given inﬁnite compact metric space. (a) D is homeomorphic to a subset of K if K is uncountable. (b) Let L be a compact subset of K. Then there exists a linear isometry T : C(L) → C(K) such that T IL = IK and T (f |L ) = f for all f ∈ C(K). (c) D is homeomorphic to K if K is perfect and totally disconnected. (d) c0 is isometric to a subspace of C(K). (e) C(K) ∼ C0 (K, k0 ) where k0 ∈ K and C0 (K, k0 ) = {f ∈ C(K): f (k0 ) = 0}. In fact, there is an absolute constant γ so that d(C(K), C0 (k, k0 )) γ . R EMARK 2.6. d(X, Y ) denotes the multiplicative Banach–Mazur distance between Banach spaces X and Y .

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P ROOF. (b), the linear form of the Tietze-extension theorem, is due to Borsuk [17]. (a) and (c) are standard topological results. To see (d) (which holds for any inﬁnite compact Hausdorff space), let U1 , U2 , . . . be disjoint non-empty open subsets of K, and for each j , choose 0 ϕj 1 in C(K) with ϕj = 1 and support ϕj ⊂ Uj for all j . One has immediately that then (ϕj ) is isometrically equivalent to the usual c0 -basis. To obtain (e), let X be a subspace of C(K) isometric to c0 . In fact, our argument shows that we may choose X ⊂ C0 (K, k0 ). Since K is separable, X is complemented in C(K) by Sobczyk’s theorem [65] Thus in fact we may choose Y a closed linear subspace of C0 (K, k0 ) with C0 (K, k0 ) = X ⊕ Y.

(2.2)

Thus we have that C(K) = [1] ⊕ C0 (K, k0 ) = [1] ⊕ X ⊕ Y ∼ [1] ⊕ c0 ⊕ Y ∼ c0 ⊕ Y ∼ X ⊕ Y = C0 (K, k0 ).

(2.3)

(Note that [1] is the same as Φ, the one-dimensional space of scalars.) Now the existence of γ may be obtained by tracing through this argument quantitatively, using Sobczyk’s result that in fact there is a projection of C(K) onto X of norm at most two. Indeed, suppose Z is a separable Banach space containing a subspace X isometric to c0 . By Sobczyk’s theorem, there is a subspace Y of Z with d(Z, c0 ⊕ Y ) 6 (where we take direct sums in the ∞ -norm). But also if Z0 is any co-dimension 1 subspace of Z, then by the Hahn–Banach theorem, d(Z0 , Z ⊕ Φ) 6. Hence since d(Z ⊕ Φ), c0 ⊕ Φ ⊕ Y ) 6, and c0 ⊕ Φ is isometric to c0 , it follows that d(Z, Z0 ) 36. R EMARK 2.7. It is unknown if 2.5(e) holds for non-metrizable compact Hausdorff spaces K. P ROOF OF M ILUTIN ’ S THEOREM ( MODULO 2.4). We ﬁrst note that for any k0 ∈ D,

C(D) ⊕ C(D) ⊕ · · · c ∼ = C0 (D, k0 ) ∼ C(D) 0

(2.4)

(where X ∼ = Y means X is isometric to Y ). Indeed this follows from Lemma 2.5(c). Thus, using Theorem 2.4 and Proposition 2.3, it sufﬁces to show that given K uncountable compact metric, then cc

C(D) → C(K).

(2.5)

Indeed, we then obtain that C(K) ∼ C(D). So of course also C(K) ∼ C([0, 1]). But 2.5 follows immediately from Lemma 2.5(a) and (b). Indeed, choose L a compact subset of K homeomorphic to D and choose T as in 2.5(b); set X = T (C(L)). Then X is isometric to C(D) and is contractively complemented in C(K) via the map: P (f ) = T (f |L ).

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R EMARK . Actually, the above argument and the proof of Milutin’s lemma gives even more isometric information; namely one has T HEOREM 2.8. Let K and L be compact metric spaces with K uncountable. Then there is a unital isometry from C(L) onto a subspace X of C(K), which is contractively complemented in C(K). The unital isometry and contractive projection are thus positive maps. cc

We now present the proof of Theorem 2.4. We ﬁrst formulate the → order as follows, leaving the simple proof to the reader. P ROPOSITION 2.9. Let X and Y be given Banach spaces. The following are equivalent. cc (a) X → Y . (b) There exist linear contractions U : X → Y and V : Y → X so that IX = V ◦ U . That is, the following diagram holds.

X

Y V U I

(2.6) X.

D EFINITION 2.10. Let L and K be compact metric spaces and ϕ : L → K be a continuous surjection. (a) ϕ 0 : C(K) → C(L) denotes the map ϕ0f = f ◦ ϕ

for all f ∈ C(K).

(b) A linear map T : C(L) → C(K) is called a regular averaging operator for ϕ if T = 1 and (T ϕ 0 )f = f for all f ∈ C(K), i.e., (2.6) holds with X = C(K), U = ϕ0, V = T . Note that ϕ 0 (C(K)) is in fact a unital subalgebra of C(L) isometric to C(K). It is easily seen (via the argument for Proposition 2.9) that ϕ 0 (C(K)) is contractively complemented in C(L) iff ϕ admits a regular averaging operator T . Thus Milutin’s lemma means one can choose a continuous surjection ϕ : D → K which admits a regular averaging operator. Milutin did this by an explicit construction, while Ditor’s argument proceeds conceptually, but indirectly. (For further results, see [24], and especially [9] for recent comprehensive work on regular averaging operators in both the metric and non-metrizable setting.) We ﬁrst deal with the basic ingredient in the proof. Given X1 , . . . , Xn topological spaces, X1 ⊕ · · · ⊕ Xn denotes their topological disjoint sum. Of course if these are compact metric spaces, so is X1 ⊕ · · · ⊕ Xn ; we may formally identify X1 ⊕ · · · ⊕ Xn with the metric

The Banach spaces C(K)

1555

space n

where dist (x, i), (y, j ) = 1 if i = j

and dist (x, i), (y, i) = dXi (x, y)

Xi × {i},

i=1

(where dXi is the metric on Xi ). L EMMA 2.11. Let K be a compact metric space, and K1 , . . . , Kn be non-empty compact subsets such that K=

n

int Ki .

(2.7)

i=1

Let τ : K1 ⊕ · · · ⊕ Kn → K be the map deﬁned by τ (k, i) = k

for all i and k ∈ Ki .

(2.8)

Then τ admits a regular averaging operator. R EMARK . We obviously may assume that int Ki = interior Ki = ∅ for all i. However we do not insist that the Ki ’s are distinct; in fact we may need repetitions .n in our application of 2.11. It is also clear that τ is a continuous surjection of i=1 Ki onto K. P ROOF OF L EMMA 2.11. We may choose ϕ1 , . . . , ϕn a partition of unity ﬁtting the open cover of K, (int Ki )ni=1 . That is, the ϕj ’s are in C(K) and satisfy 0 ϕj 1 for all j, supp ϕj ⊂ int Kj n

(2.9)

for all j,

(2.10)

ϕj ≡ 1.

(2.11)

j =1

supp ϕ = x: ϕ(x) > 0 . Next, ﬁx i, and for f ∈ C( fi (k) = f (k, i)

.n

j =1 Kj ),

if k ∈ Ki ,

deﬁne fi on K by (2.12)

fi (k) = 0 if k ∈ Ki .

(2.13)

fi · ϕi

(2.14)

Then is continuous.

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Indeed, since τ maps Ki × {i} homeomorphically into Ki , it follows that fi |Ki is continuous, and so of course (fi · ϕi )|Ki is also continuous. Since ϕi (x) = 0 for all x ∈ Ki and ϕi is continuous on K, it follows that if (xn ) is a sequence in K ∼ Ki such that xn → x with x ∈ Ki , then (fi · ϕi )(xn. ) = 0 for all n and also (fi · ϕi )(x) = 0, proving (2.14). Finally, deﬁne T : C( ni=1 Ki ) → C(K) by Tf =

n

fi ϕi

for all f ∈ C

i=1

Then ﬁxing f ∈ C( any k ∈ K

n /

Ki .

(2.15)

i=1

.n

i=1 Ki ),

we have that indeed Tf ∈ C(K) by (2.14). Moreover for

n Tf (k) fi (k)ϕi (k) i=1

maxfi (k) ϕi (k) i

f ∞ .

(2.16)

Thus T = 1 and of course T is linear. Finally, if f ∈ C(K), then for any k, n 0

τ f i (k)ϕi (k) T τ 0 f (k) = i=1

=

f (k)ϕi (k)

= f (k)

n

ϕi (k) = f (k),

(2.17)

i=1

completing the proof.

We need one more tool; inverse limit systems of topological spaces. We just formulate the special case needed here (see Lemma 2 of [23] for the general situation). L EMMA 2.12. Let (Kn )∞ n=1 be a sequence of compact metric spaces, and for each n, let ϕn : Kn+1 → Kn be a given continuous surjection. There exists a compact metrizable space K∞ satisfying the following for all n: There exists a continuous surjection ϕ˜n : K∞ → Kn ,

(2.18)

ϕn ϕ˜n+1 = ϕ˜ n .

(2.19)

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1557

Letting Yn = ϕ˜ n0 (C(Kn )), then ∞

Yn

is dense in C(K∞ ).

(2.20)

n=1

If moreover ϕn admits a regular averaging operator for each n, then ϕ˜ 1 admits a regular averaging operator. R EMARK . The space K∞ is essentially determined by (2.16) and (2.17) and is called the inverse limit of the system (Kn , ϕn )∞ n=1 . P ROOF. Let K∞ be the subset of (kj ) ∈ K∞

*∞

n=1 Kn

iff kj = ϕj (kj +1 )

deﬁned by for all j.

(2.21)

Of course the axiom of choice yields that K∞ is not only non-empty, but for all n, ϕ˜ n : K∞ → Kn is a surjection, where

ϕ˜ n (kj ) = kn

for any (kj ) ∈ K∞ .

(2.22)

* K∞ is also a closed subset of ∞ n=1 Kn , where the latter is endowed with the Tychonoff topology, it also follows immediately that ﬁxing n, then (2.19) holds. But this implies that Yn ⊂ Yn+1 .

(2.23)

Indeed, say y ∈ Yn and let y = ϕ˜n0 (f ), for a (unique) f ∈ C(Kn ). But 0

0 ϕ˜ n0 (f ) = f ◦ ϕ˜n = f ◦ ϕn ◦ ϕ˜n+1 = ϕ˜ n+1 ϕn f ∈ Yn+1 .

(2.24)

Now it follows that ∞ n=1 Yn is a unital subalgebra of C(K∞ ) separating its points, hence this is dense in C(K∞ ) by the Stone–Weierstrass theorem. Finally, if each ϕn admits a regular averaging operator, then for all n there exists a contractive linear projection Pn : Yn+1 → Yn from Yn+1 onto Yn . It follows that there exists a unique contractive linear projection P : ∞ j =1 Yj → Y1 such that for all n and y ∈ Yn+1 , P (y) = P1 P2 · · · Pn−1 Pn (y).

(2.25)

But then P uniquely extends to a unique contractive projection from C(K∞ ) onto P (Y1 ) by (2.20), completing the proof via Proposition 2.9. Finally, we give the P ROOF OF T HEOREM 2.4. Let K1 = K. We inductively deﬁne K2 , K3 , . . . , KN , . . . satisfying the hypotheses of Lemma 2.12.

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Step 1. Choose n1 > 1 and W11 , . . . , Wn11 compact subsets of K1 , each with non-empty interior and diameter less than one, such that K=

n1

int Wj1 .

(2.26)

j =1

. Set K1 = nj 1=1 Wj1 ; endow K1 with the metric described in the comments preceding Lemma 2.12, and let ϕ1 : K1 → K be the continuous surjection given by Lemma 2.11. Thus ϕj (Wj1 × {j }) = Wj1 for all j and ϕj admits a regular averaging operator by Lemma 2.11. . Step m. Assume that Km = nj m=1 Wjm has been deﬁned, and ﬁx j , 1 j nm . Thus Wj = Wjm × {j } is a natural clopen subset of Km . Choose kj > 1 and Wj,1 , . . . , Wj,kj compact subsets of Wj , each with non-empty interior, and diameter less than 1/(m + 1), with Wj =

kj

int Wj,i .

(2.27)

i=1

Next, let nm+1 =

nm

j =1 kj

Win+1 = Wψ(i)

and for 1 i nm+1 ,

(2.28)

where ψ : {1, . . . , nm+1 } → {(j, i): 1 i kj , 1 j nm } is a bijection. .n m+1 Set Kn+1 = j m+1 , endow Km+1 with the metric described preceding Lem=1 Wj ma 2.11 and choose ϕm : Km+1 → Km the continuous surjection admitting a regular averaging operator given by Lemma 2.11. Thus in fact ϕm (Wj, × {}) = Wj,i , where ψ() = (j, i) for all j and i and moreover

nm+1

Km =

int Wjm+1 .

(2.29)

j =1

This completes the inductive construction of the Km ’s and ϕm ’s. Lemma 2.12 now yields the existence of a continuous surjection ϕ : K∞ → K admitting a regular averaging operator, where K∞ satisﬁes the conclusion of Lemma 2.12. It remains only to check that K∞ is perfect and totally disconnected, hence homeomorphic to D. The details for this quite naturally involve the further structure of inverse systems. For each 1 j n, deﬁne the map ϕj,n : Kn+1 → Kj by ϕj,n = ϕj ◦ ϕn . Now in our particular construction, we have that for all m and 1 j nm , ϕm maps Wjm × {j } isometrically onto Wjm .

(2.30)

But then it follows that ϕj,m |Wjm ×{j }

is an isometry for all i m and 1 j nm .

(2.31)

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1559

Let us endow K∞ with the metric ∞

dj (xj , yj ) , d (xj ), (yj ) = 2j

(2.32)

j =1

where dj is the metric on Kj . Then it follows that for all m and 1 j nm m < diam W j

2 , m

m = ϕ˜ j W m × {j } . where W j j

(2.33)

m . But then there are points x and y in W m × {j } Indeed, suppose (xk ) and (yk ) belong to W j j such that xi = ϕi,m (x) and yi = ϕi,m (y) for all 1 i m, and have by (2.31), m ∞ 1 1 1 1 (x , y ) + + m (2.34) d d(xi , yi ) m i i j j 2 2 m 2 j =1

j =m+1

since diam Wjm < 1/m. m : 1 j nm ; m = 1, 2, . . .} is a It then follows that the family of clopen subsets {W j base for the topology of K∞ . Hence K∞ is totally disconnected. Finally, our insistence that at each stage m we “label” at least 2 sets contained in Wjm × {j } (i.e., kj > 1), insures that m contains at least two points for all m and 1 j nm , whence K∞ is indeed perfect. W j B. C(K) spaces with separable dual via the Szlenk index Of course C(K)∗ is separable if and only if K is inﬁnite countable compact metric. It is a standard result in topology that every such K is homeomorphic to C(α+) for some countable ordinal α ω. We use standard facts about ordinal numbers. An ordinal α denotes the set of ordinals β with β < α; α+ denotes α + 1. Finally, for α a limit ordinal, C0 (α) denotes the space of continuous functions on α vanishing at inﬁnity, which of course can be identiﬁed with {f ∈ C(α+): f (α) = 0}. The Banach spaces C(ωα +) arise quite naturally upon applying a natural inductive construction to c = C(ω+) (the space of converging sequences). Indeed, for any locally compact Hausdorff spaces X1 , X2 , . . . , we have that

def C0 (X1 ) ⊕ C0 (X2 ) ⊕ · · · c = Y 0

(2.35)

. is again algebraically isometric to C0 (X) where X = ∞ j =1 Xj , the “direct sum” of the spaces X1 , X2 , . . . . Of course then Y has a unique “unitization” as the space of continuous functions on the one point compactiﬁcation of X, which we’ll denote by Y ⊕ [1]. The norm here is quite explicitly given as (2.36) y ⊕ c · 1 = sup sup yj (ω) + c, j ω∈Xj

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where y = (yj ) ∈ Y . Now deﬁne families of C(K) spaces (Yα )1α<ω1 as follows. Let Y10 = c0 and Y1 = c = 0 Y1 ⊕ [1]. Let Y20 = (c ⊕ c ⊕ · · ·)c0 and Y2 = Y20 ⊕ [1]. Suppose β is a countable ordinal and Xα , Yα deﬁned for all α < β. If β is a successor, say β = α + 1, set Yβ0 = (Yα ⊕ Yα ⊕ · · ·)c0 and Yβ = Yβ0 ⊕ [1]. If β is a limit ordinal, choose αn < β with αn - β and set Yβ0 = (Yα1 ⊕ Yα2 ⊕ · · ·)c0 and Yβ = Yβ0 ⊕ [1]. It is not difﬁcult to see that then for all 1 α < ω1 ,

Yα0 is algebraically isometric to C0 ωα

and Yα is algebraically isometric to C ωα + .

(2.37)

Now the topological classiﬁcation of inﬁnite countable compact metric spaces K is known; each such space is homeomorphic to exactly one of the ordinals (ωα + 1) · n for some 1 α < ω1 and positive integer n ([45]). Indeed, one has that α = Ca(K) and n = #K (α) . (See the comments following 2.19 below for the deﬁnition of Ca(K).) We thus obtain: P ROPOSITION 2.13. Let K be an inﬁnite countable compact metric space. Then there exists a unique 1 α < ω1 and a unique n so that C(K) is (algebraically) isometric to Yα ⊕ · · ⊕< Yα , the direct sum taken in the sup norm. 9 ·:; n

The isomorphic classiﬁcation of these spaces is far more delicate. The result is as follows (Bessaga and Pełczy´nski [15]). T HEOREM 2.14. Let K be an inﬁnite countable compact metric space. α (a) C(K) is isomorphic to C(ωω +) for some countable ordinal α 0. α β (b) If 0 α < β < ω1 , then C(ωω +) is not isomorphic to C(ωω +). It turns out that the Szlenk index actually distinguishes these spaces. This index was introduced for Banach spaces with separable dual, by Szlenk [67], eight years after the seminal work of [15]. For X with X∗ separable, we denote its Szlenk index by Sz(X). The following remarkable result was established by Samuel [63], based in part on work of Alspach and Benyamini [6]. α

T HEOREM 2.15. Let 0 α < ω1 . Then Sz(C(ωω +)) = ωα+1 . We give a detailed proof of 2.14(a), but only indicate some of the ideas involved in 2.15, which of course yields 2.14(b). We ﬁrst give an alternate derivation for the Szlenk index, similar to that indicated in Section 1.3 of [10]. Fix X a separable Banach space, and let ε > 0. We deﬁne a derivation dε on the ω∗ -compact subsets K of X∗ as follows: Let δε (K) denote the set of all x ∗ ∈ K such that there exists a sequence (xn∗ ) in K with xn∗ → x ∗ ω∗ and xn∗ − x ∗ ε for all n. (2.38)

The Banach spaces C(K)

1561

Now deﬁne: dε (K) = δε (K)

ω∗

.

(2.39)

Now deﬁne a transﬁnite descending family of sets Kα,ε for 0 α < ω1 as follows. Let K0,ε = K and K1,ε = dε (K). Let γ be a countable ordinal and suppose Kα,ε deﬁned for all α < γ . If γ is a successor, say γ = α + 1, set Kγ ,ε = dε (Kα,ε ).

(2.40)

If γ is a limit ordinal, choose (αn ) ordinals with αn < γ for all n and αn → γ ; set Kγ ,ε =

∞

Kαn,ε .

(2.41)

n=1

We may now deﬁne ordinal indices as follows. D EFINITION 2.16. Let K be a ω∗ -compact subset of X∗ , with X a separable Banach space. (a) βε (K) = sup{α ω1 : Kα,ε = ∅}. (b) β(K) = supε>0 βε (K). (c) Sz(X) = β(Ba X∗ ). Now it is easily seen that in fact there is a (least) α < ω1 with Kα,ε = Kα+1,ε . Moreover one has that then Kα,ε = ∅ iff K is norm separable iff βε (K) < ω1 , and then α = βε (K) + 1. Thus one obtains that Sz(X) < ω1 iff X∗ is norm-separable. Szlenk’s index was really only originally deﬁned for Banach spaces with separable dual. In fact, however, one arrives at exactly the same ﬁnal ordinal indices as he does, assuming that 1 is not isomorphic to a subspace of X, in virtue of the 1 -theorem [58]. The derivation in [67] is given by: K → τε (K) where τε (K) is the set of all x ∗ in K so that there is a sequence (xn∗ ) in K and a weakly null sequence (xn ) in Ba(X) such that limn→∞ |xn∗ (xn )| ε. Now let Pα (ε, K) be the transﬁnite sequence of sets arising from this derivation as deﬁned in [67]. Let also ηε (K), the “ε-Szlenk index of K,” equal sup{α < ω1 : Pα (ε, K) = ∅} and η(K) = supε>0 ηε (K)}. The following result shows the close connection between Szlenk’s derivation and ours; its routine proof (modulo the 1 -theorem) is omitted. P ROPOSITION 2.17. Let X be a separable Banach space containing no isomorph of 1 , and let K be a ω∗ -compact subset of X∗ . Then for all ε > 0 and countable ordinals α, ε Pα , K ⊃ Kα,ε ⊃ Pα (2ε, K). (2.42) 2 Hence

η

ε , K βε (K) η(2ε, K) 2

and thus η(K) = β(K).

(2.43)

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H.P. Rosenthal

One may now easily deduce the following permanence properties. P ROPOSITION 2.18. Let X, Y be given Banach spaces and K, L weak* compact norm separable subsets of X∗ . (a) L ⊂ K implies β(L) β(K). (b) If T : Y → X is a surjective isomorphism, then β(T ∗ K) = β(K). (c) If Y ⊂ X and π : X∗ → Y ∗ is the canonical quotient map, then β(πK) β(K). In turn, this yields the following isomorphically invariant properties of the Szlenk index. C OROLLARY 2.19. Let X and Y be given Banach spaces with norm-separable duals. Then if Y is isomorphic to a subspace of a quotient space of X, Sz(Y ) Sz(X). R EMARK . Of course this yields that Sz(X) = Sz(Y ) if X and Y are of the same Kolmogorov dimension; i.e., each is isomorphic to a subspace of a quotient space of the other. This reveals at once both the power and the limitation of the Szlenk index. For the next consequence of our permanence properties of the Szlenk index, recall the Cantor–Bendixon index Ca(K) of a compact metrizable space K, deﬁned by the cluster point derivation. For W ⊂ K, let W denote the set of cluster points of W . Then deﬁne K (α) , the α-th derived set of K, by K (0) = K, K (α+1) = (K (α)) , and K (β) = α<β K (α) for countable limit ordinals α. Then deﬁne (for K = ∅) Ca(K) = sup 0 α ω1 : K (α) = ∅ .

(2.44)

One has that K is countable iff Ca(K) < ω1 , and of course if α is the least γ with K (γ ) = K (γ +1) , then either K (α) = ∅ and α = Ca(K) + 1, or K (α) is perfect. (Note: we are unconventional here; the index of Cantor–Bendixon is traditionally deﬁned as Ca(K) + 1.) C OROLLARY 2.20. (a) Let X be a separable Banach space and K be a countable ω∗ -compact subset of X∗ so that for some δ > 0, K is “δ-separated,” i.e., k − k δ

for all k = k in K.

(2.45)

Then βδ (K) = β(K) = Ca(K).

(2.46)

(b) Sz(C(ωα + 1)) α for any countable ordinal α. P ROOF. (a) We actually have that for any 0 < ε δ and any closed subset W of K, δε (W ) = dε (W ) = W .

(2.47)

The Banach spaces C(K)

1563

But then for all countable ordinals α, Kα,ε = K (α)

(2.48)

which immediately yields (2.46). As for (b), we have that

Ca ωα + = α.

(2.49)

But ωα + is obviously ω∗ -homeomorphic to a 2-separated subset K of Ba(C(ωα +))∗ . Thus we have

Sz C ωα + = β(K) = β2 (K) = α.

(2.50)

We may now obtain, via Theorem 2.14, the “easier” half of Theorem 2.15. C OROLLARY 2.21. Let 0 α < ω1 . Then α

Sz C ωω + ωα+1 .

(2.51)

P ROOF. For each positive integer n, we have that α

Sz C ωω ·n + ωα · n by 2.20(b). α

(2.52)

α

But C(ωn·ω +) is isomorphic to C(ωω +) by Theorem 2.14, and hence since the Szlenk index is isomorphically invariant (by Corollary 2.19) α

Sz C ωω + n · ωα

for all integers n,

(2.53)

which implies (2.51).

We next deal with (a) of Theorem 2.14. We ﬁrst give a functional analytical presentaα tion of the spaces C(ωω ), using injective tensor products. We ﬁrst recall the deﬁnitions (see [22], speciﬁcally pp. 485–486). D EFINITION 2.22. Given Banach spaces X and Y , the injective tensor norm, · ε , is deﬁned on X ⊗ Y , the algebraic tensor product of X and Y , by n n ∗ ∗ ∗ ∗ ∗ xk ⊗ yk = sup x (xk )y (yk ): x ∈ Ba X , y ∈ Ba Y k=1

ε

(2.54)

k=1

for any n, x1 , . . . , xk in X and y1 , . . . , yk in Y . The completion of X ⊗ Y under this norm ∨

is called the injective tensor product of X and Y , denoted X ⊗ Y .

1564

H.P. Rosenthal ∨

When K and L are compact Hausdorff spaces, then we have that C(K) ⊗ C(L) is canonically isometric to C(K × L), where the elementary tensor x ⊗ y in C(K) ⊗ C(L) is simply identiﬁed with the function (x ⊗ y)(k, ) = x(k)y() for all k, ∈ K × L. α We then obtain the following natural tensor product construction of the spaces C(ωω +) (where we use the “unitization” given in (2.36)). Deﬁne a family (Xα )α<ω1 as follows. Let X0 = c and also let X0,1 = c0 . Suppose Xα ∨

∨

has been deﬁned. Set Xα,n = Xα ⊗ · · · ⊗ Xα (n-times). Then set Xα+1,0 = (Xα,1 ⊕ Xα,2 ⊕ · · · ⊕ Xα,n ⊕ · · ·)c0 , and Xα+1 = Xα+1,0 ⊕ [1]. Finally, suppose β is a countable limit ordinal, and Xα has been deﬁned for all α < β. Choose (αn ) a sequence with αn - β, set Xβ,0 = (Xα1 ⊕ · · · ⊕ Xαn ⊕ · · ·)c0 ; then set Xβ = Xβ,0 ⊕ [1]. The following result now follows by transﬁnite induction. P ROPOSITION 2.23. Let α < ω1 and 1 n < ∞. Then Xα,0 is algebraically isometα α ric to C0 (ωω ), and Xα,n is algebraically isometric to C(ωω ·n +). In particular, Xα = α C(ωω +). We next give the main step in the proof of Theorem 2.14(a). T HEOREM 2.24. There exists an absolute constant κ so that for any inﬁnite countable compact metric space K,

d C(K), C(K) ⊕ C(K) ⊕ · · · c κ. 0

R EMARK 2.25. Of course, from (a) that also for any n, d(C(K), .n it follows immediately . n ∞ -direct sum of n copies of X. C(K)) κ where X denotes the 1 1 P ROOF. It sufﬁces to show this for all the spaces C(ωα +). Indeed, once this has been done, it follows from Proposition 2.13 that given K, there exists a unique α and n .n α +). But then (C(K) ⊕ C(K) ⊕ · · ·) is isometric to C(ω with C(K) isometric to c0 1 .n α +) ⊕ · · ·) , which of course is isometric to (C(ωα +) ⊕ · · ·) ; hence we obtain (C(ω c c 0 0 1 that also d(C(K), (C(K) ⊕ C(K) ⊕ C(K) ⊕ · · ·)c0 ) κ. Let (Xβ )β<ω1 and (Yβ )β<ω1 , be our transﬁnite presentation of these spaces, preceding 2.13. Surprisingly, this proof is not by transﬁnite induction. Suppose β 1, and β is a successor, β = α + 1. But then Xβ = (Yα ⊕ Yα ⊕)c0 , and so Xβ is isometric to its c0 sum with itself. Then letting γ be the constant in Lemma 2.5(e),

d Xβ ⊕ [1], Xβ γ .

(2.55)

Since also Yβ = Xβ ⊕ [1],

d (Yβ ⊕ Yβ ⊕ · · ·)c0 , Xβ γ ,

(2.56)

d (Yβ ⊕ Yβ ⊕ · · ·)c0 , Yβ γ 2 .

(2.57)

thus

The Banach spaces C(K)

1565

Now suppose β is a limit ordinal; choose (αn ) with αn - β. Then ∞ / (Xβ ⊕ Xβ ⊕ · · ·) = (Yαj ⊕ Yαj ⊕ · · ·)c0 j =1

(2.58)

c0

= (Xα1 +1 ⊕ Xα2 +2 ⊕ · · ·)c0 → (Yα1 +1 ⊕ Yα2 +2 ⊕ · · ·)c0 ∼ = Xβ .

(2.59) (2.60) (2.61)

Here αn+1 < β too, and the ﬁnal isometry follows from the invariance (isometric) in the deﬁnition of the Xβ ’s, while the “2-complementation” follows trivially, since Xαn +1 is codimension 1 in Yαn +1 for all n. That is, (2.58) yields that (Xβ ⊕ Xβ ⊕ · · ·)c0 is isometric to a 2 complemented subspace of Xβ . But then a quantitative version of the Pełczy´nski decomposition method produces an absolute constant τ so that d(Xβ ⊕ Xβ ⊕ · · · , Xβ ) τ. But then (2.55) yields immediately that

d (Yβ ⊕ · · ·)c0 , Yβ γ 2 τ. This completes the proof.

(2.62)

(2.63)

We need one ﬁnal ingredient, involving tensor products. L EMMA 2.26. Let α = 1 or α be a countable limit ordinal, and let n be a positive integer, and let K = ωα +. Then

C K n ∼ C(K). P ROOF. It really sufﬁces to prove this for n = 2. For we obtain that setting X = C(K), then ∨

X ∼ X ⊗ X.

(2.64)

But now it follows immediately from induction that X∼

n∨ 8

X,

(2.65)

1 ∨ ∨ = =n∨ ∼ n where n∨ 1 X = X ⊗ · · · ⊗ X n-times, and of course 1 X = C(K ). We ﬁrst note an immediate consequence of Lemma 2.5(b) (the Borsuk theorem). Let M be any compact metric space and L be a closed subset of M; let C0 (K, L) = {f ∈ C(K): f () = 0 for all ∈ L}. Then

C(M) ∼ C0 (M, L) ⊕ C(L).

(2.66)

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H.P. Rosenthal

Now suppose that α = 1. We wish to show that

C(ω+) ∼ C ω2 + = C (ω+) × (ω+) .

(2.67)

To avoid confusing notation, set p = ω and denote {p} just by p. Set M = (ω+) × (ω+) and let L = [(ω+) × p] ∪ [p × (ω+)]. Then C0 (M, L) is isometric to c0 , hence (2.66) yields that

C ω2 + ∼ c ⊕ c0 ∼ c = C(ω+).

(2.68)

Now let α be a countable limit ordinal, and choose (αn ) with αn < α for all n and αn → α. Let M = K × K, set p = ωα , and let L = [K × p] ∪ [p × K]. Then it follows (via Theorem 2.24) that C(L) ∼ C(K).

(2.69)

Now for each j , let Kj = ωαj +. Then C0 (M, L) ∼ = ∼ =

/

C(Kj × Kn )

1j,n<∞

∞ ∞ / /

C(Kj × Kn )

n=j

n=1

(2.70) c0

. c0

(2.71)

c0

But for each j n, cc

C(Kj × Kn ) → C(Kn × Kn ).

(2.72)

Hence

∞ /

cc

C(Kj × Kn )

n=j

→

∞ /

C(Kn × Kn )

(2.73)

n=j

c0

∼ = C0 (K, p).

(2.74)

Indeed, note that C(Kn × Kn ) ∼ = C(ωαn ·2 +) and αn · 2 < α for all n. Thus ﬁnally, by (2.70) and (2.73),

cc C0 (M, L) → C(M, L) ⊕ C0 (M, p) ⊕ · · · c

0

∼ C0 (M, p) by Theorem 2.24.

(2.75) (2.76)

The Banach spaces C(K)

1567

.∞ cc Of course, C0 (K, p) → C0 (M, L) for C0 (K, p) ∼ = ( n=1 C(1 × Kn ))c0 . Then by the Pełczy´nski decomposition method C0 (K, p) ∼ C0 (M, L)

(2.77)

and so ﬁnally by (2.66), C(M) ∼ C0 (K, p) ⊕ C(K) ∼ C(K).

(2.78)

At last, we give the P ROOF OF T HEOREM 2.14(a). Let K be as in its statement. We know there is an inﬁnite countable ordinal β with C(K) ∼ = C(β+). Deﬁne γ (2.79) α = sup γ : ωω β . Then α

ωω β < ωω Since ωω

α+1

α+1

= limn→∞ ωω

β ωω

α ·n

(2.80)

.

α ·n

, there is a positive integer n with (2.81)

.

Then evidently from (2.80) and (2.81), α cc C ωω + → C(β+)

α

cc and C(β+) → C ωω ·n .

(2.82) α

Now if α = 0, ωα = 1; otherwise, ωα is a limit ordinal. So setting M = ωω +, C(M) ∼ α C(M n ) by Lemma 2.26, and of course M n = ωω ·n +, thus α

α

(2.83) C ωω ·n + ∼ C ωω , α

whence C(K) is isomorphic to C(ωω +) by the decomposition method.

We ﬁnally make some observations about the proof of the “harder” half of Theorem 2.14. We ﬁrst give the ﬁrst of several arguments, here, showing that c = C(ω+) is not isomorphic to C(ωω +), via the Szlenk index. P ROPOSITION 2.27. Sz(c0 ) = ω. Of course then Sz(c) = Sz(C(ωn +)) = ω also (for all n < ∞), while Sz(C(ωω +)) ω2 by Corollary 2.21. The following elementary result easily yields 2.27, since of course Sz(c0 ) ω by (2.24). We identify c0∗ with 1 ; of course then a sequence in Ba(1 ) converges ω∗ precisely when it converges pointwise on N.

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H.P. Rosenthal

L EMMA 2.28. Let 0 < δ L and let f , (fn ) in L Ba(1 ) so that fn → f ω∗ and fn − f δ for all n. Then f L − δ. P ROOF. Let 0 < ε, and choose M with ∞ f (j ) < ε.

(2.84)

j =M+1

Then choose n so that M fn (j ) − f (j ) < ε.

(2.85)

j =1

But then δ−ε<

∞ ∞ ∞ fn (j ) − f (j ) < fn (j ) + f (j ) j =M+1

j =M+1

(2.86)

j =M+1

so by (2.84) ∞ fn (j ) > δ − 2ε,

(2.87)

j =M+1

hence ∞ fn (j ) < L − δ + 2ε,

(2.88)

j =M

and ﬁnally M f (j ) (L − δ) + 3ε

(2.89)

j =1

by (2.85) and (2.88). Hence ∞ f (j ) < L − δ + 4ε,

(2.90)

j =1

proving the lemma since ε > 0 is arbitrary.

C OROLLARY 2.29. Let K = Ba(1 , ω∗ ), and 0 < δ < 1. Then if f ∈ Kn,δ (as deﬁned following (2.38)), f 1 − nδ.

The Banach spaces C(K)

P ROOF. Immediate by induction on n.

1569

P ROOF OF P ROPOSITION 2.27. Again let 0 < δ < 1. It follows from the preceding result that if nδ > 1 − δ, then Kn+1,δ = ∅, hence (cf. Deﬁnition 2.16) βδ (K) n and so β(K) = Sz(c0 ) ω. The proof of Theorem 2.15 in [63] uses rather delicate properties of the ordinal numbers α and their reﬂection in properties of the spaces C(ωω +). In fact, it is ﬁrst proved in [63] β α that if α < β, then C(ωω +) is not isomorphic to a quotient space of C(ωω +), and then the desired inequality about the Szlenk index is deduced, using in part a result in [6]. Of course this result is in turn a consequence of 2.15 via the natural properties of the Szlenk index developed above. The present author “believes” a direct functional analytical proof α of 2.15 should be possible, in the spirit of the presentation of the spaces C(ωω ) given above.

3. Some Banach space properties of separable C(K)-spaces A. Weak injectivity We ﬁrst consider a separable weak injectivity result, due to Pełczy´nski [54]. T HEOREM 3.1. Let Y be a subspace of a separable Banach space X, with Y isomorphic to a separable C(K)-space. Then there exists a subspace Z of Y which is isomorphic to Y and complemented in X. We give a proof due to Hagler [31], which yields nice quantitative information. We say that Banach spaces X and Y are λ-isomorphic if there is a surjective isomorphism T : X → Y with T T −1 λ. If X ⊂ Y , we say that X is λ-complemented in Y if there is a linear projection P of Y onto X with P λ. The reader may then easily establish that if the diagram (2.6) holds with linear maps U and V satisfying U V λ, then X is λ-isomorphic to a λ-complemented subspace of Y . The quantitative version of 3.1 given in [31] then goes as follows: T HEOREM 3.2. Let K be an inﬁnite compact metric space, X a separable Banach space, and Y a subspace of X λ-isomorphic to C(K). If K is uncountable, let Ω = D, the Cantor set. If K is countable, let Ω = K. Then Y contains a subspace λ-complemented in X and λ-isomorphic to C(Ω). Of course, Theorem 3.1 follows from Theorem 3.2 and Milutin’s theorem (Theorem 2.1). We require a topological lemma (due to Kuratowski in the uncountable case, Pełczy´nski in the countable case). L EMMA 3.3. Let M and L be compact metric spaces and τ : M → L be a continuous surjection.

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H.P. Rosenthal

(a) If L is uncountable, there is a subset Ω of M homeomorphic to D with τ |Ω a homeomorphism of Ω with τ (Ω). (b) If L is countable, there is a subset Ω of M homeomorphic to L with τ |Ω a homeomorphism of Ω with τ (Ω). P ROOF OF T HEOREM 3.2. If W is a ω∗ -compact subset of the dual B ∗ of a Banach space B, let RW : B → C(W ) be the continuous map (RW b)(w) = w(b) for all b ∈ B, w ∈ W . Now choose T : C(K) → Y a surjective isomorphism with T = 1 and T −1 λ.

(3.1)

Let i : Y → X be the identity injection. Then

(iT )∗ λ Ba X∗ ⊃ Ba C(K)∗ .

(3.2)

Then regarding K as canonically embedded in C(K)∗ , we have that setting M = [(iT )∗ ]−1 (K)] ∩ λ Ba(X∗ ) then M is ω∗ metrizable and setting τ = (iT )∗ |M, then τ :M → K

is a continuous surjection.

(3.3)

If K is uncountable, choose Ω ⊂ M homeomorphic to D satisfying (a) of Lemma 2.5; if K is countable, choose Ω ⊂ M homeomorphic to K, satisfying (b) of 2.5. Set Ω = τ (Ω). Finally, let β = τ −1 . Recall (cf. Deﬁnition 2.10) that β 0 : C(Ω) → C(Ω ) is the canonical (algebraic) isometry induced by β; since β is a homeomorphism. β 0 is surjective. At last let E : C(Ω ) → C(K) be an isometric linear extension operator (as provided by Lemma 2.5(b)). We then have that the following diagram is commutative. Y T

C(K) E

i

X

RΩ

β0

C(Ω )

C(Ω)

I

C(Ω ).

(To check this, let ω ∈ Ω and set β(ω ) = ω. Thus ω = τ (ω) = (iT )∗ (ω). Let f ∈ C(Ω ). Then β 0 RΩ iT Ef (ω ) = RΩ iT Ef (βω ) = iT Ef (ω) = Ef (iT )∗ (ω) = (Ef )(ω ) = f (ω ).) Finally, let U = iT E and V = β 0 RΩ . Then U 1 and since Ω ⊂ λ Ba(X∗ ), V λ. Thus Z = U (C(Ω )) is λ-isomorphic to C(Ω ) and λ-complemented in X. Of course Z ⊂ Y , thus completing the proof. Theorem 3.1 has an interesting consequence for C(K) quotients of separable spaces.

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C OROLLARY 3.4. Let K be a compact metric space, and X be a separable Banach space. If C(K) is isomorphic to a quotient of a subspace of X, then C(K) is isomorphic to a quotient space of X. P ROOF. Let Y be a subspace of X such that C(K) is isomorphic to a quotient space of Y and let Z be a subspace of ∞ , isometric to C(K). We may thus choose a bounded linear : X → ∞ a bounded linear surjection T : Y → Z. Since ∞ is injective, we may choose T operator extending T . But then W =def T(X) is separable and of course Z ⊂ W . Hence we may choose Z ⊂ Z with Z isomorphic to Z and a bounded linear projection P from W onto Z . But then P T maps X onto Z , completing the proof. B. c0 -saturation of spaces with separable dual We next discuss a “thin” property of C(K) spaces for K countable, due to Bessaga and Pełczy´nski. D EFINITION 3.5. Let X be an inﬁnite-dimensional Banach space X. X is called c0 saturated if c0 embeds (isomorphically) into every (closed linear) inﬁnite-dimensional subspace. P ROPOSITION 3.6 ([15]). Let K be a countable inﬁnite compact metric space. Then C(K) is c0 -saturated. It is unknown if every quotient space of such a C(K)-space is c0 -saturated. Actually, to “play the devil’s advocate”, it is also unknown if 2 is isomorphic to a subspace of a quotient of C(ωω +). Proposition 3.6 is really an immediate consequence of our transﬁnite description of the C(K)-spaces and the following natural permanence property of c0 -saturated spaces. L EMMA 3.7. Let X1 , X2 , . . . be c0 saturated Banach spaces. Then X =def (X1 ⊕ X2 ⊕ · · ·)c0 is c0 saturated. P ROOF. We ﬁrst observe that for all n, X1 ⊕ · · · ⊕ Xn is c0 saturated. Indeed, it sufﬁces, using induction, to show this for n = 2. If Y is an inﬁnite-dimensional subspace of X1 ⊕ X2 which is not isomorphic to a subspace of X1 , then letting P be the natural projection of X1 ⊕ X2 onto X1 , we may choose a normalized basic sequence (yn ) in Y with P yn < ∞. It then follows that for some N , (yj )∞ j =N is isomorphic to a subspace of ∞ X2 , whence [yj ]j =N contains a subspace isomorphic to c0 . Now for each n, let Pn be the natural projection of X onto X1 ⊕ · · · ⊕ Xn , and let Y be an inﬁnite-dimensional subspace of X. Then if Y is isomorphic to a subspace of X1 ⊕ · · · ⊕ Xn , c0 embeds in Y by what we proved initially. If this is false for all n, we may choose a normalized basic sequence (yn ) in Y so that Pn (yn )| < ∞. Well, a standard travelling hump argument now yields that there is a subsequence (yn ) of (yn ) with yn equivalent (almost isometrically) to the c0 basis.

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P ROOF OF P ROPOSITION 3.6. We may just use the transﬁnite description of the spaces Yα = C(ωα +) given preceding Proposition 2.13. It sufﬁces to prove these spaces are c0 saturated, since for every inﬁnite countable ordinal β, C(β+) is isometric to the n-fold direct sum of one of these, for some n. Of course c0 is itself c0 saturated, and so then trivially so is c = Y1 . Suppose β > 1 is a countable ordinal, and it is proved that Yα is c0 -saturated for all α < β. If β is a successor, say β = α + 1, then Yβ0 = (Yα ⊕ Yα ⊕ · · ·)c0 is c0 -saturated by Lemma 3.7, and Yβ is isomorphic to Yβ0 by Lemma 2.5(e). But if β is a limit ordinal, then choose (αn ) with αn - β. So then Yβ is again isomorphic to Xβ = (Yα1 ⊕ Yα2 ⊕ · · ·)c0 , which again is c0 -saturated by Lemma 3.7. C. Uncomplemented embeddings of C([0, 1]) and C(ωω +) in themselves The last result we discuss in some depth in this section is Amir’s theorem: C(ωω +) is not separably injective [8]. By the results of the preceding section, it follows that if K is an inﬁnite compact metric space, then C(K) is separably injective only if C(K) is isomorphic to c0 (as also pointed out in [8]). (See ([33, pp. 18–19] for a short proof of the theorem that c0 is separably injective.) Of course it follows that C([0, 1]) is not separably injective. A concrete witness of this result: let φ : {0, 1}N → [0, 1] be the Cantor map, φ((εj )) = ∞ ε j 0 N N j =1 2j . Then φ (C[0, 1]) is uncomplemented in C({0, 1} ) and of course {0, 1} is homeomorphic to D the Cantor discontinuum. This uncomplementation result is due to Milutin [47]. We give a proof of both of these results, by using a classical space of discontinuous functions on [0, 1] which arises in probability theory; namely the space of all scalar-valued functions on [0, 1] which are right continuous with left limits, denoted by rcl([0, 1]). We may easily generalize this to arbitrary compact subsets of [0, 1]. D EFINITION 3.8. Let K be an inﬁnite compact subset of [0, 1]. Let rcl(K, D) denote the family of all scalar-valued functions f on K so that f is continuous for each k ∈ K ∼ D, and so that f is right continuous with left limit at each point d ∈ D. In case D = K, let rcl(K) = rcl(K, K). We shall show that if D is a countable dense subset of [0, 1], then C([0, 1]) is an uncomplemented subspace of rcl([0, 1], D); and that this yields Milutin’s result concerning the Cantor map, for D the set of all end points of dyadic intervals. Finally, we show that there is a subset K of [0, 1] homeomorphic to ωω + such that C(K) is uncomplemented in rcl(K). We ﬁrst need the following concept. D EFINITION 3.9. Let K be a subset of [0, 1], and let K(1) denote the set of two-sided cluster points of K. That is, x ∈ K(1) provided there exist sequences (yj ) and (zj ) in K with yj < x < zj for all j andlimj →∞ yj = x = limj →∞ zj . Then for n 1, let K(n+1) = (K(n) )(1) . Finally let K(ω) = ∞ n=1 K(n) . Of course we could deﬁne K(α) for arbitrary countable ordinals, but we have no need of this. Also, if K is not closed, we need not have that K(1) ⊂ K, and moreover, even if K is

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closed, K(1) may not be; e.g., [0, 1](1) = (0, 1). We do, however, have the following simple result. P ROPOSITION 3.10. Let K be a compact subset of [0, 1]. Then K(n+1) ⊂ K(n) for all n. P ROOF. Set, for convenience, K(0) = K. So the result trivially holds for n = 0. Suppose proved for n, and let x ∈ K(n+2) . Then choosing (yj ) and (zj ) as in 3.9 with (yj ), (zj ) in K(n+1) , for all n, the yj ’s and zj ’s also belong to K(n) by induction, thus x ∈ K(n+1) . Now we dig into the way in which C(K) is embedded in rcl(K), which, after all, is algebraically isometric to C(M) for some compact Hausdorff M. P ROPOSITION 3.11. Let K be a compact subset of [0, 1], and assume D is an inﬁnite countable subset of K(1) . Set B = rcl(K, D). Then B is an algebra of bounded functions and B/K is isometric to c0 . R EMARK 3.12. Without the countability assumption we still get that B/K is isometric to c0 (D). P ROOF. For each f ∈ B, d ∈ D, let f (d−) = limx↑d f (x) (i.e., the left limit of f at d). Fix f ∈ B. Now it is easily seen that f is bounded. In fact, a classical elementary argument shows that for all ε > 0, d ∈ D: f (d) − f (d−) > ε

is ﬁnite.

(3.4)

It then follows that deﬁning T : B → ∞ (D) by (Tf )(d) =

f (d) − f (d−) 2

for all f ∈ B, d ∈ D

(3.5)

then T is a linear contraction valued in c0 (D). Now for each d ∈ D, deﬁne fd ∈ B by fd (k) = 1 if k < d,

fd (k) = −1 if k d.

(3.6)

Now since d ∈ K(1) , it follows easily that dist(fd , C(K)) = 1. In fact, letting π : B → B/C(K) be the quotient map, we have that for any n, k distinct points d1 , . . . , dn , and arbitrary scalars c1 , . . . , cn , cj fdj . (3.7) cj fdj = max |cj | = T π j

This easily yields that in fact T is a quotient map, and moreover if d1 , d2 , . . . is an enumeration of D, then (πfdj ) is isometrically equivalent to the usual c0 basis and [πfdj ] = B/C(K). The next lemma is the crucial tool for our non-complementation results.

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L EMMA 3.13. Let K be a compact subset of [0, 1] so that K(n) = ∅. If K is countable, let D = K(1) . If K = [0, 1], let D be a countable dense subset of (0, 1) (the open unit interval). Assume for each d ∈ D, there is given gd ∈ C(K). Then given ε > 0, there exist d1 , . . . , dn in D and v in K so that (fd + gd )(ν) > 1 − ε j j

for all 1 j n.

(3.8)

P ROOF. For convenience, we assume real scalars. Note also that in the case K = [0, 1], K(n) = (0, 1). Let ε > 0 be ﬁxed. First choose d1 ∈ K(n) ∩ D. Now choose δ1 > 0 so that letting V1 = (d1 − δ1 , d1 + δ1 ) ∩ K, then gd (d1 ) − gd (x) < ε 1 1

for all x ∈ V1 .

(3.9)

For simplicity in notation, set a = gd1 (d1 ). Now if x ∈ V1 and x > d1 , (fd1 + gd1 )(x) < −1 + a + ε.

(3.10)

If x ∈ V1 , x < d1 , then (fd1 + gd1 )(x) > 1 + a − ε.

(3.11)

max |1 + a − ε|, |−1 + a + ε| = 1 − ε + |a| 1 − ε.

(3.12)

But

Now since d1 is a two-sided cluster point of K(n−1) , it follows that def

V11 = (d1 − δ1 , d1 ) ∩ K(n−1) ∩ D = ∅ and def

V12 = (d1 , d1 + δ1 ) ∩ K(n−1) ∩ D = ∅. 1 = V 1 or V 2 , and then Hence if follows from (3.7)–(3.11) that we may set V 1 1 1 ∩ K(n−1) ∩ D = ∅ V

(3.13)

and (fd + gd )(x) > 1 − ε 1 1

1 . for all x ∈ V

(3.14)

1 ∩ K(n−1) ∩ D, and proceed in exactly the same way as in the ﬁrst Now choose d2 ∈ V 1 an open neighborhood of d2 so that step. Thus, we ﬁrst choose V2 ⊂ V gd (d2 ) − gd (x) < ε 2 2

for all x ∈ V2 .

(3.15)

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2 an open subset of Since d2 is a right and left cluster point of K(n−2) we again choose V V2 such that (fd + gd )(x) > 1 − ε 2 2

2 for all x ∈ V

(3.16)

and so that 2 ∩ K(n−2) ∩ D = ∅. V

(3.17)

Continuing by induction, we obtain d1 , . . . , dn in D, dn+1 ∈ K, and open sets in K, 0 ⊃ V 1 ⊃ V 2 ⊃ · · · ⊃ V n , so that for all 1 j n + 1, dj ∈ V j and K =V (fd + gd )(x) > 1 − ε j j

j . for all x ∈ V

Evidently then d1 , . . . , dn and v = dn+1 satisfy the conclusion of the lemma.

(3.18)

We are now prepared for our main non-complementation result. T HEOREM 3.14. Let n > 1 and let K and D be as in Lemma 3.13. Set B = rcl(K, D). Then if P is a bounded linear projection of B onto C(K), P n − 1.

(3.19)

Hence if K = [0, 1] or if K is countable and K(n) = ∅ for all n, C(K) is an uncomplemented subspace of rcl(K, D). P ROOF. Let λ = P ; also let π : B → B/C(K) be the quotient map. Then letting Y = kernel P , standard Banach space theory yields that π(Y ) = B/C(K) and y (λ + 1)πy for all y ∈ Y.

(3.20)

Now by Proposition 3.11 and its proof, B/C(K) is isometric to c0 and in fact [π(fd )]d∈D = B/C(K) and (πfd ) is isometrically equivalent to the c0 -basis (for c0 (D)). But it follows from (3.20) that we may then choose (unique) yd ’s in Y so that πyd = πfd

for all d

(3.21)

and cd yd (λ + 1) max |cd |

(3.22)

for any choice of scalars cd with cd = 0 for all but ﬁnitely many d. But (3.21) yields that for each d ∈ D there is a gd ∈ C(K) so that yd = fd + gd

for all d.

(3.23)

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At last, given ε > 0, we choose d1 , . . . , dn in D and v ∈ K satisfying (3.13), i.e., the conclusion of Lemma 3.13. But then we may choose scalars c1 , . . . , cn with |cj | = 1 for all j , so that cj (fdj + gdj )(v) > 1 − ε

for all 1 j n.

(3.24)

Hence n n cj y d j cj (fdj + gdj ) (v) > n − nε. j =1

(3.25)

j =1

Finally, (3.22) and (3.25) yield that λ + 1 > n − nε.

(3.26)

But ε > 0 was arbitrary, so the conclusion of the theorem follows.

To complete the proof of Amir’s theorem, we only need to exhibit a subset K of [0, 1] with K homeomorphic to ωω + and K(w) = ∅. This is easily done, in the next result. P ROPOSITION 3.15. Let α = ωn + for some 1 n ω. Then there is a subset K of [0, 1] which is homeomorphic to α, so that K (j ) = K(j ) for all j ω. Moreover then rcl(K) is algebraically isometric to C(K) ⊕ C(K). P ROOF. Obviously, we may put K inside [−1, 1] or any particular interval [a, b] instead. For n = 1 let K = {1/n, −1/n, 0: n = 1, 2, . . .}. Then evidently K(1) = K (1) = {0}, K is homeomorphic to ω+, and rcl(K) is clearly algebraically isometric to c ⊕ c = C(K) ⊕ C(K). Suppose 1 n < ∞, α = ωn +, and K = K α has been constructed satisfying the conclusion of the proposition. Let now {Kj : j ∈ Z ∼ {0}} be a family of “copies” of K α , where for each j 1, Kj ⊂

1 1 , j +1 j

(3.27(i))

while if j −1 Kj ⊂

1 1 , j j +1

Finally, let K α+1 =

j ∈Z Kj

(3.27(ii)) ∪ {0} where Z = Z ∼ {0}.

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Then for any 1 i n, (i) K α+1,(i) = Kj ∪ {0} j ∈Z

=

Kj,(i) ∪ {0}

j ∈Z α+1 = K(i) .

(3.28)

In particular, for any j , Kj(n) consists of a single point, xj . Thus, α+1 = {xj , x−j , 0: j ∈ N} K α+1,(n) = K(n)

(3.29)

and of course as in the ﬁrst step K α+1,(n+1) = Kα+1,(n+1) = {0}.

(3.30)

Now letting X = {f ∈ rcl(K α+1 ): f (x) = 0 for all x 0} and Y = {f ∈ rcl(K α+1 ): f (x) = 0 for all x < 0}, then rcl K α+1 = X ⊕ Y

(3.31)

(algebraically and isometrically, ∞ direct sum). But it follows easily from our induction hypothesis that X and Y are both algebraically isometric to C(ωn+1 +), whence the ﬁnal statement of the proposition holds. This proves the result for all n < ω. Of course, for α = ω, we may now just repeat the entire procedure, this time placing inn 1 1 1 ) and ( −n+1 , −n ), a “copy” of K ω + which we have constructed side each interval ( n1 , n+1 above, thus achieving the proof. C OROLLARY 3.16. For each n > 1, there is a unital subalgebra An of C(ωn · 2+) with An algebraically isometric to C(ωn +), which is not λ-complemented in C(ωn · 2+) for any λ < n − 1. There is also a unital subalgebra B of C(ωω +) which is algebraically isometric to C(ωω +) which is uncomplemented in C(ωω +). from the preceding two results, P ROOF. The ﬁrst assertion follows immediately . .∞ for the n A ) is uncomplemented in ( ﬁnal assertion, it follows that B0 =def ( ∞ j =1 j c0 n=1 C(ω · ω 2+))c0 . But the second space is just C0 (ω ), while B0 is also algebraically isometric to C0 (ωω ). Hence just taking the unitizations of each, the result follows. R EMARK 3.17. Of course, since C(ωn +) is isomorphic to c0 , it has the separable extension property. Thus, there exists λn so that for all separable Banach spaces X ⊂ Y and of T to Y with T λn T . Our operators T : X → C(ωn +), there is an extension T argument yields that λn > n − 1. Actually, Amir proves in [8] that λn = 2n + 1 for all n = 1, 2, . . . .

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We ﬁnally deduce Milutin’s result that the Cantor map induces an uncomplemented embedding of C([0, 1]) in C(D). We identify D with {0, 1}N as above, and let ϕ : D → [0, 1] be the Cantor map deﬁned above. P ROPOSITION 3.18. Let D be the set of dyadic rationals in (0, 1); i.e., D = {k/2n : 1 k < 2n , n = 1, 2, . . .}. There exists an algebraic surjective isometry T : C({0, 1}N) → rcl([0, 1], D) such that T (ϕ 0 (C[0, 1])) = C([0, 1]). Thus ϕ 0 (C[0, 1]) is uncomplemented in C({0, 1}N ) by Theorem 3.14. P ROOF. Deﬁne a “standard” partial inverse map τ : [0, 1] → D as follows. If x ∈ [0, 1], x∈ / D, there is a unique y ∈ D with ϕ(y) = x, and deﬁne τ (x) = y.

(3.32)

If x ∈ D, then there is a unique (ε j ) ∈ D so that for a unique n 1, εj = 0 for all j > n, εn = 1, and ϕ((εj )) = x, i.e., x = nj=1 εj /2j . Now deﬁne

τ (x) = (εj ) .

(3.33)

Of course then

ϕ τ (x) = x

for all x ∈ [0, 1].

(3.34)

Now deﬁne T by

(Tf )(x) = f τ (x) for all f ∈ C(D), x ∈ [0, 1].

(3.35)

Now it easily follows that T is an algebraic isometry mapping C(D) into ∞ [0, 1]. We now easily check that

T ϕ0f = f

for all f ∈ C [0, 1] .

(3.36)

Moreover, if f ∈ C(D), then Tf is continuous at x for all x ∈ / D.

(3.37)

Finally, let x ∈ D and (εj ) = τ (x) with n as given preceding (3.33). Let (ym ) be a sequence in [0, 1] with ym → x. Suppose ﬁrst that x < ym

for all m.

Then it follows that for all m,

(m) (m) τ (ym ) = ε1 , . . . , εn , βn+1 , βn+2 , . . .

(3.38)

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and in fact then τ (ym ) → τ (x) = (ε1 , . . . , εn , 0, . . .). Hence

(Tf )(ym ) = f τ (ym ) → f τ (x) = (Tf )(x)

(3.39)

by continuity of f . Thus Tf is indeed right continuous at x. Suppose next that ym < x

for all m.

(3.40)

This time, it follows that

(m) (m) τ (ym ) = ε1 , . . . , εn−1 , 0, βn+1 , βn+2 , . . . for all m, and in fact now def

τ (ym ) → (ε1 , ε2 , . . . , εn−1 , 0, 1, 1, 1 . . .) = z. Hence now,

(Tf )(ym ) = f τ (ym ) → f (z)

(3.41)

by continuity of f , showing that Tf has a left-limit. Thus we have indeed proved that

T C(D) ⊂ rcl [0, 1], D . We may check, however that conversely if f ∈ rcl([0, 1], D), then deﬁning f˜ on D by f˜(τ x) = f (x) for all x ∈ [0, 1],

f˜(y) = f ϕ(y) −

(3.42) (3.43)

if y ∈ D ∼ τ ([0, 1]), then f˜ ∈ C(D), and hence ﬁnally T satisﬁes the conclusion of 3.18, completing the proof.

4. Operators on C(K)-spaces Throughout, K denotes a compact Hausdorff space. By an operator on C(K) we mean a bounded linear operator from C(K) to some Banach space X. Of course, a “C(K)-space” is just C(K) for some K. We ﬁrst recall the classical result of Dunford and Pettis. T HEOREM 4.1 ([25]). An operator on a C(K)-space maps weakly compact sets to compact sets. See [33], p. 62 for a proof. We note the following immediate structural consequence.

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C OROLLARY 4.2. Let T : C(K) → C(K) be a given weakly compact operator. Then T 2 is compact. Hence if T is a projection, its range is ﬁnite-dimensional. Evidently the ﬁnal statement may be equivalently formulated: every reﬂexive complemented subspace of a C(K)-space is ﬁnite-dimensional. We are interested here in non-weakly compact operators. Before focusing on this, we note the following structural result due to the author [57]. T HEOREM 4.3. A reﬂexive quotient space of a C(K)-space is isomorphic to a quotient space of an Lp (μ)-space for some 2 p < ∞. Let us note that conversely, Lp is isometric to a quotient space of C([0, 1]) for all 2 p < ∞. (Throughout, for all 1 p < ∞, Lp denotes Lp (μ), where μ is Lebesgue measure on [0, 1].) Theorem 4.3 is in reality the dual of the version of the main result in [57]: every reﬂexive subspace of L1 is isomorphic to a subspace of Lp for some 1 < p 2. We now focus on the main setting of this section – “ﬁxing” properties of various classes of non-weakly compact operators on C(K)-spaces. D EFINITION 4.4. Let X, Y, Z be Banach spaces. An operator T : X → Y ﬁxes Z if there exists a subspace Z of X with Z isomorphic to Z so that T |Z is an isomorphism. We now summarize the main results to be discussed here. The ﬁrst result is due to Pełczy´nski [52]. T HEOREM 4.5. A non-weakly compact operator on a C(K)-space ﬁxes c0 . To formulate the next result, we will need the notion of the Szlenk-index of an operator. D EFINITION 4.6. Let X and Y be separable Banach spaces and T : X → Y be a given operator; let ε > 0. The ε-Szlenk index of T , βε (T ), is deﬁned as βε (T ∗ (Ba(X∗ ))), where βε is as in Deﬁnition 2.16. Sz(T ), the Szlenk index of T , is deﬁned as:

Sz(T ) = sup βε (T ) = β T ∗ Ba Y ∗ . ε>0

The results in Section 2 show that Sz(C(ωω +)) = ω2 , it was in fact rather easy to obtain that Sz(C(ωω +)) ω2 . It follows easily that if an operator on a separable C(K)-space ﬁxes C(ωω +), its Szlenk index is at least ω2 . The converse to this, is due to Alspach. T HEOREM 4.7 ([1]). Let K be a compact metric space, X be a Banach space, and T : C(K) → X a given operator. The following are equivalent. (1) Sz(T ) ω2 . (2) βε (T ) ω for all ε > 0. (3) T ﬁxes C(ωω +). We next give another characterization of operators ﬁxing C(ωω +), due to Bourgain.

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D EFINITION 4.8. Let X, Y be Banach spaces and T : X → Y be a given operator. T is called a Banach–Saks operator if whenever (xj ) is a weakly null sequence in X, there is a subsequence (xj ) of (xj ) so that n1 nj=1 T (xj ) converges in norm to zero. X has the weak Banach–Saks property if IX is a Banach–Saks operator. It is easily seen that c0 has the Banach–Saks property, It is a classical result, due to Schreier, that C(ωω +) fails the weak Banach–Saks property [64]. Thus any operator on a C(K)-space ﬁxing C(ωω +), is not a Banach–Saks operator. Bourgain established the converse to this result in [18]. T HEOREM 4.9. A non-Banach–Saks operator on a C(K)-space ﬁxes C(ωω +). Bourgain also obtains “higher ordinal” generalizations of Theorem 4.7, which we will brieﬂy discuss. The ﬁnal “ﬁxing” result in this summary is due to the author. T HEOREM 4.10 [56]. Let K be a compact metric space, X be a Banach space, and T : C(K) → X be a given operator. Then if T ∗ (X∗ ) is non-separable, T ﬁxes C([0, 1]). The proofs of these results involve properties of L1 (μ)-spaces, for by the Riesz representation theorem, C(K)∗ may be identiﬁed with M(K), the space of scalar-valued regular countably additive set functions on B(K) the Borel subsets of K. A. Operators ﬁxing c0 Theorem 4.5 follows quickly from the following two L1 theorems, which we do not prove here. The ﬁrst is due to Grothendieck (Théorème 2, p. 146 of [30]). T HEOREM 4.11. Let W be a bounded subset of M(K). Then W is relatively weakly compact if and only if for every sequence O1 , O2 , . . . of pairwise disjoint open subsets of K, μ(Oj ) → 0 as j → ∞, uniformly for all μ in W .

(4.1)

The second is a relative disjointness result due to the author. P ROPOSITION 4.12 ([55]). Let μ1 , μ2 , . . . be a bounded sequence in M(K), and let E1 , E2 , . . . be a sequence of pairwise disjoint Borel subsets of K. Then given ε > 0, there exist n1 < n2 < · · · so that for all j ,

|μnj |(Eni ) < ε.

i=j

(For any sequence (fj ) in a Banach space, [fj ] denotes its closed linear span.)

(4.2)

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P ROOF OF T HEOREM 4.5. Assume X is a Banach space and T : C(K) → X is not weakly compact. Then also T ∗ : X∗ → C(K)∗ = M(K) is non-weakly compact, so in particular

def W = T ∗ Ba X∗ is non-weakly compact.

(4.3)

Of course then W is not relatively weakly compact, since it is closed. Thus by Gothendieck’s theorem, we may choose η > 0, a sequence O1 , O2 , . . . of disjoint open sets in K, and a sequence μ1 , μ2 , . . . in W with μj (Oj ) > η

for all j .

(4.4)

Let then 0 < ε < η. By passing to a subsequence of the Oj ’s and μj ’s, we may also assume by Proposition 4.12 that

|μj |(Oi ) < ε

for all j .

(4.5)

i=j

For each j , by (4.4) we may choose fj ∈ C(K) of norm 1 with 0 fj 1 and fj supported in Oj , so that fj dμj > η.

(4.6)

Set Z = [fj ]. It is immediate that Z is isometric to c0 ; in fact (fj ) is isometrically equivalent to the c0 basis. Thus we have that given n and scalars c1 , . . . , cn , n cj fj T max |cj |. T j

(4.7)

j =1

But for each j n n ∗ ∗ ci fi sup T x ci fi T x ∗ ∈Ba X∗ i=1 i=1 n ci fi dμj i=1 |ci | |fi | d|μj | |cj | fj dμj − i=j

|cj |η − max |ci |

|μj |(Oi )

i=j

|cj |η − max |ci |ε.

(4.8)

The Banach spaces C(K)

1583

But then taking the max over all j , we get that n ci fi (η − ε) max |ci |. T

(4.9)

i=1

(4.6) and (4.9) yield that T |Z is an isomorphism, completing the proof.

B. Operators ﬁxing C(ωω +) We will not prove 4.7. However, we will give the description of the isometric copy of C(ωω +) which is ﬁxed, in Bourgain’s proof of this result. We ﬁrst indicate yet another important description of the C(α+)-spaces, due to Bourgain, which is fundamental in his approach. D EFINITION 4.13. Let T∞ be the inﬁnitely branching tree consisting of all ﬁnite sequences of positive integers. For α, β in T∞ , deﬁne α β if α is an initial segment of β; i.e., if α = (j1 , . . . , jk ) and β = (m1 , . . . , m ), then k and ji = mi for all 1 i k. Also, let (α) = k, the length of α. The empty sequence ∅ is the “top” node of T∞ . A set T ⊂ T∞ will be called a tree if whenever β ∈ T and α ∈ T∞ , α β and α = ∅, then α ∈ T . Finally, a tree T is called well-founded if it contains no strictly increasing sequence of elements of T∞ . We now deﬁne Banach spaces associated to trees T , denoted XT , as follows. D EFINITION 4.14. Let T be a well founded tree, and let c00 (T ) denote all systems (cα )α∈T of scalars, with ﬁnitely many cα ’s non-zero. Deﬁne a norm · T on c00 (T ) by (cα ) = max cγ . T α∈T

(4.10)

γ α

Let XT denote the completion of c00 (T ) under · T . P ROPOSITION 4.15. Let T be an inﬁnite well-founded tree. Then there exists a countable limit cardinal α so that XT is either isometric to C0 (α) (if T has inﬁnitely many elements of length 1 and φ ∈ / T ) or to C(α+). Conversely, given any such ordinal α, there exists a tree T with XT isometric to C0 (α) or to C(α+). Moreover, let T be a given inﬁnite tree, and for α ∈ T let bα be the natural element of c00 (T ): (bα )(β) = δαβ . Let τ : N → T be a bijection (i.e., an enumeration) so that if τ (i) < τ (j ), then i < j . Then (bτ (j ) )∞ j =1 is a monotone basis for XT . All of the above assertions and developments are due to Bourgain [18], except for the basis assertion, which is due to Odell. The author is most grateful to Professor Odell for his personal explanations of these results.

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We next indicate the trees Tn corresponding to the spaces C(ωn +) for 0 n ω. For n ﬁnite, simply let Tn be all ﬁnite sequences of positive integers of length at most n; also let Tn0 = Tn ∼ {∅}. Finally, let Tω0 = ∞ {(n, α): α ∈ Tn−1 }, and let Tω = Tω0 ∪ {∅}. The n=1 reader should have no difﬁculty in establishing the assertions of 4.15 in this special case. In particular, for all 1 n ω, XTn0 is isometric to C0 (ωn ) and XTn is isometric to C(ωn +). There still remains the intuitive issue: what is the picture for a subspace of C(K) which is isomorphic to C(ωω +) and ﬁxed by an operator T satisfying the hypotheses of 4.7? The following elegant description gives Bourgain’s answer. D EFINITION 4.16. Let F be a family of non-empty clopen subsets of a totally disconnected inﬁnite compact metric space K. F is called regular if (a) any two elements of F are either disjoint or one is contained in the other. (b) F has no inﬁnite totally ordered subsets, under the order A B if A ⊃ B. We again leave the proof of the following motivating result to the reader. P ROPOSITION 4.17. Let F be an inﬁnite regular family of clopen subsets of K. There is a well founded tree T and an order preserving bijection τ : F → T . [F ] is a subalgebra of C(K) which is algebraically isometric to C(α+) or C0 (α) for some countable ordinal α. Moreover, identifying τ (A) with the basis elements bτ (A) of Proposition 4.15, then τ extends to a linear isometry of [F ] with XT . Conversely, given any well-founded tree T , then there exists a regular family F (for a suitable K) with F order isomorphic to T . Let us just indicate pictures for the regular families corresponding to Tn0 and Tn , and thus to C0 (ωn ) and C(ωn +). Of course, a sequence of disjoint clopen sets spans c0 isometrically. T10

◦

◦

◦

◦

···

We get T1 by putting all these inside one clopen set, which actually corresponds then to the function 1 in C(ω+) = c. T1

◦

◦

◦

◦

···

Now we can get T20 by repeating T1 inﬁnitely many times. T1

◦ ◦ ◦ ◦···

◦ ◦ ◦ ◦ ···

◦ ◦ ◦ ◦ ···

···

The Banach spaces C(K)

1585

Of course, we then put all these inside one clopen set, to obtain T2 . Finally, we obtain Tω0 by choosing a sequence of disjoint clopen sets O1 , O2 , . . . and inside On , we put the regular system corresponding to Tn0 . Bourgain proves Theorem 4.7 by establishing the following general result. T HEOREM 4.18. Let K a totally disconnected compact metric space, X a Banach space, and T : C(K) → X a bounded linear operator be given such that for some ε > 0 and countable ordinal α βε (T ) ωα . Then there is a regular system F of clopen subsets of K with Y =def [F ] isometric to C0 (ωω·α ) such that T |Y is an isomorphism. The whole point of our exposition: one must choose a regular family of clopen sets to achieve the desired copy of C(ωω +); this requires the above concepts. R EMARK 4.19. In view of Milutin’s theorem, it follows that for any compact metric space K and operator T : C(K) → X, T ﬁxes C(ωω·α +) provided βε (T ) = ωα . Thus Theorem 4.7 follows, letting α = 1. Actually, Alspach obtains in [1] that if βε (T ) ω for some ε > 0, K arbitrary, then still T ﬁxes some subspace of C(K) isometric to C0 (ωω ). We turn next to the basic connection between the weak Banach–Saks property and C(ωω +). We ﬁrst give Schreier’s fundamental example showing that C(ωω +) fails the weak Banach–Saks property; i.e., there exists a weakly null sequence in C(ωω +), such that no subsequence has its arithmetic averages tending to zero in norm. P ROPOSITION 4.20. There exists a sequence U1 , U2 , . . . of compact open subsets of ωω such that setting bj = XUj for all j , then (a) no point of ωω belongs to inﬁnitely many of the Uj ’s, (b) for all scalars (cj ) with only ﬁnitely many cj ’s non-zero r (4.11) cji : j1 < · · · < jr and r j1 . cj bj = max i=1

Before proving this, we ﬁrst show that the sequence (bj ) in 4.20 is an “anti-Banach– Saks” sequence. For convenience, we restrict to real scalars. P ROPOSITION 4.21. Let (bj ) be as in 4.20. Then bj → 0 weakly. Deﬁne a new norm on the span of the bj ’s by r |cji |: r = j1 and j1 < · · · < jr . cj bj = max i=1

Then x |||x||| 2x for all x ∈ [bj ].

(4.12)

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H.P. Rosenthal

It follows immediately that given j1 < j2 < · · ·, then norm as r → ∞. In fact, we have for all k that 2k bji i=1

1 r

r

i=1 bji

does not tend to zero in

2k 1 k bji . 2 2

(4.13)

i=1

We also see the fundamental phenomenon: any k-terms of the bj ’s past the k-th are 2-equivalent to the 1k -basis. It also follows, incidentally, that the norm-condition (4.13) alone, insures that bj → 0 weakly. We prefer however, to give the simpler argument which follows from 4.20. Finally, it follows that the sequence (bj ) in 4.20 is an unconditional basic sequence. P ROOF OF 4.21. Since no k belongs to inﬁnitely many of the Kj ’s, it follows that XKj → 0 pointwise on ωω +, which immediately yields that XKj → 0 weakly, by the “baby” version of the Riesz representation theorem. The lower estimate in (4.12) is trivial. But if we ﬁx cj ’s, r 1 and r = j1 < j2 < · · · < jr , then there is a subset F of j1 , . . . , jr , with r 1 c |cji |. i 2 i∈F

(4.14)

i=1

But if we enumerate F as i1 < · · · < ik , then trivially k r j1 i1 , hence r 1 1 ci |cji | |||cj bj |||. cj · bj 2 2 i∈F

(4.15)

i=1

P ROOF OF 4.20. We give yet one more (and last!) conceptualization of the compact countable spaces ωn + and ωω +. We identify their elements with certain ﬁnite subsets of N. Let F be a family of ﬁnite subsets of N, so that F contains no inﬁnite sequences F1 , F2 , . . . with Fn Fn+1 for all n, and such that F is closed under pointwise convergence (where Fj → F means XFj → XF pointwise on N). It follows easily that F is then a compact metric space. Now ﬁrst let Fn be the family of all subsets of N of cardinality as most n. It follows that (j ) Fn is homeomorphic to ωn +. In fact, we obtain by induction that Fn = Fn−j , so that (0) ﬁnally Fn = {∅}. Now we “naturally” obtain Fω homeomorphic to ωω + as follows. def

Fω = {∅} ∪

∞

{α ∪ n: (α) n and the least element of α n}.

(4.16)

n=1

In other words, Fω consists of all ﬁnite sets whose cardinality is at most its least element. It is clear that Fω is indeed compact in the pointwise topology, and moreover it is also clear (n) that Fω = ∅ for all n = 1, 2, . . . . Finally, it is also clear that for each n, the n-th term in

The Banach spaces C(K)

1587 (ω)

the above union is homeomorphic to ωn +, and so we have that Fω = {∅}, whence Fω is homeomorphic to ωω +. Now for each j , deﬁne Uj = {α ∈ Fω : j ∈ α}.

(4.17)

It then easily follows that Uj is a clopen subset of Fω , and of course φ ∈ / Uj for any j . It is trivial that no α ∈ Fω belongs to inﬁnitely many Uj ’s since α is a ﬁnite set. Finally, let (cj ) be a sequence of scalars with only ﬁnitely many non-zero elements; then for any α ∈ Fω , cj : j ∈ α . cj bj (α) =

(4.18)

But if α = {j1 , . . . , jr } with j1 < j2 < · · · < jr , then by deﬁnition of Fω , r j1 , and conversely given j1 < · · · jr with r j1 , {j1 , . . . , jr } ∈ Fω . Thus (4.18) yields (4.11), completing the proof. Next we discuss Banach–Saks operators on C(K)-spaces. Actually, Theorem 4.9 is an immediate consequence of Theorem 4.7 and the following remarkable result ([18], Lemma 17) (see Deﬁnition 4.8 for the ε-Szlenk index of an operator) T HEOREM 4.22. Let X and Y be Banach spaces with X separable and T : X → Y be a given operator. Then if the ε-Szlenk index of T is ﬁnite for all ε > 0, T is a Banach–Saks operator. In particular, if the ε-Szlenk index of X is ﬁnite for all ε > 0, X has the weak Banach–Saks property. Just to clarify notation, we ﬁrst give the P ROOF OF T HEOREM 4.9. Let T : C(K) → Y be a non-Banach–Saks operator. It is trivial that then, without loss of generality, we may assume that C(K) is separable, i.e., that K is compact metric. Then by Theorem 4.22, there exists an ε > 0 such that βε (T ∗ (Ba(Y ∗ ))) ω. But then T ﬁxes C(ωω +) by Theorem 4.7. The initial steps in the proof of Lemma 17 of [18] (given as a lemma there) can be eliminated, using a fundamental dichotomy discovered a few years earlier. Moreover, the details of the proof of Lemma 17 itself do not seem correct. Because of the signiﬁcance of this result, we give a detailed proof here. The following is the basic dichotomy discovered by the author in [59]; several proofs have been given since, see, e.g., [46]. T HEOREM 4.23 ([59]). Let (bn ) be a weakly null sequence in a Banach space. Then (bn ) has a subsequence (yn ) satisfying one of the following mutually exclusive alternatives: (a) n1 nj=1 yj tends to zero in norm, for all subsequences (yj ) of (yj ).

1588

H.P. Rosenthal

(b) (yj ) is a basic sequence so that any k terms past the k-th are uniformly equivalent to the 1k basis. Precisely, there is a δ > 0 so that for all k < j1 < · · · < jk and scalars c1 , . . . , ck , k k ci yji δ |ci |. i=1

(4.19)

i=1

Note that it follows immediately that if (yj ) satisﬁes (b), there is a constant η > 0 so

that for any subsequence (yj ) of (yj ), 2k j =1 yj ηk for all k, hence no subsequence of (yj ) has averages converging to zero in norm. In modern terminology, (yj ) has a spreading model isomorphic to 1 . Notice that Schreier’s sequence given in Proposition 4.20 is a witness to this general phenomenon. Now we give the P ROOF OF T HEOREM 4.22. Let T : X → Y be a given operator, and suppose T is nonBanach–Saks. We may assume without loss of generality that T = 1. Choose (xn ) a weakly null sequence in X so that the arithmetic averages of T xn do not tend to zero in norm. Assume that xn 1 for all n. Now choose (bn ) a subsequence of (xn ) so that setting yn = T bn for all n, then for some δ > 0 (yn ) satisﬁes (4.19).

(4.20)

Let K = T ∗ (Ba(Y ∗ )). We shall prove that Pm (δ, K) = ∅ for all m = 1, 2, . . . ,

(4.21)

where the sets Pm (δ, K) are those originally deﬁned by Szlenk in his derivation (as deﬁned above, preceding Proposition 2.17). We need the following fundamental consequence of (4.19). For all m and α = (j1 , . . . , jm ) with m < j1 < j2 < · · · < jm , there exists a yα∗ ∈ Ba(Y ∗ ) with ∗ y (yj ) δ

for all j ∈ α.

(4.22)

Γm, = α ⊂ { + 1, + 2, . . .}: #α = m

(4.23)

α

Given m, 1, we set

(i.e., Γm, is just all m element subsets of N past the -th term). We now prove the following claim by induction on m. C LAIM . For all m, m and families {yα∗ : α ∈ Γm, } with yα∗ satisfying (4.22) for all α ∈ Γm, , there is a weak∗ -cluster point of {T ∗ (yα∗ ): α ∈ Γm, } belonging to Pm (δ, K).

The Banach spaces C(K)

1589

(This is a more delicate version of the apparently incorrect argument in [18]; the author nevertheless greatly admires the ingenuity of Bourgain’s discussion there). The case m = 1 is really immediate, just using the ω∗ -compactness of K. After all, given any n > , then by deﬁnition, ∗ ∗

T y (bn ) = y ∗ (yn ) δ (n) (n)

(4.24)

∗ ) lies in P (δ, K) since (b ) is weakly null in Ba(X). hence any ω∗ -cluster point of (T ∗ y(n) 1 n Now suppose the claim is proved for m, let m + 1, and let {yα∗ : α ∈ Γm+1, } be given with yα∗ satisfying (4.22) for all α ∈ Γm+1, . Fix n > , and deﬁne y˜α∗ for each α ∈ Γm,n by ∗ . y˜α∗ = yα∪{n}

But then for all α ∈ Γm,n , y˜α∗ satisﬁes (4.22), hence by our induction hypothesis, there exists a ω∗ -cluster point xn∗ of {T ∗ (y˜α∗ ): α ∈ Γm,n } which belongs to Pm (δ, K). But since α ∪ {n} ∈ Γm+1, , we have that ! ∗ ∗ " ! ∗ T y˜ , bn = y α

α∪{n} , yn

" δ

(4.25)

for all α ∈ Γm,n ; then also ∗ x (bn ) δ. n

(4.26)

∗ ∗ But then if x ∗ is a weak∗ -cluster point of (xn∗ )∞ n=+1 , x ∈ Pm+1 (δ, K), and of course x is ∗ ∗ ∗ indeed a weak -cluster point of {T (yα ): α ∈ Γm+1, }. This completes the induction step of the claim which then shows (4.21), so the proof of Theorem 4.22 is complete.

C. Operators ﬁxing C([0, 1]) We ﬁnally treat Theorem 4.10. We shall sketch the main ideas in the proof given in [56]. We ﬁrst note, however, that there are two other proofs known, both conceptually different from each other and from that in [56]. Weis obtains this result via an integral representation theorem for operators on C(K)-spaces [68]. For extensions of this and further complements in the context of Banach lattices see [29] and [26]. Finally, the general result 4.18 also yields 4.10, as noted by Bourgain in [18]. Indeed, suppose T : C(K) → X are given as in the statement of 4.10, where K is totally disconnected. Then 4.18 yields that T ﬁxes C(L) for every countable subset L of [0, 1]. But the family of all compact subsets L of [0, 1] such that T ﬁxes C(L), forms a Borel subset of the family of all compact subsets L of [0, 1] in the Hausdorff metric; the countable ones, however are not a Borel set by classical descriptive set theory. Hence there is an uncountable compact L ⊂ [0, 1] so that F ﬁxes L, and then C(L) is isomorphic to C([0, 1]) by Milutin’s theorem. We also note that a reﬁnement of the arguments in [56] yields the following generalization of 4.10, due jointly to Lotz and Rosenthal [44]: let E be a separable Banach lattice with E ∗ weakly sequentially complete, X be a Banach space, and T : E → X be an operator with T ∗ (X∗ ) non-separable. Then T ﬁxes C([0, 1]). For extensions of this and further complements in the context of Banach lattices, see [29] and [26].

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H.P. Rosenthal

We ﬁrst introduce some (standard) terminology. Let X be a Banach space, Y be a subspace of X, and W be a subset of Ba(X∗ ). We say that W norms Y if there exists a constant λ 1 so that (4.27) y λ sup w(y) for all y ∈ Y . w∈W

If (4.27) holds, we say that W λ-norms Y . Now let K be a compact metric space. Theorem 4.10 then follows immediately from the following stronger statement: T HEOREM 4.24. A non-separable subset of Ba(C(K)∗ ) norms a subspace of C(K) isometric to C(D). Indeed, we may obviously assume that T : C(K) → X has norm one. Then assuming T ∗ X∗ is non-separable, so is W = T ∗ (Ba(X∗ )), and thus 4.24 yields a subspace Y of C(K) with Y isometric to C(D) and T |Y an isomorphism; then T ﬁxes C(D), and so of course C([0, 1]), which is isometric to a subspace of C(D). The proof of 4.24 proceeds by reduction to the following almost isometric result. L EMMA 4.25. Let Z be a subspace of C(K)∗ with Z isometric to L1 . Then for every ε > 0, Ba(Z) (1 + ε)-norms a subspace of C(K) which is isometric to C(D). We shall sketch some of the ideas in the proof of 4.25 later on. We ﬁrst note that the actual proof of Theorem 4.24 yields the following dividend. C OROLLARY 4.26. Let Z be a non-separable subspace of C(K)∗ . Then for all ε > 0, Ba(Z) (1 + ε)-norms a subspace of C(K) which is isometric to C(D). Of course 4.26 has the following immediate consequence. C OROLLARY 4.27. Let X be a quotient space of C(K) with X∗ non-separable. Then X contains a subspace (1 + ε)-isomorphic to C(D) for all ε > 0. R EMARK 4.28. The conclusion badly fails for subspaces X of C(D) which are themselves isomorphic to C(D). Indeed, it is proved in [42] that for every λ > 1 there exists a Banach space X which is isomorphic to C(D) but contains no subspace λ-isomorphic to C(D); of course X is isometric to a subspace of C(D). We now take up the route which leads to Lemma 4.25. Say that elements μ and ν of C(K)∗ are pairwise disjoint if μ and ν are singular, regarding μ, ν as complex Borel measures on K. The next result is proved by a two-step transﬁnite induction. L EMMA 4.29. Let L be a convex symmetric non-separable subset of Ba(C(K))∗ . Then there is a δ > 0 so that for all 0 < ε < δ, there exists an uncountable subset {α }α∈Γ of L and a family {μα }α∈Γ of pairwise disjoint elements of Ba(C(K)∗ ) so that for all α, μα − α ε

and μα δ.

Moreover if L is the unit ball of a subspace of C(K)∗ , one can take δ = 1.

(4.28)

The Banach spaces C(K)

1591

Now of course the family {μα /μα : α ∈ Γ } is isometrically equivalent to the usual 1 (Γ ) basis. But this is also an uncountable subset of a compact metrizable space, Ba(C(K)∗ ) in the ω∗ -topology. So it follows that we may choose α1 , α2 , . . . distinct el∗ ements of Γ with (fn )∞ n=1 ω -dense in itself where fn = μαn /μαn for all n; note that ∞ (fn )n=1 is isometrically equivalent to the usual 1 basis. A variation of an argument of Stegall [66] now yields P ROPOSITION 4.30. Suppose X is a separable Banach space and (fn ) in X∗ is isometrically equivalent to the 1 basis and ω∗ -dense in itself. Then there exists a subspace U of X∗ , isometric and ω-isomorphic to C(D)∗ , such that for all x ∈ X, (4.29) sup u(x) supfn (x). n

u∈Ba(U )

Of course U is obtained as T ∗ (C(D))∗ where T is constructed to be a quotient map of X onto C(D)). For our purposes, we only need that U contains a subspace isometric to L1 . We now complete the proof of Theorem 4.24, using 4.25, 4.29 and 4.30. Let L be as in 4.24. We may assume that L is convex and symmetric, for if L is the closed convex hull of L ∪ −L, |(x)| for all x ∈ C(K). Let δ satisfy the conclusion of Lemma then sup∈L |(x)| = sup∈ L 4.29, and let 0 < ε so that 1−ε−

ε > 0. δ

(4.30)

Now let (α )α∈Γ and (μα )α∈Γ satisfy the conclusion of 4.29. Choose α1 , α2 , . . . distinct ∗ α’s so that (fn )∞ n=1 is ω -dense in itself, where fn = μαn /μαn for all n. Also let yn = αn and δn = μαn for all n. Next, choose Z a subspace of C(K)∗ isometric to L1 such that for all x ∈ C(K), (4.31) sup z(x) supfn (x) n

z∈Ba Z

thanks to Proposition 4.30. Finally, choose X a subspace of C(K) with X isometric to C(D)) so that (4.32) (1 − ε)x sup z(x) for all x ∈ X, z∈Ba Z

by Lemma 4.25. Now by our deﬁnition of the yn ’s and δn ’s, we have for all n that (by (4.28)) fn − yn /δn

ε ε . δn δ

Thus ﬁnally ﬁxing x ∈ X with x = 1, we have 1 − ε supfn (x) by (4.31) and (4.32) n

(4.33)

(4.34)

1592

H.P. Rosenthal

ε 1 supyn (x) + δ n δ

by (4.33).

(4.35)

Hence ε 1−ε− δ supyn (x). δ n

(4.36)

Thus letting λ = ((1 − ε − εδ )δ)−1 , we have proved that L λ-norms X. Finally, if L is the unit ball of a non-separable subspace of C(K)∗ , we may choose δ = 1 by 4.29; but then it follows that since ε may be chose arbitrarily small, λ is arbitrarily close to 1, and this yields Corollary 4.26. We ﬁnally treat Lemma 4.25. Let then Z be a subspace of C(K)∗ which is isometric to L1 . Standard results yield that there exists a Borel probability measure μ on K, a Borel measurable function θ with |θ | = 1, a compact subset S of K with μ(S) = 1, and a σ -subalgebra S of the Borel subsets of S such that (S, S, μ|S) is a purely non-atomic measure space and Z = θ · L1 (μ|S). (We adopt the notation: θ · Y = {θy: y ∈ Y }.) The desired isometric copy of C(D) which is (1 + ε)-normed by Z is now obtained through the following construction. L EMMA 4.31. Let μ, E, and S be as above, and let ε > 0. Then there exist sets Fin ∈ S and compact subsets Kin of S satisfying the following properties for all 1 i 2n and n = 0, 1, . . . . (i) Kin ∩ Kin = Fin ∩ Fin = ∅ for any i = i. n+1 n+1 n+1 ∪ K2i and Fin = F2i−1 ∪ F2in+1 . (ii) Kin = K2i−1 n n (iii) Ki ⊂ Fi . (iv) (1 − ε)/2n μ(Kin ) and μ(Fin ) 1/2n . (v) θ |K10 is continuous relative to K10 . (This is Lemma 1 of [56], with condition (v) added as in the correction to [56].) We conclude our discussion with the P ROOF OF L EMMA 4.25. Let F = K10 and let A denote the closure of the linear span of {χKin : 1 i 2n , n = 0, 1, 2, . . .} in C(F ). Then A is a subalgebra of C(F ) algebraically isometric to C(D). Hence also, def Y = θ¯ · A

is a subspace of C(F ) isometric to C(D).

(4.37)

(θ¯ denotes the complex conjugate of θ , in the case of complex scalar.) Now let E : C(F ) → C(K) be an isometric extension operator, as insured by the Bosuk theorem (Lemma 2.5(b) above). Finally, set X = E(Y ). So, evidently X is a subspace of C(K), isometric to C(D). We claim that Ba(Z)

1 − norms X 1 − 2ε

(4.38)

The Banach spaces C(K)

1593

which yields 4.25. Of course, it sufﬁces to show that for a dense linear subspace X0 of X sup z(x) (1 − 2ε)x for all x ∈ X0 .

(4.39)

z∈Ba Z

We take X0 to be the linear span of the functions E(θ¯ · χKin ). So, ﬁx n and let n

φ = θ¯

2

ci χKin

for scalars c1 , . . . , c2n with maxi |ci | = 1.

(4.40)

i=1

˜ = 1. Of course we identify the elements of Z with the comFinally, let φ˜ = E(φ). So φ plex Borel measures in θ · L1 (μ|S). So choose i with |ci | = 1 and let f = θ χFin /μ(Fin ). Then f L1 (μ) = 1, so f · μ as an element of Z, also has norm 1. We have that f φ˜ dμ

Kin

f φ˜ dμ −

Fin ∼Kin

f φ˜ dμ

f φ˜ dμ

=

μ(Kin ) − μ(Fin )

μ(Kin ) μ(Fin ) − μ(Kin ) − μ(Fin ) μ(Fin )

1 − 2ε

Fin ∼Kin

since φ˜ 1

by Lemma 4.31.

This concludes the proof, and our discussion of Theorem 4.10.

(4.41)

5. The complemented subspace problem In its full generality, this problem, (denoted the CSP), is as follows: let K be a compact Hausdorff space and X be a complemented subspace of C(K). Is X isomorphic to C(L) for some compact Hausdorff space L? We ﬁrst state a few results which hold in general, although most of them are easily reduced to the separable case anyway. All Banach spaces, subspaces, etc., are taken as inﬁnite-dimensional. T HEOREM 5.1 ([52]). Every complemented subspace of a C(K)-space contains a subspace isomorphic to c0 . This is an immediate consequence of Corollary 4.2 and Theorem 4.5. The next result reﬁnes Milutin’s theorem to the non-separable setting. P ROPOSITION 5.2 ([23]). A complemented subspace of a C(K)-space is isomorphic to a complemented subspace of C(L) for some totally disconnected L.

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Let us point out, however: it is unknown if a (non-separable) C(K)-space itself is isomorphic to C(L) for some totally disconnected L. Later on, we shall give results characterizing c0 (or rather c) as the smallest of the complemented subspaces of C(K)-spaces. Of course Theorem 4.10 characterizes C([0, 1]) as the largest separable case. T HEOREM 5.3 ([56]). Let X be a complemented subspace of a separable C(K)-space with X∗ non-separable. Then X is isomorphic to C([0, 1]). P ROOF. Assume then X is complemented in C(K) with C(K) separable, i.e., K is metrizable. Then K is uncountable, since C(K)∗ itself must be non-separable. Thus by Milutin’s theorem, C(K) is isomorphic to C([0, 1]). By Theorem 4.10, X contains a subspace isomorphic to C([0, 1]). By Pełczy´nski’s weak injectivity result (Theorem 3.1), X contains a subspace Y isomorphic to C([0, 1]) with Y complemented in C(K). Thus by the decomposition method (applying Proposition 1.2 to C(D) instead), X is isomorphic to C([0, 1]). For the next result, recall that a Banach space X is called primary if whenever X is isomorphic to Y ⊕ Z (for some Banach spaces Y and Z), then X is isomorphic to Y or to Z. The following result is due to Lindenstrauss and Pełczy´nski. C OROLLARY 5.4 ([42]). C([0, 1]) is primary. P ROOF. Suppose C([0, 1]) is isomorphic to X ⊕ Y . Then X∗ or Y ∗ is non-separable, and hence either X or Y is isomorphic to C([0, 1]) by the preceding result. R EMARK 5.5. Actually, the stronger result is obtained in [42]: let X be a subspace of C([0, 1]). Then C([0, 1]) embeds in either X or C([0, 1])/X. Also, it is established in [5] and independently, in [16], that C(K) is primary for all countable compact K. Thus, all separable C(K)-spaces are primary. We next give characterizations of C([0, 1]) which follow from Theorem 5.3 and some rather deep general Banach space principles. We assume K is general, although the result easily reduces to the metrizable case. T HEOREM 5.6. Let X be a complemented subspace of C(K). The following are equivalent. (1) C([0, 1]) embeds in X. (2) 1 embeds in X. (3) L1 embeds in X∗ . (4) X∗ has a sequence which converges weakly but not in norm. P ROOF. The implications (2) ⇒ (3) and (1) ⇒ (3) are due to Pełczy´nski (for general Banach spaces X) [53]. (Actually, (3) ⇒ (2) is also true for general X, by [53] and a

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reﬁnement due to Hagler [32].) Of course (1) ⇒ (2) is obvious, and so is (3) ⇒ (4), since 2 is isometric to a subspace of L1 . We show (4) ⇒ (2) ⇒ (1) to complete the proof. Let then (xn∗ ) in X∗ tend weakly to zero, yet for some δ > 0, ∗ x > δ n

for all n.

(5.1)

Then for each n, choose xn in Ba(X) with ∗ x (xn ) > δ. n

(5.2)

Now since X is complemented in C(K), X has the Dunford–Pettis property (i.e., X satisﬁes the conclusion of Theorem 4.1). But then (xn ) has no weak-Cauchy sequence.

(5.3)

Indeed, if a Banach space Y has the Dunford Pettis property, then yn∗ (yn ) → 0 as n → ∞ whenever (yn∗ ) is weakly null in Y and (yn ) is weak-Cauchy in Y ; so (5.3) follows in virtue of (5.2). But then by the 1 -theorem [58], (xn ) has a subsequence equivalent to the 1 -basis, hence (2) holds. (2) ⇒ (1) Let P : C(K) → X be a projection and let Y be a subspace of X isomorphic to 1 . Let Z be the conjugation-closed norm-closed unital subalgebra of C(K) generated by Y . Then by the Gelfand–Naimark theorem (which holds in this situation for real scalars also), Z is isometric to C(L) for some compact metric space L. Let T = P |Z. Since T |Y = I |Y , T ∗ (Z ∗ ) is non-norm-separable. Hence (1) holds by Theorem 4.10. For the remainder of our discussion, we assume the separable situation. Thus, K denotes a compact metric space; a “C(K)-space” refers to C(K) for some K, so it is separable. Now of course Theorem 5.3 reduces the CSP to the case of spaces X complemented in C(K) with X∗ separable. If the CSP has an afﬁrmative answer, such an X must be c0 saturated (see Proposition 3.5). This motivates the following special case of the CSP, raised in the 70’s by the author. P ROBLEM 1. Let X be a complemented subspace of C(K) so that X contains a reﬂexive subspace. Is X isomorphic to C([0, 1])? Although this remains open, it was solved in such special cases as: 2 embeds in X, by Bourgain, in a remarkable tour-de-force. T HEOREM 5.7 ([19]). Let X be a Banach space and let T : C(K) → X ﬁx a subspace Y ∗ ∗ of C(K) so that Y does not contain ∞ n ’s uniformly. Then T (X ) is non-separable. Of course then T ﬁxes C([0, 1]) by Theorem 4.10, and so we have the C OROLLARY 5.8. Let X be complemented in C(K) and assume X contains a subspace Y which does not contain ∞ n ’s uniformly. Then X is isomorphic to C([0, 1]).

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For the remainder of our discussion, we focus on spaces X with separable dual. The next result is due to Benyamini, and rests in part on a deep lemma due to Zippin ([70,71]) which we will also discuss. T HEOREM 5.9 ([13]). Let X be a complemented subspace of C(K). Then either X is isomorphic to c0 or C(ωω +) embeds in X. The following result is an immediate consequence, in virtue of the decomposition method and weak injectivity of C(ωω +), i.e., Theorem 3.1. C OROLLARY 5.10. A complemented subspace of C(ωω +) is isomorphic to c0 or to C(ωω +). Now of course, Theorem 5.9 implies Zippin’s remarkable characterization of separably injective spaces, since if C(ωω +) embeds in X, it also embeds complementably, and hence X cannot be separably injective by Amir’s theorem [8], obtained via Theorem 3.14 above. In reality, Theorem 5.9 rests fundamentally on the main step in [70],which may be formulated as follows [13]. (Let us call βε (Ba X∗ ) the ε-Szlenk index of X, where βε is given in Deﬁnition 1.11.) L EMMA 5.11 ([70]). Let X be a Banach space with X∗ separable, and let 0 < ε < 1/2. There is a δ > 0 so that if W is a ω∗ -compact totally disconnected (1 + δ)-norming subset of Ba(X∗ ) and if γ < ωα+1 with α the δ-Szlenk index of X, then there exists a subspace Y of C(W ) with Y isometric to C(γ +) so that for all x ∈ X, there exists a y ∈ Y with iW x − y (1 + ε)iW x.

(5.4)

(Here, (iW x)(w) = w(x) for all w ∈ W . Also, iW = i if W = Ba X∗ .) Zippin also proved in [70] the interesting result that for any separable Banach space X and δ > 0, there is a (1 + δ)-norming totally disconnected subset of Ba(X∗ ). Benyamini establishes the following remarkable extension of this in the main new discovery in [13]. T HEOREM 5.12. Let X be a separable Banach space and ε > 0. There exists a ω∗ compact (1 + ε)-norming subset W of Ba(X∗ ) and a norm one operator E : C(W ) → C(Ba(X∗ )) so that EiW x − ix εx

for all x ∈ X.

(5.5)

The preceding two rather deep results hold for general Banach spaces X. In particular, the non-linear approximation resulting from Zippin’s Lemma (Lemma 5.11) shows that in a sense, the C(K)-spaces with K countable play an unexpected role in the structure of general X. The next quite simple result, however, needed for Theorem 5.9, bears solely on the structure of complemented subspaces of C(K)-spaces. It yields (for possibly nonseparable) X that if X is isomorphic to a complemented subspace of some C(K)-space, then X is already complemented in C(Ba X∗ ) and moreover, the best possible norm of the projection is found there.

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P ROPOSITION 5.13 ([14]). Let X be given, let L = Ba(X∗ ), and suppose λ 1 is such that for some compact Hausdorff space Ω, there exist operators U : X → C(Ω) and V : C(Ω) → X with IX = V ◦ U

and U V λ.

(5.6)

Then i(X) is λ-complemented in C(L). P ROOF. Without loss of generality, U = 1. Let Ω be regarded as canonically embedded in C(Ω)∗ . Thus letting ϕ = U ∗ |Ω, ϕ maps Ω into L. So of course ϕ ◦ maps C(L) into C(Ω). We now simply check that V ϕ ◦ i(x) = x

for all x ∈ X.

(5.7)

Then it follows that V ϕ ◦ is a projection from C(L) onto iX, and of course ◦ V ϕ V ϕ ◦ λ.

(5.8)

R EMARK 5.14. Theorem 5.12 and the preceding proposition may be applied to C(K)spaces themselves to obtain that C(K) is (1 + ε)-isomorphic to a (1 + ε)-complemented subspace of C(D), for all ε > 0. Thus the main result in [13] yields another proof of Milutin’s theorem. We prefer the exposition in Section 2, however, for the above result “loses” cc the isometric fact that C(K) → C(D) for all K. The next remarkable result actually yields most of the known positive results in our present context. T HEOREM 5.15 ([13]). Let X∗ be separable, with X a Banach space isomorphic to a complemented subspace of some C(K)-space. There exists a δ > 0 so that if α is the δSzlenk index of X and γ < ωα+1 , then X is isomorphic to a quotient space of C(γ +). P ROOF. By the preceding result, i(X) is already complemented in C(L) where L = Ba(X∗ ) in its ω∗ -topology. Let P : C(L) → i(X) be a projection and let λ = P . Now let 0 < ε be such that 1 ελ < . 2

(5.9)

Choose ε > δ > 0 satisfying the conclusion of Zippin’s lemma. Now choose W a (1 + δ)norming totally disconnected ω∗ -compact subset of L and a norm one operator E satisfying the conclusion of Theorem 5.12; in particular, (5.5) holds. Finally let x ∈ X, and choose y ∈ Y satisfying (5.4). Then since E = 1, Ey − EiW x εiW x.

(5.10)

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Then by (5.5)

Ey − ix ε iW x + x 2εx.

(5.11)

Since P ix = ix, we have P Ey − ix 2εP x 2ελx.

(5.12)

Since 2ελ < 1 by (5.6) and of course ix = x, it follows ﬁnally by (5.12) that P E|Y maps Y onto X, completing the proof. We now obtain the P ROOF OF T HEOREM 5.9. Suppose ﬁrst that the ε-Szlenk index of X is ﬁnite for all ε > 0. Then by Theorem 5.15, there is a positive integer n so that X is isomorphic to a quotient space of C(ωn +). But in turn, C(ωn +) is isomorphic to c0 , and so X is thus isomorphic to a quotient space of c0 . Finally, every quotient space of c0 is isomorphic to a subspace of c0 by a result of Johnson and Zippin [35]. But X is also a L∞ -space by a result of Lindenstrauss and the author [41], and hence also by the results in [35], X is isomorphic to c0 . If the ε-Szlenk index of X is at least ω for some ε > 0, then X contains a subspace isomorphic to C(ωω +) by Alspach’s result, Theorem 4.7. Recall that a Banach space X is called an L∞ -space if there is a λ > 1 so that for all ﬁnite-dimensional E ⊂ X, there exists a ﬁnite-dimensional F with E ⊂ F ⊂ X so that

(5.13) d F, ∞ n λ, where n = dim F. If λ works, X is called an L∞,λ -space. Using partitions of unity, it is not hard to see that a C(K)-space is an L∞,1+ -space, i.e., it is an L∞,1+ε -space for all ε > 0. However, a Banach space X is an L∞,1+ -space if and only if it is an L1 (μ)-predual; i.e., X∗ is isometric to L1 (μ) for some μ. We shall discuss these brieﬂy later on. The result of [41] mentioned above: a complemented subspace of an L∞ -space is also an L∞ -space. In general, L∞ -spaces are very far away from C(K)spaces; however the following result due to the author [60], shows that small ones are very nice. (The result extends that of [35] mentioned above.) P ROPOSITION 5.16 ([60]). Let X be a L∞ -space which is isomorphic to a subspace of a space with an unconditional basis. Then X is isomorphic to c0 . (This was subsequently extended in [29] to L∞ -spaces which embed in a σ –σ Banach lattice.) Theorem 5.15 actually yields that if X is as in its statement, there exists a countable compact K so that X and C(K) have the same Szlenk index, with X isomorphic to a quotient space of C(K). Remarkably, Alspach and Benyamini prove in [6] that for any L∞ -space X with X∗ separable, one has that C(K) is isomorphic to a quotient space of X, K as above. So in particular, using also a result from Section 2, we have

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T HEOREM 5.17 ([13,6]). Let X be isomorphic to a complemented subspace of a C(K)space with X∗ separable. Then the Szlenk index of X is ωα+1 for some countable ordinal α α and then X and C(ωω +) are each isomorphic to a quotient space of the other. Despite the many positive results discussed so far, the eventual answer to the CSP seems far from clear. We conclude this general discussion with two more problems on special cases. Let then X be isomorphic to a complemented subspace of a C(K)-space with X∗ separable. P ROBLEM 2. Does X embed in C(α+) for some countable ordinal α? What if Sz(X) = ω2 ? 2

Finally, what is the structure of complemented subspaces of C(ωω +)? Speciﬁcally, 2

P ROBLEM 3. Let X be a complemented subspace of C(ωω +) with Sz(X) = ω2 . Is X 2 isomorphic to C(ωω +)? If Sz(X) = ω3 , is X isomorphic to C(ωω +) itself? We next indicate complements to our discussion. Alspach constructs in [1] a quotient space of C(ωω +) which does not embed in C(α+) for any ordinal α; thus Problem 2 cannot be positively solved by just going through quotient maps. The remarkable ﬁxing results Theorems 4.7 and 4.9 cannot be extended without paying some price. Alspach proves in 2 2 [4] that there is actually a surjective operator on C(ωω +) which does not ﬁx C(ωω +). α+1 This result has recently been extended by Gasparis [28] to the spaces C(ωω +) for all ordinals α and an even wider class of counterexamples is given by Alspach in [4]. Thus an afﬁrmative answer even to Problem 3 must eventually use the assumption that one has a projection, not just an operator. We note also results of Wolfe [69], which yield rather complicated necessary and sufﬁcient conditions that an operator on a C(K)-space ﬁxes C(α+). Some of the original motivation for the concept of L∞ -spaces was that these might characterize C(K)-spaces by purely local means. However Benyamini and Lindenstrauss discovered this is not the case even for L∞,1+ -spaces. They construct in [14] a Banach space X with X∗ isometric to 1 , such that X is not isomorphic to a complemented subspace of C([0, 1]). We note in passing, however, that the CSP is open for separable spaces X (in its statement) which are themselves L1 (μ) preduals. It is proved in [36] that separable L1 (μ) preduals X are actually isometric to quotient spaces of C([0, 1]). Hence if X is such a space and X∗ is non-separable, X contains for all ε > 0 a subspace (1 + ε)isomorphic to C([0, 1]), by the results of [56] discussed above. Also, it thus follows by Theorem 4.5 that separable L1 (μ) preduals contain isomorphic copies of c0 . Bourgain and Delbaen prove in [20] that separable L∞ -spaces are not even isomorphic to quotients of C([0, 1]) in general; they exhibit, for example, an L∞ -space such that every subspace contains a further reﬂexive subspace. For further counterexample L∞ -spaces of a quite general nature, see [21]. Here are some positive results on the structure of separable L∞ -spaces, which of course yield results on complemented subspaces of C(K)-spaces. Results of Lewis and Stegall

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[39] and of Stegall [66] yield that if X is a separable L∞ -space, then X∗ is isomorphic to 1 or to C([0, 1])∗ . Thus in particular, the duals of complemented subspaces of separable C(K)-spaces are classiﬁed. It is proved in [34] that every separable L∞ -space X has a basis which is moreover shrinking in case X∗ is separable. A later reﬁnement in [49] yields that the basis (bj ) may be chosen with d([bj ]nj=1 , ∞ n ) λ for all n (for some λ); of course this characterizes L∞ -spaces. We conclude with a brief discussion of the positive solution to the CSP problem in the isometric setting. Let L be a locally compact 2nd countable metrizable space and let X be a contractively complemented subspace of C0 (L). Then X is isomorphic to a C(K)space. The reason for this: such spaces X are characterized isometrically as Cσ -spaces. This is proved for real scalars in [43], and for complex scalars in [27]. For real scalars, X is a Cσ subspace of C(L) provided there exists an involutive homeomorphism σ : L → L such that X = {f ∈ C(L): f (σ x) = −f (x) for all x ∈ X}. See [27] for the complex scalar case. It follows by results of Benyamini in [11] that such spaces are isomorphic to C(K)spaces; in fact it is proved in [11] that separable G-spaces are isomorphic to C(K)-spaces. This family of spaces includes closed sublattices of C(K)-spaces. It then follows (using the known structure of Banach lattices) that if a complemented subspace X of a separable C(K)-space is isomorphic to a Banach lattice, Xis isomorphic to a C(K)-space. On the other hand, it remains an open question, if complemented subspaces of Banach lattices are isomorphic to Banach lattices. We note ﬁnally that Benyamini later constructed a counterexample to his result in the non-separable setting, obtaining a non-separable sublattice of a C(K)-space which is not isomorphic to a complemented subspace of C(L) for any compact Hausdorff space L [12]. For further complements on the CSP in the non-separable setting, see [71]; for properties of non-separable C(K)-spaces, see [48] and [72]. We note ﬁnally the following complement to the isometric setting [7]. If a Banach space X is (1 + ε)-isomorphic to a (1 + ε)-complemented subspace of a C(K)-space for all ε > 0, then X is contractively complemented in C(L) where L = (Ba(X)∗ , ω∗ ), hence X is a Cσ -space.

References [1] D.E. Alspach, Quotients of C[0, 1] with separable dual, Israel J. Math. 29 (1978), 361–384. [2] D.E. Alspach, A quotient of C(ωω ) which is not isomorphic to a subspace of C(α), α < ω, Israel J. Math. 33 (1980), 49–60. [3] D.E. Alspach, C(K) norming subsets of C[0, 1]∗ , Studia Math. 70 (1981), 27–61. [4] D.E. Alspach, Operators on C(ωα ) which do not preserve C(ωα ), Fund. Math. 153 (1997), 81–98. [5] D.E. Alspach and Y. Benyamini, Primariness of spaces of continuous functions on ordinals, Israel J. Math. 27 (1977), 64–92. [6] D.E. Alspach and Y. Benyamini, C(K) quotients of separable L∞ spaces, Israel J. Math. 32 (1979), 145– 160. [7] D.E. Alspach and Y. Benyamini, A geometrical property of C(K) spaces, Israel J. Math. 64 (1988), 179– 194. [8] D. Amir, Projections onto continuous functions spaces, Proc. Amer. Math. Soc. 15 (1964), 396–402. [9] S.A. Argyros and A.D. Arvanitakis, A characterization of regular averaging operators and its consequences, Studia Math. 151 (3) (2002), 207–226.

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[42] J. Lindenstrauss and A. Pełczy´nski, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249. [43] J. Lindenstrauss and D.E. Wulbert, On the classiﬁcation of the Banach spaces whose duals are L1 spaces, J. Funct. Anal. 4 (1969), 332–349. [44] H.P. Lotz and H.P. Rosenthal, Embeddings of C(Δ) and L1 [0, 1] in Banach lattices, Israel J. Math. 31 (1978), 169–179. [45] S. Mazurkiewicz and W. Sierpinski, Contribution à la topologie des ensembles dénombrables, Fund. Math. 1 (1920), 17–27. [46] S. Mercourakis, On Cesàro summable sequences of continuous functions, Mathematika 42 (1) (1995), 87– 104. [47] A.A. Milutin, Isomorphisms of spaces of continuous functions on compacts of power continuum, Tieoria Func. (Kharkov) 2 (1966), 150–156 (Russian). [48] S. Negrepontis, Banach spaces and topology, Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan, eds, Elsevier (1984), Chapter 23, 1045–1142. [49] N.J. Nielsen and P. Wojtaszczyk, A remark on bases in Lp -spaces with an application to complementably universal L∞ -spaces, [50] T. Oikhberg and H.P. Rosenthal, Extension properties for the space of compact operators, J. Funct. Anal. 179 (2) (2001), 251–308. [51] A. Pełczy´nski, Projections in certain Banach spaces, Studia Math. 19 (1960), 209–228. [52] A. Pełczy´nski, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Polish Acad. Sci. Math. Astr. Phys. 10 (1962), 265–270. [53] A. Pełczy´nski, On Banach spaces containing L1 (M), Studia Math. 30 (1968), 231–246. [54] A. Pełczy´nski, On C(S)-subspaces of separable Banach spaces, Studia Math. 31 (1968), 231–246. [55] H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math. 37 (1970), 13–36; Correction, ibid., 311–313. [56] H.P. Rosenthal, On factors of C([0, 1]) with non-separable dual, Israel J. Math. 13 (1972), 361–378. [57] H.P. Rosenthal, On subspaces of Lp , Ann. Math. 97 (1973), 344–373. [58] H.P. Rosenthal, A characterization of Banach spaces containing 1 , Proc. Nat. Acad. Sci. USA 71 (1974), 2411–2413. [59] H.P. Rosenthal, Normalized weakly null sequences with no unconditional subsequences, Durham Symposium on the Relations Between Inﬁnite-Dimensional and Finite-Dimensional Convexity, Bull. London Math. Soc. 8 (1976), 22–24. [60] H.P. Rosenthal, A characterization of c0 and some remarks concerning the Grothendieck property, Longhorn Notes, The University of Texas Functional Analysis Seminar (1982–83), 95–108. [61] H.P. Rosenthal, The complete separable extension property, J. Operator Theory 43 (2000), 324–374. [62] H.P. Rosenthal, Banach and operator space structure of C ∗ -algebras, to appear. [63] C. Samuel, Indice de Szlenk des C(K) (K espace topologique compact dé nombrable), Seminar on the Geometry of Banach Spaces, Vols. I, II, Publ. Math. Univ. Paris VII, Paris (1983), 81–91. [64] J. Schreier, Ein Gegenbeispiel zur Theorie der schwachen Konvergence, Studia Math. 2 (1930), 58–62. [65] A. Sobczyk, Projection of the space (m) on its subspace (c0 ), Bull. Amer. Math. Soc. 47 (1941), 938–947. [66] C. Stegall, Banach spaces whose duals contain 1 (7), with applications to the study of dual L1 (M)-spaces, Trans. Amer. Math. Soc. 176 (1973), 463–477. [67] W. Szlenk, The non-existence of a separable reﬂexive Banach space universal for all separable Banach spaces, Studia Math. 30 (1968), 53–61. [68] L.W. Weis, The range of an operator in C(X) and its representing stochastic kernel, Arch. Math. 46 (1986), 171–178. [69] J. Wolfe, C(α) preserving operators on C(K) spaces, Trans. Amer. Math. Soc. 273 (1982), 705–719. Elsevier, Amsterdam (to appear). [70] M. Zippin, The separable extension problem, Israel J. Math. 26 (3–4) (1977), 372–387. [71] M. Zippin, Extensions of bounded linear operators, Handbook of Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1703–1736 (this Handbook). [72] V. Zizler, Nonseparable Banach spaces, Handbook of Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2003), 1737–1810 (this Handbook).

CHAPTER 37

Concentration, Results and Applications Gideon Schechtman∗ Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel E-mail: [email protected]

Contents 1. Introduction: approximate isoperimetric inequalities and concentration 2. Methods of proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Isoperimetric inequalities, Brunn–Minkowski inequality . . . . . . 2.2. Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Product spaces. Induction . . . . . . . . . . . . . . . . . . . . . . . 2.4. Spectral methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Bounds on Gaussian processes . . . . . . . . . . . . . . . . . . . . 2.6. Other tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Dvoretzky-like theorems . . . . . . . . . . . . . . . . . . . . . . . 3.2. Fine embeddings of subspaces of Lp in lpn . . . . . . . . . . . . . 3.3. Selecting good substructures . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. 1605 . 1606 . 1606 . 1610 . 1614 . 1619 . 1624 . 1625 . 1625 . 1625 . 1627 . 1631 . 1632

Abstract Concentration inequalities are estimates for the degree of approximation of functions on metric probability spaces around their mean. It turns out that in many natural situations one can give very good such estimates, and that these are extremely useful. We survey here some of the main methods for proving such inequalities and give a few examples to the way these estimates are used.

∗ The author was partially supported by The Israel Science Foundation founded by The Academy of Science and Humanities.

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1. Introduction: approximate isoperimetric inequalities and concentration Let (Ω, F , μ) be a probability space where F is the Borel σ -ﬁeld with respect to a metric d on Ω. The isoperimetric problem for the probability metric space (Ω, F , μ, d) is: given 0 < a < 1 and ε > 0, what is inf μ(Aε ); A ∈ F , μ(A) = a ? and for what A is it attained. Here Aε , the ε neighborhood of A, is deﬁned as Aε = {ω ∈ Ω; d(ω, A) < ε}. There are relatively few interesting cases, some of which will be described below, in which the answer to this question is known. However, it turns out that for many applications a solution to a somewhat weaker question is sufﬁcient: instead of ﬁnding the actual inﬁmum of the quantity above it is enough to ﬁnd a good lower bound to μ(Aε ), subject to μ(A) = a. We shall refer to such a lower bound as a solution to the approximate isoperimetric inequality (for the given space and parameters) provided the solution is optimal except for absolute constants in the “right places”. Let us illustrate the above by the example most relevant for us. The space under question will be (S n−1 , F , μ, d). Here S n−1 is the unit sphere in Rn , d the geodesic distance, F the Borel σ -ﬁeld and μ the normalized Haar measure (the unique probability measure on S n−1 which is invariant under the orthogonal group). Lévy [38] stated and sketched a proof of the isoperimetric inequality for this space. For every a and ε the minimal set is an (arbitrary) cap (i.e., a d-ball) of measure a. For a cap B of measure 1/2, Bε is a cap of radius π/2 + ε. A standard computation then implies that, for a = 1/2, say, and √ 2 any ε μ(Aε ) μ(Bε ) 1 − π/8 e−ε n/2 for any Borel set A ⊂ S n−1 of measure 1/2. 2 Any inequality, μ(Aε ) 1 − e−cε n , holding for all A with μ(A) = 1/2, with c an absolute constant, will be referred to as an approximate isoperimetric inequality (for sets of measure 1/2) in this case. As we shall see below these inequalities are extremely powerful, the value of the constant c is of little importance for the applications we have in mind, and it is much easier to prove the approximate inequality than the isoperimetric one. Moreover, several proofs of the approximate isoperimetric inequality in this case (and there are many of them) can be generalized to other situations in which no isoperimetric inequality is known. The importance of the approximate isoperimetric inequalities stems from the fact that they imply the following concentration phenomenon. In the setup above, if μ(Aε ) 1 − η/2 for all A with μ(A) 1/2 and if f : Ω → R is a function with Lipschitz constant 1, i.e., |f (x) − f (y)| d(x, y) for all x, y ∈ Ω, then μ({x; |f (x) − M| ε}) η. Here M denotes the median of the function f , i.e., is deﬁned by μ({f M}), μ({f M}) 1/2. This is easily seen (and ﬁrst noticed by Lévy in the setting of S n−1 ) by applying the inequality μ(Aε ) 1 − η/2 once for the set {f M} and once for {f M}. If η is small this is interpreted as “any such f is almost a constant on almost all of Ω”. For example, in the example above we get that any Lipschitz function of constant one, 2 f : S n−1 → R, satisﬁes μ({x ∈ S n−1 ; |f (x) − M| ε}) 2 e−ε n/2 , which is quite counterintuitive. The median M can be replaced by the expectation of f , Ef = S n−1 f dμ provided we change the constants 2, 1/2 to other absolute constants. Furthermore, each of these two

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concentration inequalities is also equivalent (with a change of constants) to

2 μ × μ (x, y) ∈ S n−1 × S n−1 ; f (x) − f (y) ε C e−cε n . This holds not only in this particular example but in great generality (see, for example, [45], V. 4). The opposite statement to the one in the second to last paragraph also holds. Concentration implies approximate isoperimetric inequality: if μ({x; |f (x) − M| ε}) η for all Lipschitz functions with constant one then μ(Aε ) > 1 − η for all sets A of measure at least 1/2. This follows easily by considering the function f (x) = d(x, A). Milman realized the relevance of Lévy’s concentration inequality to problems in Geometry and Functional Analysis. Using it he found in [43] a new proof of Dvoretzky’s theorem [11] on Euclidean section of convex bodies which was much more accessible than the complicated original proof. Much more importantly, his proof is subject to vast variations and generalizations. See Section 3.1 for this proof. Except for using the idea of concentration in many instances himself, Milman also promoted the search for new concentration inequalities and new applications of them. We refer the interested reader to an expository article [44] written by Milman on the subject. In this article we survey many (but not all) of the methods of proof of concentration and approximate isoperimetric inequalities. We tried to concentrate mostly on methods which are quite general or that we feel were not explored enough and should become more general. There are many different such methods with some overlap as to the inequalities they prove. Section 2 contains this survey. In Section 3 we give a sample of applications of concentration inequalities. There are many more such applications. At some points our presentation is very sketchy since on one hand many of the applications need the introduction of quite a lot of tools not directly connected to the main theme here and on the other hand some of the subjects dealt with in this application section are also dealt with, with more details in other articles in this Handbook. We hope we give enough to wet the reader’s appetite to search for more in the original sources or the other articles of this Handbook. We would like to emphasize that this is far from being a comprehensive survey of the topic of concentration. This author has a soft point for new ideas in proofs and in many instances below preferred to give a glimpse into these ideas by treating a special case or a version of the relevant result which is not necessarily the last word on it rather than to give all the details on the subject. There are two recent books related to the subject matter here. Ledoux’s book [34] is very much in the spirit of Sections 2.3 and 2.4. Chapter 3 12 in Gromov’s book [19] presents a different point of view on the subject of concentration. 2. Methods of proof 2.1. Isoperimetric inequalities, Brunn–Minkowski inequality We start by stating two forms of the classical Brunn–Minkowski inequality. Here | · | denotes Lebesgue measure in Rn and A + B denotes the Minkowski’s addition of sets in Rn ; A + B = {a + b; a ∈ A, b ∈ B}.

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T HEOREM 1. (1) For every n and every two nonempty measurable subsets of Rn A and B, |A + B|1/n |A|1/n + |B|1/n .

(1)

(2) For every n, every two non-empty measurable subsets of Rn A and B and every 0 < λ < 1, λA + (1 − λ)B |A|λ |B|1−λ .

(2)

Equality in either inequality holds if and only if A and B are homothetic. Theorem 1 has many different proofs. We refer to [56] for two of them and for an extensive discussion concerning this theorem. A variation of this theorem was proved by Prékopa and Leindler [50,36]. One possible proof of their theorem is by induction on the dimension (see, e.g., [49]). Theorem 1 is a simple consequence of this theorem. T HEOREM 2. Let f, g, h be integrable non-negative valued functions on Rn and let 0 < λ < 1. Assume

h λx + (1 − λ)y f (x)λ g(y)1−λ ,

for all x, y ∈ Rn ,

(3)

then

Rn

h

λ Rn

f

1−λ Rn

g

.

(4)

Theorem 1 provides a simple proof of the classical isoperimetric inequality in Rn . To avoid restricting ourselves to bodies whose surface area is deﬁnable we prefer to state it as: for every 0 < a < ∞ and every ε > 0, among all bodies of volume a in Rn the ones for which the volume of Aε is minimal are exactly balls of volume A. Maurey [42] noticed that Theorem 2 can be used to give a simple proof of the approximate isoperimetric inequality on the sphere (or equivalently for the canonical Gaussian measure on Rn ). Recently, Arias-de-Renya, Ball and Villa [4] discovered an even more direct proof of the approximate isoperimetric inequality on the sphere, using Theorem 1. Their proof actually establishes a far reaching generalization originally due to Gromov and Milman [21]. We refer to [25] for a discussion of the notion of uniform convexity. We only recall the following (equivalent) deﬁnition for the modulus of convexity δ of a normed space (X, · ): x + y ; x, y 1, x − y ε . δ(ε) = inf 1 − 2

(5)

Given a norm · on Rn we consider, in the following theorem, the set S = {x ∈ Rn ; x = 1} with the metric d(x, y) = x − y and the Borel probability measure μ(A) = |{tA; 0 t 1}|/|{x; x 1}|.

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T HEOREM 3. Let · be a norm on Rn and let δ be the modulus of convexity of (Rn , · ). Then for any Borel set A ⊂ S and any ε > 0, μ(Aε ) > 1 − 2μ(A)−1 e−2nδ(ε/2) .

(6)

P ROOF. Let K = {x; x 1} and ν the normalized Lebesgue measure on K. By considering the set {tA; 1/2 t 1} it is clearly enough to prove that, for B ⊂ K, ν(Bε ) > 1 − ν(B)−1 e−2nδ(ε) . Put C = {x ∈ K; d(x, B) ε} then, for all x ∈ B, y ∈ C, (x + y)/2 1 − δ(ε), i.e.,

B +C ⊂ 1 − δ(ε) K 2 therefore, by the Brunn–Minkowski inequality,

2n ν(B)ν(C) 1 − δ(ε) e−2nδ(ε) .

Since for the Euclidean norm on Rn , δ(ε) ε2 /8, we get a simple proof of the approximate isoperimetric inequality for the sphere S n−1 (with the Euclidean or geodesic distance and Haar measure) discussed in the introduction. C OROLLARY 4. If A ⊂ S n−1 and ε > 0 then μ(Aε ) > 1 − 2μ(A)−1 e−nε

2 /16

.

Consequently, if f : S n−1 → R is a function with Lipschitz constant 1 then

2 μ x; |f (x) − M| ε 8 e−nε /16 . There are several ways to prove the isoperimetric inequality (as opposed to approximate isoperimetric inequalities) on the sphere. Some of them generalize to give isoperimetric inequalities in other situations. We refer to Appendix I in [45] in which Gromov presents a generalization based on Levy’s original proof and proves an isoperimetric inequality for Riemannian manifolds in term of their Ricci curvature. A particularly useful instance of this generalization is the case of O(n) equipped with its Haar measure and Euclidean metric (i.e., the Hilbert–Schmidt norm). [14] contains a relatively easy and self-contained proof of the isoperimetric inequalities on the sphere by symmetrization. It seems however to be very special to S n−1 . We now sketch very brieﬂy a proof by another method of symmetrization which is not very well known and which we think deserves to be better known. It seems to have the potential to generalize to other situations, see, for example, the last paragraph of this subsection. The method is due to Baernstein II and Taylor [6] and is written in detail with indications towards generalizations in [7]. S KETCH OF PROOF OF L EVY ’ S ISOPERIMETRIC INEQUALITY . Given a hyperplane H through zero in Rn we denote S0 = S n−1 ∩ H and by S+ and S− the two open half spheres

Concentration, results and applications

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in the complement of H . Let also σ = σH be the reﬂection with respect to H . Of course σ is an isometry with respect to the (Euclidean or geodesic) metric on S n−1 , it satisﬁes σ 2 = identity and preserves the Haar measure. It also satisﬁes that if x, y ∈ S+ then d(x, y) d(x, σ (y)). Given a set A ⊆ S n−1 we deﬁne its two point symmetrization A∗ with respect to the above decomposition as

A∗ = A ∩ (S+ ∪ S0 ) ∪ A ∩ S− ∩ σ (A ∩ S+ ) ∪ σ (A ∩ S− \ σ (A ∩ S+ )) , i.e., we “push up” elements of A ∩ S− into S+ using σ whenever there is space available. The term symmetrization seems a bit misleading since we desymmetrize as far as symmetry with respect to H is concerned. The point of course is that A∗ is closer to cap than A is and in that sense is more symmetric. Note that if A is Borel, μ(A∗ ) = μ(A). It is also easy to prove that for every ε > 0 and for every A ⊆ S n−1 ∗

A ε ⊆ (Aε )∗ . In particular,

μ A∗ ε μ (Aε )∗ μ(Aε ). The deﬁnition of the symmetrization procedure and the last property hold for any metric probability space (K, μ) admitting an isometric and measure preserving involution σ and any partition of the complement of K0 = {x; x = σ (x)} into K− , K+ provided this involution and partition satisfy the following properties: K+ = σ (K− ) and d(x, y) d(x, σ (y)) for all x, y ∈ K+ . To prove the isoperimetric inequality we would like to apply the operation A → A∗ with respect to many hyperplanes, reach a set so that no farther application of this operation improves μ(Aε ) and prove that such a set must be a cap. We’ll sketch in a minute how to do that for S n−1 but we would like to emphasize again that this seems plausible in other situations as well and we think it deserves further investigation. Consider the metric space C of all closed subsets of S n−1 with the Hausdorff metric. Fix A ∈ C and consider the set B ⊆ C of all sets B ∈ C satisfying: • For all ε > 0 μ(Bε ) μ(Aε ) and • μ(B) = μ(A). One checks that the set B is closed in C. Fix a point x0 ∈ S n−1 and let C be the closed cap centered at x0 with measure μ(A). It is enough to prove that C ∈ B. For any hyperplane H with x0 ∈ / H we denote by S+ the open half sphere containing x0 . One now proves that B → μ(B ∩ C) is upper semicontinuous on C. Consequently, μ(B ∩ C) attains its maximum on B, say at B. We shall show that B ⊇ C which will prove the claim. If this is not the case then μ(B \ C) = μ(C \ B) > 0. Let x ∈ B \ C and y ∈ C \ B be points of density of the respective sets and let H be the hyperplane perpendicular to the segment [x, y] and crossing it at the midpoint (x + y)/2. Let B(x, r) ⊂ S− , B(y, r) ⊂ S+ be small balls such that μ(B(x, r) ∩ (B \ C)) >

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0.99μ(B(x, r)) and μ(B(y, r) ∩ (C \ B)) > 0.99μ(B(y, r)). Applying the symmetrization B → B ∗ with respect to this hyperplane, most of B(x, r) will be transferred into B(y, r) while no point of C ∩ B is transferred to a point which is not in C. Thus, μ(B ∗ ∩ C) > μ(B ∩ C). Since B ∗ also belongs to B we get a contradiction. With a bit more effort the proof above can be adjusted to show that caps are the only solutions to the isoperimetric problem in S n−1 . The method of proof presented here can be used to prove other isoperimetric-like inequalities. [13] contains an explicit example in which the setting is similar to the one above (i.e., dealing with subsets of S n−1 ) with the difference that one measures the ε-boundary of a set A in a different way: the measure of the set of all pairs (x, y) with x ∈ A, y ∈ /A and the distance between x and y is at most ε. In [13] this is used to solve a (discrete!) problem concerning an efﬁcient algorithm for approximating the maximal cut in a graph.

2.2. Martingales Recall that for f ∈ L1 (Ω, F , P ) and for G, a sub σ -algebra of F , the conditional expectation, E(f |G), of f given G is the unique h ∈ L1 (Ω, G, P|G ) satisfying

h dP = A

f dP

for all A ∈ G.

(7)

A

(h is the Radon–Nikodým derivative of the measure ν(A) = A f dP on G with respect to P|G .) The correspondence f → E(f |G) is a linear positive operator of norm one on all the spaces Lp (Ω, F , P ), 1 p ∞. Some additional properties of this operator are: • If G ⊂ G is a sub σ -algebra then E(E(f |G)|G ) = E(f |G ). • If g ∈ L∞ (Ω, G, P ) then E(fg|G) = gE(f |G). • For the trivial σ -algebra G = {∅, Ω}, E(f |G) = Ef , the expectation of f . Given a ﬁnite or inﬁnite increasing sequence of σ -algebras, F0 , F1 , . . . , a sequence of elements of L1 (Ω, F , P ), f0 , f1 , . . . , is said to be a martingale with respect to F0 , F1 , . . . if fi = E(fj |Fi ) for all i j . We shall always assume here that F0 is the trivial σ -algebra {∅, Ω} and that the sequence is ﬁnite with the last terms being fn = f and Fn = F . Then, fi = E(f |Fi ), i = 0, 1, . . . , n. We also denote di = fi − fi−1 , i = 1, 2, . . . , n, and call the sequence {di }ni=1 the martingale difference sequence. One set of examples of a martingale is the following: let Xi be a sequence of mean zero independent random variables and put fi = ij =0 Xj , then {fi } is a martingale with respect to {Fi } where Fi is the smallest σ -algebra with respect to which X0 , . . . , Xi are measurable. In a lot of senses a general martingale resembles this particular set of examples. There are many inequalities estimating the probability of the deviation of f = fn from f0 = Ef in terms of the behavior of the sequence {di }. In the next proposition we gather some of them. (1) is due to Azuma [5] or [57], p. 238. (2) and (3) are due to Pisier [47], (2) was ﬁrst used in [26]. (4) is a generalization to the martingale case of Prokhorov’s inequality. In a somewhat weaker form it ﬁrst appears in [29]. The form here is from [23].

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P ROPOSITION 5. (1) For all t > 0, n

2 2 di ∞ . P ω; |f (ω) − Ef | t 2 exp −t /2

(8)

i=1

(2) For all 1 0, P

q

ω; |f (ω) − Ef | t K exp −δ t/ di ∞ p,∞ ,

(9)

where q −1 + p−1 = 1, K and δ depend only on p and {ai }ni=1 p,∞ = max1j n j 1/p aj∗ with {aj∗ } denoting the decreasing rearrangement of the sequence {|aj |}. (3) For all t > 0, P

ω; |f (ω) − Ef | t K exp − exp δt/ di ∞ 1,∞ ,

(10)

where K and δ are absolute constants. (4) Put M = max1in di ∞ and S 2 = ni=1 E(di2 |Fi−1 )∞ . Then, for all t > 0, P

Mt t · arc sinh ω; |f (ω) − Ef | t 2 exp − . 2M 2S 2

(11)

The proofs of these and similar inequalities are usually quite simple. Let us sketch the proof of (1). If Fi is “rich” enough, extreme points in the set {d ∈ L∞ (Ω, Fi , P ); E(d|Fi−1 ) = 0, |d| a} have constant absolute value equal to a. Consequently for all λ ∈ R,

2 2 E eλdi |Fi−1 cosh λdi ∞ eλ di ∞ /2 .

(12)

Extending Fi (to become rich enough) if necessary, this inequality holds always. It follows that E eλ

n

i=1 di

2 n−1 2 = E E eλ i=1 di |Fn−1 eλ dn ∞ /2 .

(13)

Iterating this (by applying E(·|Fn−2 ), then E(·|Fn−3 ) . . .) we get E eλ(f −Ef ) eλ

2 n d 2 /2 i ∞ i=1

(14)

.

Applying Chebyshev’s inequality we get, for positive λ, P

ω; f (ω) − Ef t P ω; eλ(f (ω)−Ef )−λt 1 e−λt E eλ(f −Ef ) e−λt +λ

2

n

2 i=1 di ∞ /2

.

Minimizing over positive λ and repeating this with negative λ we get the result.

(15)

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G. Schechtman

Yurinski [64] was probably the ﬁrst to use martingale inequalities in the context of Banach space valued random variables. The point is that if Xi are independent Banach space valued random variables and we form the martingale fi = E( nj=1 Xj |Fi ) then the martingale differences satisfy |di | Xi . This can be used to estimate the tail behavior of nj=1 Xj . Maurey [41] noticed that martingale deviation inequalities can be used to prove approximate isoperimetric inequality for the interesting case of the permutation group. We present a somewhat simpliﬁed version of his proof with some abstractization ([51,45]). The length of a ﬁnite metric space (Ω, d) is deﬁned as the inﬁmum of = ( ni=1 ai2 )1/2 over all sequences a1 , . . . , an of positive numbers satisfying: there exists a sequence {Ωk }nk=0 of partitions of Ω with • Ω0 = {Ω} and Ωn = {{ω}}ω∈Ω . • Ωk reﬁnes Ωk−1 , k = 1, . . . , n. • If k = 1, . . . , n, A ∈ Ωk−1 , B, C ⊂ A and B, C ∈ Ωk then there is a one to one map h from B onto C such that d(ω, h(ω)) ak for all ω ∈ B. The two basic examples we shall deal with are the Hamming cube, Hn , and the permutation group, Πn . The Hamming cube is the set {0, 1}n with the metric d((εi )ni=1 , (δi )ni=1 ) = #{i; εi = δi }. Πn is the set of permutations of {1, √ d(π, ϕ) = √ 2, . . . , n} with the metric #{i; π(i) = ϕ(i)}. The length is smaller or equal n in the ﬁrst case and 2 n − 1 in the second. Let us illustrate this in the second example. Fix 1 k n − 1 and i1 , i2 , . . . , ik distinct elements of {1, 2, . . . , n}. Put Ai1 ,i2 ,...,ik = π ∈ Πn ; π(1) = i1 , . . . , π(k) = ik

(16)

and let Ωk be the partition whose atoms are all the sets Ai1 ,i2 ,...,ik where (i1 , i2 , . . . , ik ) ranges over all n!/(n − k)! possibilities. It is clear that the ﬁrst two requirements from {Ωk }n−1 k=0 are satisﬁed (with n − 1 replacing n). To show that the third one is satisﬁed with ai = 2 for i = 1, . . . , n − 1, let A = Ai1 ,i2 ,...,ik−1 ∈ Ωk−1 and B = Ai1 ,i2 ,...,ik−1 ,r , C = Ai1 ,i2 ,...,ik−1 ,s ∈ Ωk and deﬁne h : B → C by h(π) = (r, s) ◦ π (where (r, s) is the transposition of r and s). We are now ready to state the main theorem of this section. T HEOREM 6. Let (Ω, d) be a ﬁnite metric space of length at most . Let P be the normalized counting measure on Ω. Then, (1) Let f : Ω → R satisfy |f (x) − f (y)| d(x, y) for all x, y ∈ Ω. Then for all t > 0, P

ω; |f (ω) − Ef | t

2 exp −t 2 /22 .

(17)

(2) Let A ⊂ Ω with P (A) 1/2 then for all t > 0

P (At ) 1 − 2 exp −t 2 /82 .

(18)

S KETCH OF PROOF. Let = ( ni=1 ai2 )1/2 with ai and Ωi , i = 0, . . . , n, as in the deﬁnition of length. Let Fi be the ﬁeld generated by Ωi and form the martingale fi = E(f |Fi ), i = 0, . . . , n. Note that fi is constant on each atom B of Ωi and that this constant is

Concentration, results and applications

1613

fi|B = Avex∈B f (x). If B, C are two atoms of Ωi contained in an atom A of Ωi−1 then by the third property of the sequence of partitions, |fi|B − fi|C | = |B|

f (x) − f h(x) ai .

−1

(19)

x∈B

Since fi−1|A is the average of fi|B over all atoms B of Fi which are subsets of A, we get from (19) that |fi−1|A − fi|C | ai and since this holds for all such A and C, di ∞ ai . Now apply 5(1). This proves (1). (2) follows from (1) as explained in the introduction. C OROLLARY 7. Let (Ω, d) be either Hn or Πn . (1) Let f : Ω → R satisfy |f (x) − f (y)| d(x, y) for all x, y ∈ Ω. Then for all t > 0, P

ω; |f (ω) − Ef | t 2 exp −t 2 /8n .

(20)

(2) Let A ⊂ Ω with P (A) 1/2 then for all t > 0

P (At ) 1 − 2 exp −t 2 /32n .

(21)

By considering a ball in the Hamming metric it is easy to see that, except for the choice of the absolute constants involved, the result for Hn is best possible. In this case, the exact solution to the isoperimetric problem is known as well (and, for sets of measure 2k /2n , is a ball) [22,15]. For sets of measure of the form 2k /2n this can also be deduced from the method of two-point symmetrization introduced in the previous section. For Πn the solution to the isoperimetric problem is not known. However, again except for the absolute constants involved, the corollary gives the right result: E XAMPLE 8. Let n be odd and deﬁne A ⊂ Π2n by A = π; π(i) n for more than n/2 indices i with 1 i n .

(22)

Then, μ(A) = 1/2 and for all k < n/2,

P (Ack )

1 = (2n)! 1 = 2n

n

[ n2 −k]+1

l=0

n n! n! n! l (n − l)! l!

n2 . l

[ n2 −k]+1

(23)

l=0

For k with k/n bounded away from 0 and 1/2, a short computation shows that this is larger 2 than e−δk /n .

1614

G. Schechtman

It is also not hard to see that, at least for some a and t, balls are not the solution to the isoperimetric problem inf{P (At ); P (A) = a} on Πn . We wonder whether there is an equivalent, with constants independent of n (and hopefully natural), metric on Πn for which one can solve the isoperimetric problem. The advantage of the method described above is in its generality; in principle, whenever we have a metric probability space we can estimate its length by trying different sequences of partitions and get some approximate isoperimetric inequality. In reality it turns out that in most speciﬁc problems, and in particular when the space is naturally a product space, one gets better results by other methods.

2.3. Product spaces. Induction In [58] Talagrand introduced a relatively simple but quite powerful method to prove concentration inequalities which works in many situations in which the probability space is a product space with many components. The proofs, as naive as they may look, are by induction on the number of components. The monograph [62] contains many more instances in which variants of this method work. Another feature in Talagrand’s work is the deviation from the traditional way of measuring distances; the “distance” of a point from a set is not always measured by a metric. We start with a small variation on the original theorem of Talagrand taken from [27]. T HEOREM 9. Let Ωi ⊂ Xi , i = 1, . . . , n, be compact subsets of normedspaces with diam(Ωi ) 1. Consider Ω = Ω1 × Ω2 × · · · × Ωn as a subset of the 2 sum ( ni=1 ⊕Xi )2 . Let μi be a probability measure on Ωi , i = 1, . . . , n, and put P = μ1 × μ2 × · · · × μn . For a compact A ⊂ Ω denote the convex hall of A byconv(A) and for x ∈ Ω put ϕ(x, A) = dist(x, conv(A)) (with respect to the metric in ( ni=1 ⊕Xi )2 ). Then eϕ

(1)

2 (x,A)/4

1 . P (A)

(24)

In particular, for all t > 0, P

x; ϕ(x, A) > t

1 2 e−t /4 . P (A)

(2) If f : Ω → R is convex and Lipschitz (with respect to the metric of ( with constant 1 then 2 P x; f (x) − f > t 4 e−ct

(25) n

i=1 ⊕Xi )2 )

(26)

for all t > 0 and some universal c > 0. S KETCH OF PROOF. The proof of the ﬁrst assertion of (1) is by induction. The second assertion of (1) and also (2) (with a bit more effort) follow as in (15). The other

Concentration, results and applications

1615

theorems in this section are proved similarly. We shall illustrate the induction step. 2 Assume that eϕ (x,A)/4 dP (x) 1/P (A) for all compact A ⊂ Ω = Ω1 × · · · × Ωn and let A ⊂ Ω × Ωn+1 . For ω ∈ Ωn+1 put A(ω) = {x ∈ Ω; (x,ω) ∈ A} (where, for x = (x1 , . . . , xn ) ∈ Ω, (x, ω) = (x1 , . . . , xn , ω)). Put also B = ω∈Ωn+1 A(ω). Fix a y = (x, ω) ∈ Ω × Ωn+1 and notice that ϕ(y, A) ϕ(x, A(ω)) provided A(ω) = ∅. Also, ϕ(y, A) ϕ(x, B) + 1. From these two inequalities it is easy to deduce that, for all 0 λ 1, ϕ 2 (y, A) λϕ 2 (x, A(ω)) + (1 − λ)ϕ 2 (x, B) + (1 − λ)2 . Using Hölder’s inequality and the induction hypothesis, one gets, for all ω ∈ Ωn+1 , eϕ

2 ((x,ω),A)/4

Ω

2 e(1−λ) /4 P (A(ω)) −λ . P (B) P (B)

(27)

We now use a numerical inequality (which can serve as a good Calculus exercise). For all 0 p 1, inf p−λ e(1−λ)

0p1

2 /4

2 − p.

Using this inequality with p = P (A(ω))/P (B) and integrating (27) over ω, we get P × μn+1 (A) 1 1 ϕ 2 ((x,ω),A)/4 e 2− . (28) P (B) P (B) P × μn+1 (A) Ωn+1 Ω Note that if Xi = {−1, 1} with the uniform measure for each i then by Corollary 7 the same conclusion as in Theorem 9(2) holds for any (i.e., not necessarily convex) function satisfying |f (x) − f (y)| n−1/2 |x i − yi |. However, for a convex function, Theorem 9 gives a much better result since n−1/2 |xi − yi | ( |xi − yi |)1/2 . The theorem above has the disadvantage that, because of the convexity assumption, it applies only to Ωi ’s which lie in a linear space. This is taken care of in the next theorem from [62] which surprisingly is extremely applicable. Given * (Ωi , Fi , μi ), i = 1, . . . , n, form the product space (Ω, P ) with * probability spaces Ω = ni=1 Ωi and P = μi . For x, y ∈ Ω let U (x, y) be the sequence in {0, 1}n which realizes the Hamming distance between x and y, i.e., has 0 exactly in the coordinates i where xi = yi . For a subset A of Ω and for x ∈ Ω we set U (x, A) to be the subset of {0, 1}n consisting of all sequences U (x, y) for some y ∈ A, i.e., U (x, A) = {εi }ni=1 ∈ {0, 1}n ; for some y ∈ A, yi = xi iff εi = 0 . For x ∈ Ω and A ⊂ Ω let ϕ(x, A) = d(0, conv(U (x, A))). It should be noted that, in general, ϕ(x, A) is not induced by a metric. i.e., there is no metric d on Ω such that ϕ(x, A) = inf{d(x, y); y ∈ A}. This is easily seen to be the case for Ω = {0, 1}n , for example. T HEOREM 10. Let A ⊂ Ω then 1 2 eϕ (x,A)/4 . P (A)

(29)

1616

G. Schechtman

In particular, for all t > 0, P

x; ϕ(x, A) > t

1 2 e−t /4 . P (A)

Using the Hahn–Banach theorem one can show that ϕ(x, A) = sup inf αi ; y ∈ A .

αi2 =1

(30)

(31)

{i;yi =xi }

Notice that, if h denotes the Hamming distance on Ω, i.e., h(x, y) = #{i; yi = xi }, then formula (31) implies that ϕ(x, A) h1/2 (x, A). Using this inequality and the martingale 2 method of Section 2.2 one gets only P (ϕ(x, A) > t 1/2 ) C e−ct /n while Theorem 10 2 gives P (ϕ(x, A) > t 1/2 ) < 4 e−t /4 4 e−t /4n for t in the relevant range, 0 < t < n. This illustrates the possible advantage of this inequality over Corollary 7 for Hn . Theorem 10 has many applications. We refer to [62] for some of them. A variant of Theorem 9 and particularly of (26) was recently proved by Ledoux ([32] or [33]). The difference is that the convexity assumption on f is weakened to convexity of each variable separately but the conclusion is only a one-sided deviation inequality: 2 P x; f (x) − f > t 4 e−ct . (32) It is unknown whether a similar lower deviation inequality also holds. The next result was ﬁrst proved by Talagrand in [59]. The original proof was very complicated but in [62] Talagrand presented a much simpler*inductive proof * which we shall sketch here. Consider a product probability space (Ω = ni=1 Ωi , P = ni=1 μi ). Given a q ∈ N and q + 1 elements of Ω, x, y 1, . . . , y q , we deﬁne the “Hamming distance” of x from the q-tuple y 1 , . . . , y q by

q (33) / yi1 , . . . , yi . h x; y 1, . . . , y q = # i; xi ∈ Given q subsets A1 , . . . , Aq of Ω, we deﬁne

h(x; A1, . . . , Aq ) = inf h x; y 1, . . . , y q ; y 1 ∈ A1 , . . . , y q ∈ Aq . T HEOREM 11. q h(x;A1,...,Aq ) *q

1 . j =1 P (Aj )

(34)

(35)

In particular, P

for all k ∈ N.

x; h(x; A1, . . . , Aq ) k

1 q −k P (A ) j j =1

*q

(36)

Concentration, results and applications

1617

S KETCH OF PROOF OF THE INDUCTION STEP. For A1 , . . . , Aq ⊂ Ω × Ωn+1 and ω ∈ Ωn+1 put Aj (ω) = y ∈ Ω; (y, ω) ∈ Aj ,

j = 1, . . . , q,

(37)

and

Bj =

Aj (u),

j = 1, . . . , q.

(38)

u∈Ωn+1

Fix ω ∈ Ω and k ∈ {1, . . . , q} and put also Cj =

Bj Ak (ω)

if j = k, if j = k.

(39)

One then shows that

h (x, ω); A1, . . . , Aq min 1 + h(x; B1 , . . . , Bq ), h(x; C1, . . . , Cq ) .

(40)

It then follows from the induction hypothesis that P (Bk ) 1 min q, min . 1kq P (Ak (ω)) j =1 P (Bj )

q h((x,ω);A1,...,Aq ) *q

(41)

If 0 hi 1, i = 1, . . . , q, are functions on a probability space then

−1 q $ −1 min q, min hi . hi 1iq

(42)

i=1

This follows easily from the inequality h−1 ( h)q 1 which holds for every function h satisfying q −1 h 1. Using (42) and integrating (41) over Ωn+1 , we get the assertion for n + 1. We shall see in a minute the big advantage of this theorem over the concentration inequality for the Hamming metric. Although it looks like there is not much difference between h(·; A, A), say, and the Hamming distance of a point from a set (d(·, A) of Section 5), it turns out that the last theorem gives much better concentration when it applies. Theorem 11 is still looking for good applications. As far as we know Theorem 11 has basically one application dealing with the tail behavior of norms of sums of independent Banach space valued random variables. This is the original application which led Talagrand to prove this result (see [59] and [62], Section 13). This particular application also has a different proof [31].

1618

G. Schechtman

To illustrate the advantage of Theorem 11 over the basic inequality for the Hamming metric we deﬁne a class of functions and state a corollary which amounts to a deviation inequality for this class of functions. For I ⊂ {1, . . . , n} denote ΩI =

$

Ωi

and Ω ∗ =

i∈I

ΩI

I ⊂{1,...,n}

and let f : Ω ∗ → R+ . We say that f is monotone if I ⊂ J ⊆ {1, , . . . , n}

implies f (xi )i∈I f (xj )j ∈J

(43)

for all (xj )j ∈J ∈ ΩJ . We say that f is subadditive if for all I, J disjoint subsets of {1, . . . , n} and all (xi )i∈I ∪J ∈ ΩI ∪J ,

f (xi )i∈I ∪J f (xi )i∈I + f (xj )j ∈J .

(44)

Here is an example of such afunction: let Ωi be subsets of a normed space (X, · ) and put f ((xi )i∈I ) = Aveεi =±1 i∈I εi xi . For x ∈ ΩI , y ∈ ΩJ we shall denote by h(x, y) the number of coordinates in which xi = yi including coordinates in which one or both of xi , yi are not deﬁned. C OROLLARY 12. Let f : Ω ∗ → R+ be monotone, subadditive and satisfy |f (x)−f (y)| h(x, y) for all x, y ∈ Ω ∗ . Then, for all a > 0, 1 k n and q ∈ N, P

x ∈ Ω; f (x) (q + 1)a + k

P (f a)−q q −k .

(45)

For a being the median of f and q = 2, say, one gets P (f 3a + k) 42−k . If a ) k ) n this is much better than what one gets for a general Lipschitz function from, e.g., 2 the martingale method. There one gets P (f a + k) 2 e−k /4n . Note the resemblance with the situation concerning Theorem 9: in both cases we evaluate the probability of deviation of f from its expectation (or median), a quantity which depends only on the behavior of f on Ω (since the probability measure is supported there). However, by extending f to a larger set (in Theorem 9 the convex hull of Ω, here Ω ∗ ), if possible, using its Lipschitz constant on the larger set and some additional properties of the extended function (there convexity, here monotonicity and subadditivity) we get, in some cases a stronger concentration result than the basic one. P ROOF OF C OROLLARY 12. For 1 i q put Ai = A = {x ∈ Ω; f (x) a}. Then

f (x) (q + 1)a + k ⊆ h(x; A1, . . . , Aq ) k .

Indeed, if h(x; A1, . . . , Aq ) < k, let y 1 , . . . , y q ∈ A be such that, putting I = q {i; xi ∈ / {yi1 , . . . , yi }}, #I < k. The complement of I can be written as kj =1 Jj with

Concentration, results and applications

1619

j

Jj ⊆ {1, . . . , n} satisfying xi = yi for i ∈ Jj . Then, assuming I is not empty, f (x) f (x|I ) +

q

f (x|Jj )

j =1

j

1 + f y|Jj f x|I + y|J 1 q

j =1

q 1 j

#I + f y|J1 + f y|Jj j =1 q

j

f y #I + f y 1 + j =1

< k + (q + 1)a.

(46)

The corollary follows now immediately from Theorem 11.

The paper [62] also contains a generalization of the concentration inequality for the permutation group, Corollary 7. The (inductive) proof of this result is a bit harder than the other proofs surveyed in this section and we shall not reproduce it. This result also awaits good applications. Equip the symmetric group Sn with its natural probability measure, μ. For σ ∈ Sn and A ⊆ Sn let n 2 f (σ, A) = inf si ; (s1 , . . . , sn ) ∈ VA (σ ) , (47) i=1

where VA (σ ) is the convex hall of the set (s1 , . . . , sn ) ∈ {0, 1}n ; ∃τ ∈ A s.t. ∀i n, si = 0 ⇒ τ (i) = σ (i) . T HEOREM 13. For every A ⊂ Sn , t > 0

μ σ ; f (σ, A) > t

1 e−t /16. μ(A)

(48)

The manuscript [62] contains many reﬁnements of Theorems 10, 11 and 13 which we do not reproduce here.

2.4. Spectral methods Let (Ω, F , μ) be a probability space, A some set of measurable functions and E : A → R+ some function (which we shall refer to as energy function). For f ∈ L2 (Ω) denote by

1620

G. Schechtman

σ 2 (f ) the variance of f , σ 2 (f ) =

and, for f ∈ L2 log L (i.e.,

f log f dμ − 2

f 2 dμ −

2 f dμ

(49)

f 2 log+ f dμ < ∞), denote by ε(f ) the entropy of f 2 ,

ε(f ) =

(f − Ef )2 dμ =

2

2 f dμ log f dμ 2

(50)

(which is necessarily ﬁnite). We say that (A, E) satisfy a Poincaré inequality with constant C if σ 2 (f ) CE(f )

for all f ∈ A.

(51)

We say that (A, E) satisfy a logarithmic Sobolev inequality with constant C if ε(f ) CE(f )

for all f ∈ A.

(52)

The main example of an energy function E is related to the gradient or generalization of it. If d is a metric on Ω (and F the Borel σ -ﬁeld), deﬁne the norm of the gradient at x ∈ Ω by ∇f (x) = lim sup |f (x) − f (y)| . d(x, y) y→x

(53)

Note that ∇f (x) by itself is not deﬁned. The reason for this notation is of course that if (Ω, d) is a Riemannian manifold (in particular if it is Rn with the Euclidean distance) and if f is differentiable at x then |∇f (x)| is the Euclidean norm of the gradient of f at x. Deﬁne now 2 (54) E(f ) = ∇f (x) dμ(x). The classical Poincaré (or Rayleigh–Ritz) inequality says that, in the case of a compact Riemannian manifold, (51) is satisﬁed with C = λ−1 1 , λ1 being the ﬁrst positive eigenvalue of the Laplacian on L2 (Ω, μ). We shall only deal here with the energy function (54). [33] contains many other examples and a comprehensive treatment of the subject of this section. If A is the set of bounded Lipschitz functions on (Ω, d), the norm of the gradient satisﬁes the chain rule: if φ ∈ C 1 (R) and f ∈ Ω then φ ◦ f ∈ Ω and

∇(φ ◦ f )(x) ∇f (x)φ f (x)

(55)

Concentration, results and applications

1621

and consequently E(φ ◦ f ) f 2Lip

φ f (x) 2 dμ(x),

(56)

where f Lip denotes the Lipschitz constant of f . The next theorem, basically due to Gromov and Milman, shows that Poincaré inequality implies concentration. T HEOREM 14. Let (Ω, F , μ, d) be a probability metric space. Let A be the set of bounded Lipschitz functions on (Ω, d) and let E be deﬁned √by (54). Assume that (A, E) satisﬁes the Poincaré inequality (51). Then for all |λ| < 2/ C and every bounded f with Lipschitz constant 1 E eλ(f −Ef )

240 . 4 − Cλ2

(57)

In particular

% − C2 t

P |f − Ef | > t 240 e

for all t > 0.

(58)

P ROOF. By (51) and (56)

2

C E eg − E eg/2 CE eg/2 gLip E eg 4 for any g ∈ A. In particular, for any λ, λ 2 Cλ2 λf Ee E eλf − E e 2 f 4 or E eλf

1 1−

Cλ2

λ f 2 Ee2 .

4

Iterating we get for every n, E eλf

n−1 $ k=o

1 1−

2k

Cλ2 4k+1

λ 2n E e 2n f

which tends to ∞ $ k=o

1 1−

Cλ2 4k+1

2k eλEf .

1622

G. Schechtman

R EMARK 15. (1) A simple limiting argument shows now that the assumption that f is bounded is superﬂuous. (2) The simple example of the exponential distribution on R shows that (except for the absolute constants involved) one can’t improve the concentration function e−ct . As we shall see below, what looks like a slight change, logarithmic Sobolev inequality instead of 2 Poincaré inequality, changes the behavior of the concentration function from e−ct to e−ct . The next theorem is apparently due to Herbst. T HEOREM 16. Let (Ω, F , μ, d) be a probability metric space. Let A be the set of bounded Lipschitz functions on (Ω, d) and let E be deﬁned by (54). Assume that (A, E) satisﬁes the logarithmic Sobolev inequality (52) then for all λ ∈ R and every bounded f with Lipschitz constant 1 E eλ(f −Ef ) eCλ

2 /4

(59)

.

In particular

2 P |f − Ef | > t 2 e−t /C

for all t > 0.

(60)

P ROOF. Put h(λ) = E eλf , then

ε eλf/2 = Eλf eλf −E eλf log E eλf = λh (λ) − h(λ) log h(λ) .

(61)

Also, from (56), we get,

λ2 λ2 E eλf/2 E eλf = h(λ). 4 4

(62)

Combining (61), (62) and (52) we get

λ2 C λh (λ) − h(λ) log h(λ) h(λ) 4 or, putting k(λ) = λ−1 log h(λ) (and, by continuity, k(0) = Ef ), k (λ) =

1 1 h (λ) C − 2 log h(λ) , λ h(λ) 4 λ

for all λ ∈ R.

It follows that k(λ) − k(0) Cλ/4 and thus E eλ(f −Ef ) = eλ(k(λ)−k(0)) eCλ

2 /4

.

R EMARK 17. A simple limiting argument shows that here too the assumption that f is bounded is superﬂuous.

Concentration, results and applications

1623

Both Poincaré inequality and logarithmic Sobolev inequality carry over nicely to product spaces in the following sense: for i = 1, 2, . . . , n, let (Ωi , Fi , μi ) be a probability space, + Ai some set *nof measurable functions on Ωi and Ei : Ai → R some energy function. Put (Ω, P ) = i=1 (Ωi , μi ). Given a function f on Ω we denote by fi the same function considered as a function of the i-th variable only, keeping all other variables ﬁxed. De ﬁne E(f ) = EP ni=1 Ei (fi ). Let A denote the set of all functions f such that (for all x1 , . . . , xn ) and for all i, fi is in Ai . One can prove that σ 2 (f ) EP

n

σ 2 (fi )

and ε(f ) EP

i=1

n

ε(fi ),

(63)

i=1

from which the following proposition easily follows. P ROPOSITION 18. Assume (Ai , Ei ), i = 1, . . . , n, all satisfy Poincaré inequality (resp. logarithmic Sobolev inequality) with a common constant, C. Then (A, E) satisﬁes Poincaré inequality (resp. logarithmic Sobolev inequality) with the same constant, C. E XAMPLE 19. The symmetric exponential measure on R, i.e., the measure with density 1 e−|t | , satisﬁes Poincaré inequality with constant 4. Consequently, the same is true for the 2 measure on Rn which is the n-fold product of this measure. The canonical Gaussian measure on R and thus on Rn satisﬁes logarithmic Sobolev inequality with constant 2. The proof of both statements can be found in [33]. The second one is due to Gross and, in view of Theorem 16, implies the concentration inequality for γn , the Gaussian measure on Rn : if f : Rn → R is Lipschitz with constant one with respect to the Euclidean metric then 2 γn f − f dγn > t C e−ct . From this it is not hard to get the concentration inequality for S n−1 . One uses Lemma 22 below. We would also like to state a theorem ﬁrst proved by Talagrand [61] which “interpolates” between the last two theorems. See [8] and [33] for a relatively simple proof along the lines of the proofs of the last two theorems. We state it only for a speciﬁc probability measure P on Rn , the product of the measures with density 12 e−|t | on R. See [33] for generalizations. T HEOREM 20. Let f : Rn → R be a function satisfying f (x) − f (y) αx − y2

and f (x) − f (y) βx − y1 .

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Then, with the probability introduced above,

P f (x) − Ef > r C exp −c min r/β, r 2 /α 2

(65)

for some absolute positive constants C, c and all r > 0. R EMARK 21. Although it deals with a different probability measure, Theorem 20 also implies the concentration inequality for the Gaussian measure on Rn (and thus, via Lemma 22 below, also for the Haar measure on S n−1 ). This follows from a simple transference of the Gaussian measure to the product of the symmetrized exponential measure discussed above. Thus, Theorem 20 can be considered as a strengthening of these inequalities. We refer to [61] and [33] for that and further discussion. Although the methods in this and the previous section are specialized to product measures, there is a way to transfer such results to some other situations. In particular to the case of unit balls of np spaces equipped with the normalized Lebesgue measure. The basic A;0t 1}| tool is the following simple result: consider the measure μ(A) = |{t |{x;xp 1}| on the surface of the np ball, 0 < p < ∞. Consider also n independent random variables X1 , X2 , . . . , Xn p each with density function cp e−|t | , t ∈ R. (Note that necessarily cp = p/2Γ (1/p).)

L EMMA 22. Put S = ( ni=1 |Xi |p )1/p . Then ( XS1 , XS2 , . . . , XSn ) induces the measure μ on ∂Bpn . Moreover, ( XS1 , XS2 , . . . , XSn ) is independent of S. See [54] for a proof. This lemma is used there to compute the tail behavior of the q norm on the np ball. Recently ([55]) this result was strengthen, in the case p = 1, q = 2, to give a concentration inequality for general Lipschitz functions, with respect to the Euclidean metric, on the n1 ball B1n . The proof combines most of the results of this section and we shall not give it here. T HEOREM 23. There exist positive constants C, c such that if f : ∂B1n → R satisﬁes |f (x) − f (y)| x − y2 for all x, y, ∈ ∂B1n then, for all t > 0,

μ x; |f (x) − Ef | > t C exp(−ctn).

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2.5. Bounds on Gaussian processes As we shall see below, in the application sections, concentration inequalities are used mostly to ﬁnd a point ω, in the metric probability space under consideration, in which a big collection of functions {Gt (ω)}t ∈T are each close to its mean. There may be other ways to reach such a conclusion. Assuming the means of all the functions under consideration are zero, it would be enough, for example, to prove that E supt ∈T |Gt | is small (then, for a set of ω’s of measure at least 1/2, supt ∈T |Gt (ω)| is at most 2×small).

Concentration, results and applications

1625

When T is a metric space and Gt a Gaussian process (meaning that any ﬁnite linear combination of the Gt ’s has a Gaussian distribution) the evaluation of E supt ∈T |Gt | is an extensively studied subject in Probability (having to do with the existence of a continuous version of the process). See, for example, [35]. There are well studied connections between the quantity E supt ∈T |Gt | and the entropy (or covering) function of the metric space T as well as with other properties of T . A recent achievement in this area is Talagrand’s majorizing measure theorem which relates the boundedness of E supt ∈T |Gt | to the existence of a certain measure (called majorizing measure) on T and gives new ways to estimate this quantity. A recent book treating this subject is [10]. We’ll not get into it any further here; we only remark that the proofs in this area are very much connected with concentration properties of Gaussian variables.

2.6. Other tools We dealt above mostly with geometric and probabilistic tools to prove concentration and approximate isoperimetric inequalities. There are many other methods and results that are not discussed here for lack of space. In particular we didn’t discuss at all combinatorial methods. For example, the (exact) isoperimetric inequality for the Hamming cube (from which Corollary 7 for that case follows) was ﬁrst proved by Harper [22] (see also [15] for a simpler proof) by combinatorial methods. There are also geometrical and probabilistic methods we didn’t discuss. [48] contains a yet another nice probabilistic proof of Corollary 4 due to Maurey and Pisier. It uses properties very special to Gaussian variables and thus does not seem to generalize much. [53] contains a generalization of Corollary 4 to harmonic measures on S n−1 . The proof is by reduction to the Haar measure. A new probabilistic method for proving concentration inequalities which emerged recently is that of transportation cost, see [40], where it was initiated by Marton, and the followup in [63]. [34] devotes a chapter to this subject. This method seems very much related to Kantorovich’s solution of Monge’s “mass transport” problem although, as far as I know, no concrete relation has been found yet. The “localization lemma” proved by Lovasz and Simonovits in [39] is a way to reduce certain integral inequalities in Rn to integral inequalities involving functions of one variable. It can be used to prove certain approximate isoperimetric inequalities as is explored in [30]. The list above is far from exhausting all the sources on this vast subject.

3. Applications 3.1. Dvoretzky-like theorems The introduction of the method(s) of concentration of measure into Banach Space Theory was initiated by Milman in his proof [43] of Dvoretzky’s theorem concerning spherical sections of convex bodies [11]. Although this topic is extensively reviewed in the article

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G. Schechtman

[16] in this Handbook, I would like to begin the applications section with a statement of the theorem and a brief description of its proof. T HEOREM 24. For all ε > 0 there exists a constant c = c(ε) > 0 such that for any n-dimensional normed space X there exists a subspace Y of dimension k c log n such that the Banach–Mazur distance d(Y, k2 ) 1 + ε. See [25] for the deﬁnition of the Banach–Mazur distance. The one-to-one correspondence between n-dimensional normed spaces and n-dimensional symmetric convex bodies (and the fact that every 2n-dimensional ellipsoid has an n-dimensional section which is a multiple of the canonical Euclidean ball) easily shows that the theorem above is equivalent to the following geometrical statement. By a convex body in Rn we mean a compact convex set with non-empty interior. T HEOREM 25. For all ε > 0 there exists a constant c = c(ε) > 0 such that every centrally symmetric convex body K admits a k c log n dimensional central section K0 and a positive number r satisfying rB ⊂ K0 ⊂ (1 + ε)rB, where B is the canonical Euclidean ball in the subspace spanned by K0 . S KETCH OF PROOF. Since the statement of each of the two theorems is invariant under invertible linear transformations, we may assume that the unit ball K of X = (Rn , · ) satisﬁes B2n ⊂ K and the canonical Euclidean ball B2n in Rn is (the) ellipsoid of maximal volume among all ellipsoids inscribed in K. (It is a theorem of F. John that the maximal volume ellipsoid is uniquely determined but we do not need this fact here.) A relatively easy theorem of Dvoretzky and % Rogers [12] (see also [45], p. 10) implies now that E = E · = S n−1 x dμ(x) > c logn n for some absolute constant c. Denoting by ν the normalized Haar measure on the orthogonal group O(n) and applying Corollary 4 to the function x → x, which is Lipschitz with constant one, we get that, for every ﬁxed x ∈ S n−1 ,

ν u; ux − E > εE = μ x ∈ S n−1 ; x − E > εE < e−cε

2 E2 n

< e−cε

2 log n

.

Fix a k-dimensional subspace V0 ⊂ Rn and an ε net N in V0 ∩ S n−1 of cardinality smaller than (3/ε)k . The existence of such a net follows from an easy volume argument (see [45], 2 p. 7). It then follows that if (3/ε)k e−cε log n < 1, i.e., if k is no larger than a constant depending on ε times log n, then

ν u; ux − E > εE, for some x ∈ N < 1 which implies that there is a u ∈ O(n) such that (1 − ε)E ux (1 + ε)E,

for all x ∈ N .

Concentration, results and applications

1627

It now follows from a successive approximation argument that similar inequalities hold for all x ∈ S n−1 which implies the conclusion of the theorem for the subspace uV0 . We next state another application of the concentration inequality on the Euclidean sphere. This lemma of Johnson and Lindenstrauss is much simpler but has a lot of applications including “real life” ones like efﬁcient algorithms for detecting clusters. T HEOREM 26. Let x1 , x2 , . . . , xn be points in some Hilbert space. If k c > 0 an absolute constant), then there are y1 , y2 , . . . , yn ∈ k2 satisfying

c ε2

log n (with

xi − xj yi − yj (1 + ε)xi − xj

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for all 1 i = j n. S KETCH OF PROOF. We may assume that the points xi are in n2 . Fix a k < n and a rank k orthogonal projection P0 on n2 . When u ranges over O(n), P = uP0 u−1 ranges over all rank k orthogonal projections. It √ is not hard to check that, for all x ∈ S n−1 , −1 E = O(n) uP0 u x dν(u) is of the order k/n and thus, for every x ∈ S n−1 ,

2 ν u; uP0 u−1 x − E > εE = μ x; P0 x − E > εE < e−cε k . It follows that, if k

C ε2

log n, there is a u ∈ O(n) for which

xi − xj −1 (1 + ε)E (1 − ε)E uP0 u xi − xj for all i = j . The range of uP0 u−1 is k-dimensional. Take yi =

uP0 u−1 xi (1−ε)E .

3.2. Fine embeddings of subspaces of Lp in lpn When specializing the proof of Theorem 24 to the case of X = nr , one sees quite easily that if 1 r < 2 then for all ε > 0 there exists a constant c = c(r, ε) > 0 such that for all n there exists a subspace Y of nr of dimension k cn whose Banach–Mazur distance to Euclidean space, d(Y, k2 ) 1 + ε. (For 2 < r < ∞ the same holds with k cn2/r .) This subject is extensively reviewed in [16]. Since it is known (and follows from the existence of p-stable random variables, see below) that, for r < p 2, p embeds isometrically into Lr [0, 1], it is natural to ask whether a similar statement holds with 2 replaced by p, i.e., whether, for r < p < 2, kp (1 + ε)embeds into nr for k proportional to n. Noticing that Gaussian variables are very different from p-stable ones for p < 2 (the ﬁrst decay exponentially while the latter only polynomially), and that the concentration inequality behind the proof of Theorem 24 has very much to do with the exponential decay of Gaussian variables, one’s ﬁrst guess would be that the

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answer to the question above is negative (and probably that k can only be some logarithmic function of n). It turns out, however, that the answer to the question above is positive. It was proved in [26] that for 1 p < 2 and for every n and ε, n1 contains a subspace Y with d(Y, kp ) < 1 + ε where k c(p, ε)n. This was the ﬁrst result concerning “tight embeddings” that didn’t deal with Euclidean spaces. It was proved using certain approximation of p-stable random variables and concentration inequalities for martingales as discussed in Section 2.2. This result lead to a series of generalizations and results of similar nature. We refer to [28] for a survey of this topic. Here we only deal with two such examples of generalizations. We would ﬁrst like to mention a result of Pisier [47], generalizing the result above from the side of the containing space, n1 . p Recall that a random variable h whose characteristic function is given by E eit h = e−c|t | , for some positive constant c, is called (symmetric) p-stable. Lévy proved the existence of such variables for 0 2.) A p-stable variable has r-th moment for all r < p but doesn’t have p-th moment. For 1 < p < 2 we’ll denote from now on by h the p-stable variable whose ﬁrst moment is equal to 1. This deﬁnes its distribution uniquely. If h, h1 , . . . , hn are independent and identically distributed then it is easy to see (compute the characteristic function) that ni=1 αi hi also has the same distribution as h as long as ni=1 |αi |p = 1. In particular the span of h1 , . . . , hn in L1 [0, 1] is isometric to np . For 1 < p < 2, the stable type p constant of a Banach space X, denoted STp (X), is the smallest constant C such that, E hi xi Cn1/p sup xi

(68)

1in

deﬁnition for all ﬁnite sequences {x1 , . . . , xn } of elements of X. (This is an equivalent to the more common one where n1/p sup1in xi is replaced with ( ni=1 xi p )1/p .) Pisier’s result is: T HEOREM 27. For each 1 0 there is a positive constant c = c(p, ε) such that any Banach space X contains a subspace Y satisfying d(Y, kp ) as long as k < cSTp (X)p/(p−1) .

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Since it is easy to see that STp (n1 ) n(p−1)/p , this implies the result of [26] referred to above. A BRIEF SKETCH OF THE PROOF. Pick a ﬁnite sequence, x1 , x2 , . . . , xn , of elements of X for which max xi = 1 and E hi xi 12 n1/p STp (X). Let u1 , u2 , . . . be a sequence of independent random variables each uniformly distributed over the set of 2n j elements {±x1, ±x2 , . . . , ±xn }. Put also Γj = i=1 Ai , j = 1, 2, . . . , where the Aj ’s are independent (and independent of the sequence {ui }) canonical exponential variables, i.e., P (Ai > t) = e−t , t > 0. We shall use a representation theorem for p-stable variables, due

Concentration, results and applications

1629

to LePage, Woodroofe and Zinn [37] which says in particular that, for some constant cp depending only on p, S=

∞

−1/p

Γj

uj has the same distribution as cp n−1/p

j =1

n

hi xi

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i=1

and in particular, ES 2p STp (X). Note that for any functional x ∗ , x ∗ (S) is a p-stable variable. If S1 , . . . , Sk are independent and all have the same distribution as S then it is easily seen that if ki=1 |αi |p = 1 then ki=1 αi Si has the same distribution as S and in particular E ki=1 αi Si = ES. −1/p The next step is to replace the random coefﬁcients {Γj } with the deterministic se −1/p u and let R , . . . , R be independent and all have quence {j −1/p }. Put R = ∞ j j 1 k j =1 the same distribution as R. A computation using the explicit distribution of Γj shows that c

C = ESi − Ri < ∞ and it follows that, if

k

i=1 |αi |

p

= 1,

k k k αi Si − E αi Ri C |αi | E i=1

i=1

i=1

Ck

(p−1)/p

k

1/p |αi |

p

i=1

< Cc(p−1)/p STp (X)

(71)

by the choice of k. Note that ki=1 |αi |p = 1 implies that {αi j −1/p }p,∞ = 1 and thus Proposition 5(2) implies that for all such {αi } and for all t > 0,

P αi Ri − E αi Ri > t K exp −δt p/(p−1) .

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This last equation is of course the place where the method of concentration enters, which was the main thing we wanted to illustrate here. The rest of the proof goes along similar lines to the end ofthe proof of Theorem 24: note that it follows from (71), that, for c small enough, E αi Ri is of order STp (X). Choose an ε net in the sphere of kp of cardinality smaller than (3/ε)k . Then, with high probability, αi Ri is of order STp (X) for all sequences {αi } in the net. By a successive approximation the same holds now for all sequences {αi } in the sphere of np which completes the proof. Another way to generalize the result of Schechtman and Johnson [26] (that cn p nicely embeds in n1 ) is from the side of the embedded space, np . After some initial work by

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Schechtman (mostly [52]) on embedding ﬁnite-dimensional subspaces of Lp [0, 1] in lowdimensional nr spaces in which a new class of “random embeddings” (which were not related to p stable variables) were introduced, Bourgain, Lindenstrauss and Milman [9] proved that, for 0 < r < 2, every k-dimensional subspace of Lr [0, 1] (1 + ε)-embeds in nr provided n/k is at least a certain power of log n times a constant (depending only on r and ε). See also [27] for a different proof. All the proofs involved use concentration in one way or another. The result of [9] mentioned above was improved and simpliﬁed by Talagrand [60]. Since his proof has to do with bounds on Gaussian processes and is related to Section 2.5, we would like to brieﬂy review it. As we have already advertised, the article [28] has more on that subject. Here we shall deal only with the case r = 1. T HEOREM 28. For every ε there is a constant C(ε) such that for all n, every Cn log n n-dimensional subspace Y of L1 [0, 1] is (1 + ε)-isomorphic to a subspace of 1 . We remark in passing that one of the main open problems in this area is whether the factor log n is needed. Besides concentration inequalities the proof uses some other heavy tools and is discussed in [28]. We shall only touch the idea involving bounds on Gaussian processes. T HE IDEA OF THE PROOF. By crude approximation we may assume that Y is a subspace of m 1 for some ﬁnite (but huge) m. We would like to show that a restriction to a “random” subset of cardinality Cn log n of the coordinates is a good isomorphism when restricted to Y . Of course this is wrong in general (for instance if Y has an element which is supported on only one coordinate, this element would most probably be sent to zero by such a restriction). The idea is to ﬁrst “change the density” and send Y to an isometric subspace whose elements are “spread out” over the m coordinates. The idea that this may work was the point of [52]. It will be dealt with in [28] and will not be discussed here any further. We’ll concentrate in describing how to evaluate the norm of the random restriction on Y and the norm of its inverse assuming Y is already in good position. We do it inductively, restricting ﬁrst to a random set of about half the coordinates where each coordinate is chosen with probability 1/2 and the different choices are independent. Equivalently, let {εi }m i=1 be independent variables each taking the values −1 and 1 with probability 1/2 each. We would like to evaluate the restriction to the set A = {i; εi = 1}. If we could show that sup

m 2 < ε(n, m) |x | − |x | i i

x∈Y,x1

i∈A

(73)

i=1

with ε(n, m) “very small” when n/m is small, then this would mean that (2 times) the restriction to A is very close to an isometry. Iterating this would lead, depending on the behavior of ε(n, m), to the desired random restriction onto a small set of coordinates. Note that the quantity in (73) is equal to supx∈Y,x1 | m i=1 εi |xi || and in particular is the same

Concentration, results and applications

1631

for A and its complement. Since we are interested in only one set A, of cardinality at most m/2 satisfying (73), it is enough to establish m E sup εi |xi | ε(n, m). (74) x∈Y,x1 i=1

This quantity is dominated by a similar one with independent standard Gaussian variables gi ’s replacing the εi ’s. So the problem reduces to estimating m E sup gi |xi |, x∈Y,x1 i=1

i.e., the expectation of the supremum of a speciﬁc Gaussian process. This makes the connection with Section 2.5. We shall not go into more details here. Theorem 28 has a nice geometrical interpretation which is obtained by looking at the polar body to the unit ball of Y . C OROLLARY 29. Let K be the (Minkowski) sum of segments in Rn (or limit of such bodies, these are called zonoids). Then, for every ε, there is a body L in Rn which is the sum of at most C(ε)n log n segments and which ε approximates K in the sense that L ⊂ K ⊂ (1 + ε)L.

3.3. Selecting good substructures Given a sequence of independent vectors {x1 , x2 , . . . , xn } in a normed space X and an ε > 0, what is the largest cardinality k such that there are k disjoint blocks y1 , y2 , . . . , yk which are (1 + ε)-unconditional or (1 + ε)-symmetric? Recall that by disjoint blocks we mean vectors of the form yi = j ∈σi aj xj , i = 1, . . . , k, with σ1 , σ2 , . . . , σk disjoint subsets of 1, 2, . . . , n. y1 , y2 , . . . , yk is said to be (1 + ε)-unconditional (resp. (1 + ε)-symmetric) if k k εi bi yi (1 + ε) bi yi i=1

i=1

for all signs {εi } and all coefﬁcients {bi }. (resp. if k k εi bi yπ(i) (1 + ε) bi yi i=1

i=1

for all signs {εi }, all permutations π of 1, 2, . . . , k and all coefﬁcients {bi }.)

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These problems and various variations thereof were treated quite successfully by concentration of measure methods. The point is that, ﬁxing a partition σ1 , σ2 , . . . , σk of {1, 2, . . . , n} and coefﬁcients {{aj }j ∈σi }ki=1 , the norms k k bi ei = Aveε bi εj aj xj i=1

and

u

i=1

j ∈σi

k k bi ei = Aveε,π bi εj aπ(j ) xj i=1

s

i=1

j ∈σi

on Rn are 1-unconditional and 1-symmetric respectively. If we can ﬁnd signs {{εj }j ∈σi }ki=1 such that, for all {bi }, ki=1 bi j ∈σi εj aj xj / ki=1 bi ei u is appropriately close to one, then we found disjoint blocks of length k which are (1 + ε)-unconditional. A similar statement holds for the symmetric case. For lack of space we shall not review all that is known about this subject. The unconditional case was ﬁrst treated by Amir and Milman in [2,3]. Gowers improved some of their quantitative estimates ([17,18]) and in some instances got, except for possible log factors, the best possible estimates. The symmetric case was treated by Maurey [41] and was the motivation for proving Corollary 7 (for Πn ). We were dealing here only with applications to functional analysis and convexity. There are many applications to other areas which we shall not expand on. There are applications to graph theory (see, e.g., the construction of expander graphs in [1]), to other combinatorial questions, computer science, mathematical physics and probability (in particular to estimating the tail behavior of random variables of the form εi Xi for independent vector valued random variables {Xi }). [62] contains many applications of the material of Section 2.3. References [1] N. Alon and V.D. Milman, λ1 , isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B 38 (1985), 73–88. [2] D. Amir and V.D. Milman, Unconditional and symmetric sets in n-dimensional normed spaces, Israel J. Math. 37 (1980), 3–20. [3] D. Amir and V.D. Milman, A quantitative ﬁnite-dimensional Krivine theorem, Israel J. Math. 50 (1985), 1–12. [4] J. Arias-de-Reyna, K. Ball and R. Villa, Concentration of the distance in ﬁnite dimensional normed spaces, Mathematika 45 (1998), 245–252. [5] K. Azuma, Weighted sums of certain dependent random variables, Tôhoku Math. J. 19 (1967), 357–367. [6] A. Baernstein II and B.A. Taylor, Spherical rearrangements, subharmonic functions, and ∗-functions in n-space, Duke Math. J. 43 (1976), 245–268. [7] Y. Benyamini, Two point symmetrization, the isoperimetric inequality on the sphere and some applications, Longhorn Notes, Texas Funct. Anal. Seminar, Univ. of Texas (1983–1984), 53–76. [8] S. Bobkov and M. Ledoux, Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution, Probab. Theory Related Fields 107 (1997), 383–400.

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[9] J. Bourgain, J. Lindenstrauss and V.D. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73–141. [10] R.M. Dudley, Uniform Central Limit Theorems, Cambridge Stud. Adv. Math. 63, Cambridge Univ. Press, Cambridge (1999). [11] A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Sympos. on Linear Spaces (Jerusalem, 1961), 123–160. [12] A. Dvoretzky and C.A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. USA 36 (1950), 192–197. [13] U. Feige and G. Schechtman, On the optimality of the random hyperplane rounding technique for MAX CUT, Random Structures Algorithms 20 (2002), 403–440. [14] T. Figiel, J. Lindenstrauss and V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53–94. [15] P. Frankl and Z. Füredi, A short proof for a theorem of Harper about Hamming-spheres, Discrete Math. 34 (1981), 311–313. [16] A.A. Giannopoulos and V.D. Milman, Euclidean structure in ﬁnite-dimensional normed spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 707–779. [17] W.T. Gowers, Symmetric block bases in ﬁnite-dimensional normed spaces, Israel J. Math. 68 (1989), 193– 219. [18] W.T. Gowers, Symmetric block bases of sequences with large average growth, Israel J. Math. 69 (1990), 129–151. [19] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics 152, Birkhäuser, Boston (1999). [20] M. Gromov and V.D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), 843–854. [21] M. Gromov and V.D. Milman, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263–282. [22] L.H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Combin. Theory 1 (1966), 385–393. [23] P. Hitczenko, Best constants in martingale version of Rosenthal’s inequality, Ann. Probab. 18 (1990), 1656– 1668. [24] W.B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Contemp. Math. 26 (1984), 189–206. [25] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. [26] W.B. Johnson and G. Schechtman, lpm into l1n , Acta Math. 149 (1982), 71–85. [27] W.B. Johnson and G. Schechtman, Remarks on Talagrand’s deviation inequality for Rademacher functions, Functional Analysis (Austin, TX, 1987/1989), Lecture Notes in Math. 1470, Springer, Berlin (1991), 72–77. [28] W.B. Johnson and G. Schechtman, Finite dimensional subspaces of Lp , Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 837–870. [29] W.B. Johnson, G. Schechtman and J. Zinn, Best constants in moment inequalities for linear combinations of independent and exchangeable random variables, Ann. Probab. 13 (1985), 234–253. [30] R. Kannan, L. Lovász and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), 541–559. [31] S. Kwapie´n and J. Szulga, Hypercontraction methods in moment inequalities for series of independent random variables in normed spaces, Ann. Probab. 19 (1991), 369–379. [32] M. Ledoux, On Talagrand’s deviation inequalities for product measures, ESAIM Probab. Statist. 1 (1995/97), 63–87. [33] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilitiés XXXIII, Lecture Notes in Math. 1709, Springer, Berlin (1999), 120–216. [34] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monographs 89, Amer. Math. Soc., Providence, RI (2001).

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[35] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Isoperimetry and Processes, Springer, Berlin (1991). [36] L. Leindler, On a certain converse of Hölder’s inequality II, Acta Sci. Math. (Szeged) 33 (1972), 217–223. [37] R. LePage, M. Woodroofe and J. Zinn, Convergence to a stable distribution via order statistics, Ann. Probab. 9 (1981), 624–632. [38] P. Lévy, Problèmes Concrets D’Analyse Fonctionnelle, 2nd ed., Gauthier-Villars, Paris (1951) (French). [39] L. Lovász and M. Simonovits, Random walks in a convex body and an improved volume algorithm, Random Structures Algorithms 4 (1993), 359–412. [40] K. Marton, A measure concentration inequality for contracting Markov chains, Geom. Funct. Anal. 6 (1996), 556–571. [41] B. Maurey, Construction de suites symétriques, C.R. Acad. Sci. Paris Sér. A-B 288 (1979), 679–681 (French). [42] B. Maurey, Some deviation inequalities, Geom. Funct. Anal. 1 (1991), 188–197. [43] V.D. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Funct. Anal. Appl. 5 (1971), 28–37. [44] V.D. Milman, The heritage of P. Lévy in geometrical functional analysis, Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987), Astérisque 157–158 (1988), 273–301. [45] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, New York (1986). [46] V.D. Milman and G. Schechtman, An “isomorphic” version of Dvoretzky’s theorem, C.R. Acad. Sci. Paris Sér. I Math. 321 (1995), 541–544. [47] G. Pisier, On the dimension of the lpn -subspaces of Banach spaces, for 1 p < 2, Trans. Amer. Math. Soc. 276 (1983), 201–211. [48] G. Pisier, Probabilistic methods in the geometry of Banach spaces (CIME, Varenna, 1985), Lecture Notes in Math. 1206, Springer (1986). [49] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press (1989). [50] A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. [51] G. Schechtman, Lévy type inequality for a class of ﬁnite metric spaces, Martingale Theory in Harmonic Analysis and Banach Spaces (Cleveland, Ohio, 1981), Lecture Notes in Math. 939, Springer, New York (1982), 211–215. [52] G. Schechtman, More on embedding subspaces of Lp in lrn , Compositio Math. 61 (1987), 159–169. [53] G. Schechtman and M. Schmuckenschläger, A concentration inequality for harmonic measures on the sphere, Geometric Aspects of Functional Analysis (Israel, 1992–1994), Oper. Theory Adv. Appl. 77 (1995), 255–273. [54] G. Schechtman and J. Zinn, On the volume of the intersection of two Lnp balls, Proc. Amer. Math. Soc. 110 (1990), 217–224. [55] G. Schechtman and J. Zinn, Concentration on the lpn ball, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1745, Springer, Berlin (2000), 245–256. [56] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications 44, Cambridge Univ. Press, Cambridge (1993). [57] W.F. Stout, Almost Sure Convergence, Probab. Math. Statist. 24, Academic Press (1974). [58] M. Talagrand, An isoperimetric theorem on the cube and the Kintchine–Kahane inequalities, Proc. Amer. Math. Soc. 104 (1988), 905–909. [59] M. Talagrand, Isoperimetry and integrability of the sum of independent Banach-space valued random variables, Ann. Probab. 17 (1989), 1546–1570. [60] M. Talagrand, Embedding subspaces of L1 into l1N , Proc. Amer. Math. Soc. 108 (1990), 363–369. [61] M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, Geometric Aspects of Functional Analysis (Israel, 1989–90), Lecture Notes in Math. 1469, Springer (1991), 94–124. [62] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 73–205. [63] M. Talagrand, Transportation cost for Gaussian and other product measures, Geom. Funct. Anal. 6 (1996), 587–600. [64] V.V. Yurinskii, Exponential bounds for large deviations, Theor. Probab. Appl. 19 (1974), 154–155.

CHAPTER 38

Uniqueness of Structure in Banach Spaces Lior Tzafriri Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel E-mail: [email protected]

Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1. Uniqueness of general and unconditional bases . . . . . 2. Uniqueness of symmetric bases . . . . . . . . . . . . . 3. Uniqueness of unconditional bases, up to a permutation 4. Uniqueness in ﬁnite-dimensional spaces . . . . . . . . 5. Uniqueness of rearrangement invariant structures . . . 6. Uniqueness of bases in non-Banach spaces . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Uniqueness of structure in Banach spaces

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0. Introduction In this chapter, we consider Banach spaces which can be represented as spaces of sequences or functions with some speciﬁc properties, and study the natural question whether such a representation is unique, up to a notion of equivalence which can vary from case to case. The typical space of sequences is derived froma Banach space X with a normalized ∞ Schauder basis {en }∞ n=1 an en ∈ X, there corresponds the n=1 , where to each element x = sequence of coefﬁcients (a1 , a2 , . . . , an , . . .). Of particular interest in this context will be the spaces with a normalized unconditional or symmetric basis. The notion of uniqueness mentioned above means a different thing in each of the cases under consideration. A Banach space X will be said to have a unique general or unconditional or symmetric basis, up to equivalence, if, for any two normalized bases of the same type, there exists an automorphism T of X which takes one basis into the other. The existence of a normalized Schauder basis {en }∞ n=1 in a Banach space X allows the representation of X as a space of ordered sequences. In the special case where {en }∞ n=1 is an ∞ is again a basis which need not unconditional basis, any permutation {eπ(n) }∞ of {e } n n=1 n=1 ∞ be equivalent to {en }∞ n=1 for every π , unless {en }n=1 is a symmetric basis. Therefore, in the case of spaces with a normalized unconditional basis (which is not symmetric) we have the option of considering a different type of uniqueness of the normalized unconditional basis, namely that of uniqueness up to a permutation and equivalence. More precisely, a space X has a unique normalized unconditional basis, up to a permutation (and equivalence) whenever, for each pair of normalized unconditional bases of X, there exists an automorphism T of X which takes the ﬁrst basis into a permutation of the second. Of course, both notions of uniqueness coincide in the case of symmetric bases. In the continuous case, we typically consider Banach lattices which can be represented as spaces of measurable functions on a suitable measure space. Most interesting is the special case of the so-called rearrangement invariant (r.i.) function spaces over a measure space (Ω, Σ, μ). The main property of such a space X of functions is that automorphisms of the underlying measure space transform elements of X into elements of X with the same norm. In the Basic Concepts article such spaces are called symmetric lattices. The notion of uniqueness can be also studied in the context of ﬁnite-dimensional spaces, but, in this case, we have to be more careful since for spaces of the same dimension all the structures are obviously unique. However, for families of Banach spaces {Xn }∞ n=1 , with dim Xn = n for all n, it makes perfect sense to inquire whether, for any two normalized bases of Xn , of the same type, there exists an automorphism Tn of Xn which takes the ﬁrst basis into the second and, most importantly, the quantity Tn · Tn−1 is bounded by a constant independent of n but possibly dependent on the structure constant (by structure constant we mean either the basis constant or unconditional or symmetric constant, according to the case under consideration).

1. Uniqueness of general and unconditional bases It is quite easy to prove that a separable Hilbert space has a unique normalized unconditional basis, up to equivalence. Indeed, if {un }∞ n=1 is a normalized K-unconditional basis

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L. Tzafriri

in 2 then, by the parallelogram identity, N 2 N Ave an σn un = |an |2 σn =±1 n=1

n=1

N

for every choice of N and scalars {an }2n=1 . This of course implies that K

−1

∞

1/2 |an |

2

n=1

∞ ∞ 1/2 2 a n un K |an | , n=1

n=1

∞ for any choice of {an }∞ n=1 , i.e., {un }n=1 is K-equivalent to the unit vector basis of 2 . It turns out that a separable Hilbert space does not have unique basis, up to equivalence. This fact, which is less trivial, was ﬁrst proved by Babenko [2]. His construction is based on the simple observation that the characters {eint }n∈Z , in the order {1, eit , e−it , e2it , e−2it , . . .}, form a basis in the space Lp (T, W (t)), where W (t) is an integrable weight function on the circle T, if and only if the Riesz projection, deﬁned by

P+

+∞

an e

int

=

n=−∞

∞

an eint

n=0

is bounded on this space. This latter question has been extensively studied in harmonic analysis and a well known necessary and sufﬁcient condition for the boundedness of the Riesz projection (or, as a matter of fact, of the Hilbert transform) in the space Lp (T, W (t)) is the so-called Ap -condition (see, e.g., [22, p. 254])

1 sup μ(I ) I

W (t) dt I

1 μ(I )

I

1 W (t)

p−1

1/(p−1) dt

< ∞,

where μ is the Lebesgue measure and the supremum is taken over all intervals I ⊂ T of positive measure. The weight function considered by Babenko is Wα (t) = |t|2α , where α is a suitable number. In order to ensure the integrability of Wα (t), one has to require that 2α + 1 > 0, i.e., that α > −1/2. Furthermore, it is easily veriﬁed that the A2 -condition is satisﬁed by Wα iff 1 − 2α > 0, i.e., when α < 1/2. It follows that, for −1/2 < α < 1/2, the characters {eint }n∈Z , in the order described above, form a basis of the Hilbert space L2 (T, Wα (t)). Equivalently, the sequence

2α + 1 2π 2α+1

1/2

|t|α eint n∈Z

Uniqueness of structure in Banach spaces

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forms a normalized Schauder basis in L2 (T), for any value of −1/2 < α < 1/2. For α = 0, we recover the (unique) unconditional basis of the separable Hilbert space while, for the remaining values of α, we obtain mutually non-equivalent conditional bases of the Hilbert space. Indeed, for α = β in the interval (−1/2, 1/2) and 0 < λ < π , we can ﬁnd scalars {an }n∈Z so that f (t) =

+∞

an eint = χ[0,λ) (t)

n=−∞

in L2 (T) and a.e. on T. Then the fact that f |t|α = λα−β ; f |t|β

0 < λ < π,

shows that the bases {|t|α eint }n∈Z and {|t|β eint }n∈Z are not equivalent. Another construction of conditional bases in a separable Hilbert space, due to McCarthy and Schwartz [42], is presented in detail in [37, p. 74]. The discussion above can be summarized in the following proposition. P ROPOSITION 1.1. A separable Hilbert space has, up to equivalence, a unique normalized unconditional basis and uncountably many mutually non-equivalent conditional basis. In fact, the following more general result was proved by Pelczynski and Singer [49]. T HEOREM 1.2. Any Banach space with an inﬁnite Schauder basis has uncountably many mutually non-equivalent bases. It turns out that also the spaces 1 and c0 have a unique unconditional basis, up to equivalence, but this fact, which was proved by Lindenstrauss and Pelczynski [35], is more difﬁcult and requires some consequences of the famous Grothendieck inequality. We refer to well-known corollaries of this inequality that every bounded linear operator T : c0 → 1 is 2-absolutely summing and its 2-summing norm π2 (T ) satisﬁes the inequality π2 (T ) KG T , and that every bounded linear operator U : 1 → 2 is absolutely summing and its 1-summing norm π1 (U ) satisﬁes π1 (U ) KG U . In both inequalities above, KG denotes as usual the Grothendieck constant. The precise statement of Grothendieck’s inequality together with a simple proof can be found in the Basic Concepts article. In order to prove, e.g., that 1 has a unique unconditional basis, up to equivalence, let {xn }∞ n=1 be a normalized K-unconditional basis ∞of 1 , for some K 1, and ﬁx a senotice that quence of scalars {an }∞ n=1 such that the series n=1 an xn converges. Then ∞ → 1 , deﬁned by T t = ∞ the operator T : c0 n=1 an tn xn , for t = {tn }n=1 ∈ c0 , is of norm T 2K ∞ n=1 an xn (in the real case, 2K can be replaced by K). Hence, by the aforementioned estimate of the 2-summing norm of T , we conclude that π2 (T )

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2KG K ∞ n=1 an xn . Then the deﬁnition of π2 (T ), applied for the unit vectors in c0 , together with the fact that {xn }∞ n=1 is assumed to be normalized imply that

∞

1/2 |an |

=

2

n=1

∞

1/2 T en

2

n=1

∞ 2KG K an xn . n=1

This inequality shows that the operator U : 1 → 2 , deﬁned by U ( ∞ n=1 an xn ) = ∞ {an }∞ n=1 an xn ∈ 1 , is bounded by 2KG K. Then, by usn=1 , for any convergent series ing the estimate, discussed above, ofthe 1-summing norm π1 (U ) of U , it follows that ∞ 2 K. Hence, for any x = π1 (U ) 2KG n=1 an xn ∈ 1 , ∞ n=1

∞ εn an xn εn =±1

∞ U (an xn ) π1 (U ) sup |an | = n=1

2 2 4KG K an xn ,

n=1

n=1

which completes the proof of the fact that {xn }∞ n=1 is equivalent to the unit vector basis of 1 . The case of c0 is proved by using a simple duality argument. The summary of this discussion is P ROPOSITION 1.3. Both spaces 1 and c0 have, up to equivalence, a unique normalized unconditional basis. An alternative proof of Proposition 1.3, which does not use Grothendieck’s inequality, is presented in Section 5 of the Basic Concepts article. It turns out that 2 , 1 and c0 are the only Banach spaces with a unique unconditional basis, up to equivalence. This result was proved by Lindenstrauss and Zippin [39]. T HEOREM 1.4. A Banach space has, up to equivalence, a unique unconditional basis if and only if it is isomorphic to one of the spaces 2 , 1 or c0 . Instead of the original proof from [39], we present a shorter proof based on an argument due to W.B. Johnson. Suppose that a space X has, up to equivalence, a unique normalized ∞ unconditional basis {xn }∞ n=1 . Since, for any choice of εn = ±1, n = 1, 2, . . . , {εn xn }n=1 is ∞ ∞ an unconditional basis of X it must be equivalent to {xn }n=1 and, thus, {xn }n=1 is symmetric. Let now {um }∞ m=1 be an arbitrary normalized block basis with constant coefﬁcients of {xn }∞ and denote by U its span. n=1 If we prove that X is isomorphic to X ⊕ U , then {x1 , u1 , x2 , u2 , . . . , xn , un , . . .}

Uniqueness of structure in Banach spaces

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forms a normalized unconditional basis of a space isomorphic to X. Hence, by the unique∞ ∞ ∞ ness of {xn }∞ n=1 , {un }n=1 is equivalent to {xn }n=1 , i.e., {xn }n=1 is a perfectly homogeneous basis in the terminology of [58]. Thus, by the well known result of Zippin [58], {xn }∞ n=1 is equivalent to the unit vector basis of p ; 1 p < ∞, or of c0 . In order to complete the proof, recall, e.g., that the Rademacher functions {rn }∞ n=1 span a Hilbert space in Lp (0, 1), for any value of p 1, and their span [rn ]∞ in complemented in Lp (0, 1), whenever n=1 n p > 1. In particular, 2p contains a uniformly complemented copy of n2 , for any n and n p > 1. Hence, p contains a complemented copy of the direct sum ( ∞ n=1 ⊕2 )p and, ∞ n by the decomposition method of Pelczynski, p ≈ ( n=1 ⊕2 )p , for p > 1, which shows that the unit vector basis of p is not the unique unconditional basis of this space, up to equivalence, for p > 1, p = 2. In order to prove that X ≈ X ⊕ U , Johnson suggested to use the decomposition method, as follows: let V be the span of a sequence of normalized blocks with constant coefﬁcients of {xn }∞ n=1 which has the universality property that every possible normalized block with constant coefﬁcients of {xn }∞ n=1 appears inﬁnitely many times in the sequences generating V . Then, since any block basis with constant coefﬁcients spans a complemented subspace in a space with a symmetric basis (see, e.g., [34] or [37, p. 123]), X ⊕ V is isomorphic to a complemented in X, i.e., that X ≈ X ⊕ V ⊕ W, for some space W . Since V ⊕ U ≈ V because of the above universality property, it follows that X ≈ X ⊕ V ⊕ U ⊕ W ≈ X ⊕ U, which completes the argument. R EMARK . It can be shown that any space with an unconditional basis which is not unique, up to equivalence, actually has uncountably many mutually non-equivalent unconditional bases. Suppose now that X is a space with a unique, up to equivalence, normalized Schauder ∞ ∞ basis {xn }∞ n=1 . Then {xn }n=1 is equivalent to {εn xn }n=1 , for every choice of εn = ±1, ∞ n = 1, 2, . . . , i.e., {xn }n=1 , is unconditional and thus equivalent to the unit vector basis of 2 , 1 or c0 . However, in each of these three cases, X also has a conditional basis. Simple examples of conditional bases are {en − en−1 }∞ (where e0 = 0 and {en }∞ n=1 in 1 n=1 ∞ denotes the unit vector basis in 1 ) and the summing basis { ∞ i=n en }n=1 in c, which of course is isomorphic to c0 . These facts together with Proposition 1.1 provide a partial proof of Theorem 1.2. 2. Uniqueness of symmetric bases An immediate consequence of Theorem 1.3 is the fact that 2 , 1 and c0 have also a unique symmetric basis, up to equivalence. It turns out that there are considerably more Banach spaces with a unique, up to equivalence, symmetric basis.

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T HEOREM 2.1. The spaces c0 and p ; 1 p < ∞, have, up to equivalence, a unique symmetric basis. ∗ ∞ P ROOF. Fix p > 1, let {en }∞ n=1 denote the unit vector basis of p , {en }n=1 the biorthogonal ∞ ∞ sequence associated to {en }n=1 and let {xm }m=1 be another normalized symmetric basis of ∞ p . By a simple diagonalization argument, one can ﬁnd a subsequence {xmi }∞ i=1 of {xm }m=1 ∗ such that limi→∞ en xmi exists for any choice of n. If all these limits are equal to zero then one can ﬁnd a further subsequence of {xm }∞ m=1 , which is equivalent to a block basis of ∞ . Since any normalized block basis of {en }∞ {en }∞ n=1 n=1 is equivalent to {en }n=1 itself in p ∞ ∞ it follows that {xm }m=1 is equivalent to {en }n=1 . In case there exists a value of n for which limi→∞ en∗ xmi = α = 0 then one can easily ﬁnd an inﬁnite subsequence of {xm }∞ m=1 which is equivalent to the unit vector basis of 1 , thus completing the proof.

The class of spaces with a unique symmetric basis is considerably larger than that of p -spaces and it contains, e.g., all the Orlicz sequence spaces M for which the limit limt →0 tM (t)/M(t) exists. A concrete such function is M(t) = t p /(1 + |log t|); p 1. Another class of spaces with a unique symmetric basis is that of Lorentz sequence spaces is a non-increasing sequence of positive weights d(w, p) where p 1 and w = {wn }∞ n=1 satisfying w1 = 1, limn→∞ wn = 0 and ∞ n=1 wn = ∞. Recall that d(w, p) in the space of all sequences α = {αn }∞ n=1 of scalars so that

∞ p α α = sup π(n) wn π

1/p < ∞,

n=1

where the supremum is taken over all permutation π of the integers. The proofs that these two classes of spaces do have a unique symmetric basis are not very hard but we omit them. They can be found, e.g., in [37, Section 4]. An example of a space with “many” mutually non-equivalent symmetric bases is the so-called space U1 of Pelczynski [47], which is universal for all unconditional bases in the sense that it has an unconditional basis {un }∞ i=1 , and, quite remarkably, every unconditional ∞ basic sequence {vj }∞ j =1 in any Banach space V is equivalent to a subsequence of {un }n=1 . A simple application of the decomposition method shows that the property deﬁning U1 characterizes it up to isomorphism. A simple way of constructing this space was suggested by Schechtman [52]. To this end, let {fn }∞ n=1 be a sequence of continuous functions which is dense in the space C(0, 1) of all continuous functions on [0, 1] and, for any sequence a = {an }∞ n=1 which is eventually zero, i.e., a ∈ c00 , deﬁne ∞ |||a||| = sup εn an fn ; εn = ±1, n = 1, 2, . . . . n=1

C(0,1)

The unit vectors {un }∞ n=1 clearly form an unconditional basis in the completion U1 of c00 relative to the norm ||| · |||. If {vk }∞ k=1 is an unconditional basic sequence in an arbitrary

Uniqueness of structure in Banach spaces

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Banach space V one can assume without loss of generality that V is a subspace of C(0, 1). ∞ Hence, one can ﬁnd a subsequence {fnk }∞ k=1 of {fn }n=1 so that vk − fnk C(0,1) → 0, ∞ as k → ∞, “fast enough” as to imply that {vk }k=1 is equivalent to {fnk }∞ k=1 . Therefore, ∞ , which completes the proof. {fnk }∞ is unconditional and thus equivalent to {u } nk k=1 k=1 In order to show that U1 has a symmetric basis, a fact which is not a priori obvious, one needs an interpolation argument of Davis [13]: let {xn }∞ n=1 be a normalized 1-unconditional basis in a Banach space X and, for m > 1 and p 1, deﬁne the Orlicz function Mp (t) = t p /(1 + |log t|); t > 0, and the norm

1/2 ; αm,p = inf β2Mp + γ 2p

α = βm−1 + γ m, with β ∈ Mp and γ ∈ p ,

for all α ∈ Mp . Then, whenever {mn }∞ n=1 is a sequence of numbers > 1 satisfying the ∞ −1 condition n=1 mn < ∞, the expression α

(p)

∞ = αmn,p xn , n=1

X

p whose unit vectors form a 1-symmetric basis in the compledeﬁnes a norm on a space Y p so that tion Yp of Y Kp−1 αMp α(p) Kp αp , ∞ for all α ∈ p . It turns out that {mn }∞ n=1 can be selected as to ensure that {xn }n=1 is equivalent to a block basis with constant coefﬁcients of the unit vector basis of Yp . This proves that X is isomorphic to a complemented subspace of the space Yp , which of course has a symmetric basis. This assertion can be also proved by using the fact from [34] that every space with an unconditional basis is isomorphic to a complemented subspace of a space with a symmetric basis. The advantage of the present proof is that it shows that X can be complementably embedded in uncountably many spaces with mutually non-equivalent symmetric bases. By applying this argument to U1 , one constructs, for each value of p 1, a space Zp (p) with a symmetric basis {zn }∞ n=1 so that U1 is isomorphic to a complemented subspace of Zp . Since both U1 and Zp are isomorphic to their own square one can apply the decomposition method and conclude that U1 ≈ Zp , for any p 1. This further implies that (p) U1 has, quite surprisingly, a symmetric basis which is equivalent to {zn }∞ n=1 . As we have (p) ∞ (q) ∞ noticed before, for p = q, {zn }n=1 is not equivalent to {zn }n=1 , i.e., U1 has even a continuum of mutually non-equivalent symmetric bases. With some additional effort, one can construct more “natural” spaces, namely Orlicz sequence spaces, which also have a continuum of mutually non-equivalent symmetric bases (cf. [36] or [37, p. 153]). The fact that all the examples of spaces with a symmetric basis

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considered so far have either a unique symmetric basis, up to equivalence, or uncountably many mutually non-equivalent symmetric bases lead inevitably to the question whether this is always the case. It turns out that the answer is negative: Read [50] has constructed, for every n = 1, 2, . . . or n = ℵ0 , a space Xn with a symmetric basis which has precisely n different symmetric bases. It should be added that we are very far form being able to characterize the class of spaces with a unique symmetric basis, up to equivalence. In fact, we do not have even a reasonable conjecture.

3. Uniqueness of unconditional bases, up to a permutation The notion of uniqueness of the unconditional basis of a Banach space X can be interpreted as asserting that X can be represented in a unique manner as a space of sequences a = {an }∞ n=1 ∈ X with the property that ∞ {an }∞ n=1 ∈ X ⇒ {εn an }n=1 ∈ X,

for any choice of εn = ±1; n = 1, 2, . . . . Obviously, in this representation X is considered as a set of ordered sequences. Now, if instead we consider X as a space of unordered sequences with the property described above then uniqueness of the representation has a different meaning: uniqueness up to permutation and equivalence. Recall that, as it was mentioned in the introduction, a Banach space X with a normalized unconditional basis {xn }∞ n=1 is said to have unique unconditional basis, up to equivalence and permutation, if, whenever {yn }∞ n=1 is another ∞ , for some normalized unconditional basis of X, then {yn }∞ is equivalent to {x } π(n) n=1 n=1 permutation π of the integers. It turns out that there are considerably more spaces with a unique unconditional basis, up to permutation, than the three space 2 , 1 and c0 , which are known to have, up to equivalence, a unique unconditional basis. One such class of spaces was found by Èdelšte˘ın and Wojtaszczyk [19], who showed that ﬁnite direct sums of 2 , 1 and c0 have the uniqueness property mentioned before. T HEOREM 3.1. Each of the Banach spaces 1 ⊕ 2 , 1 ⊕ c0 , 2 ⊕ c0 and 1 ⊕ 2 ⊕ c0 has, up to permutation and equivalence, a unique unconditional basis. In order to describe the ideas used in the proof, consider, e.g., the case of the space 1 ⊕ 2 and assume that {zn = xn + yn }∞ n=1 , where xn ∈ 1 and yn ∈ 2 for all n, is a normalized K-unconditional basis of this space. Put and N2 = n; xn > 1/2K N1 = n; xn 1/2K and notice that both [zn ]n∈N1 and [zn ]n∈N2 are complemented subspaces in 1 ⊕ 2 whose direct sum is the whole space. In order to study these two complemented subspaces of 1 ⊕ 2 , we need a result which “rotates” any complemented subspace of a “nice” direct

Uniqueness of structure in Banach spaces

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sum into a “correct” position. To this end, recall that an operator T from a Banach space X into a space Y is strictly singular if the restriction of T to any inﬁnite-dimensional subspace of X is not an isomorphism. Compact operators are of course strictly singular but these two notions are very different. Clearly, every bounded linear operator from 1 into 2 or, viceversa, from 2 into 1 is strictly singular. Now, we can state the theorem of Èdelšte˘ın and Wojtaszczyk [19] which provide the “rotation” into a “correct” position. T HEOREM 3.2. Suppose that X and Y are two Banach spaces so that every operator from X into Y is strictly singular, and let P be a bounded projection form X ⊕ Y onto a subspace Z. Then one can ﬁnd an automorphism S of X ⊕ Y and complemented subspaces X0 of X and Y0 of Y such that SZ = X0 ⊕ Y0 . We omit the proof of the theorem which is based on a good understanding of several facts on Fredholm operators. We return now to the proof of Theorem 3.1 in the case of the direct sum 1 ⊕ 2 . The proof will be completed once we show, e.g., that [zn ]n∈N1 ≈ 2 and [zn ]n∈N2 ≈ 1 since both 1 and 2 have a unique unconditional basis, up to equivalence. Notice that if xn =

∞

an,j zj

and yn =

j =1

∞

bn,j zj

j =1

then an,n + bn,n = 1, for all n. However, for n ∈ N1 , |an,n | Kxn 1/2, and thus |bn,n | > 1/2. Now consider the linear operator U from the subspace [εn yn ]n∈N1 of L∞ (2 ), where εn (t); n = 1, 2, . . . , denote the Rademacher functions, into [zn ]n∈N1 which maps εn yn to zn , for all n ∈ N1 . Since both {εn yn }n∈N1 and {zn }n∈N1 are unconditional bases, a well known diagonal argument shows that the corresponding diagonal operator is bounded by KU . This means that cn bn,n zn KU sup cn ε n y n , n∈N1

εn =±1 n=1,2,...

n∈N1

for any choice of {cn }n∈N1 . By Theorem 3.2, if [zn ]n∈N1 is not isomorphic to 2 then it contains a complemented subspace isomorphic to 1 . Hence, one can ﬁnd a normalized block basis wj = ∈σj d z ; j = 1, 2, . . . , of {zn }n∈N1 , which is equivalent to the unit vector basis in 1 . By passing to a subsequence if necessary, one can assume that the subspaces [y ]∈σj ; j = 1, 2, . . . , are “almost” supported on mutually orthogonal subspaces of 2 in the sense that the unit

1646

L. Tzafriri

vector basis {en }∞ n=1 of 2 can be split into disjoint subsets {en }n∈nj , j = 1, 2, . . . , so that, essentially speaking, [y ]∈σj ⊂ [en ]n∈ηj , for all j . Therefore, for any sequence {cj }∞ j =1 2 < ∞ and any choice of ε = ±1, for all n, with ∞ |c | j n j =1 ∞ ∞ ∞ 1/2 2 1/2 2 2 cj d ε y ∼ |cj | d ε y K |cj | . j =1

j =1

∈σj

j =1

∈σj

Hence, the series ∞ j =1 cj ∈σj d ε y converges unconditionally and so does the series ∞ ∞ ∈σj b, d z . This implies the convergence of the series j =1 cj j =1 cj wj , whenever ∞ ∞ 2 j =1 |cj | < ∞, which of course contradicts the fact that {wj }j =1 is equivalent to the unit vector basis of 1 . Consequently {zn }n∈N1 is equivalent to the unit vector basis of 2 . In a similar manner but using also a duality argument, one proves that [zn ]n∈N2 ≈ 1 , and thus that {zn }n∈N2 is equivalent to the unit vector basis of 1 . While the ﬁnite direct sums constructed out of the three spaces 1 , 2 and c0 have, up to equivalence and permutation, a unique unconditional basis, this is not always the case for inﬁnite direct sums constructed out of the same set of spaces. On the positive side we have, e.g., the following result from [6]. T HEOREM 3.3. Every normalized unconditional basis of an inﬁnite-dimensional complemented subspace of the direct sum (2 ⊕2 ⊕· · ·⊕2 ⊕· · ·)0 , is equivalent to a permutation of the unit vector basis of one of the following six spaces: 2 , c0 , 2 ⊕ c0 ,

∞

⊕n2

, 2 ⊕

n=1

∞

⊕n2

, (2 ⊕ 2 ⊕ · · · ⊕ 2 ⊕ · · ·)0 .

n=1

0

0

A similar statement holds for complemented subspaces of the dual space (2 ⊕ 2 ⊕ · · · ⊕ 2 ⊕ · · ·)1 . Consequently, the six spaces appearing above and their duals have, up to permutation and equivalence, a unique unconditional basis. The proof of Theorem 3.3 is not extremely difﬁcult but still beyond the scope of this article and therefore we omit it here. Another result from [6] is: T HEOREM 3.4. Every normalized unconditional basis of an inﬁnite-dimensional complemented subspace of the direct sum (1 ⊕ 1 ⊕ · · · ⊕ 1 ⊕ · · ·)0 is equivalent to a permutation of the unit vector basis of one of the following six spaces: c0 , 1 , c0 ⊕ 1 ,

∞ n=1

⊕n1

, 1 ⊕ 0

∞ n=1

⊕n1

, (1 ⊕ 1 ⊕ · · · ⊕ 1 ⊕)0 . 0

Consequently, each of these six spaces has, up to permutation and equivalence, a unique unconditional basis.

Uniqueness of structure in Banach spaces

1647

Though Theorems 3.3 and 3.4 have a similar formulation, the proof of 3.4 in [6] is considerably harder than that of 3.3. This is reﬂected also by the fact that the function M = M(K) (so that if {xn }∞ n=1 is a normalized K-unconditional basis of one of the spaces X appearing above, then {xn }∞ n−1 is M(K)-equivalent to a permutation of the unit vector basis of X) behaves differently in the two cases discussed above. While the proof of 3.3 gives M(K) as a power of K, the proof of 3.4 yields M = M(K) as an exponential function of K and examples show that an exponential function is really needed. Very recently, Casazza and Kalton [10] provided a simpler proof of the uniqueness, up to a permutation, of the unit vector basis in the space (1 ⊕ 1 ⊕ · · · ⊕ 1 ⊕ · · ·)0 and its dual. Quite surprisingly, the other inﬁnite direct sums which can be constructed out the the three spaces 1 , 2 and c0 do not have the the uniqueness property exhibited in Theorems 3.3 and 3.4. T HEOREM 3.5 (cf. [6]). The direct sums

∞

⊕n∞

n=1

, c0 ⊕ 2

∞ n=1

⊕n∞

, (c0 ⊕ c0 ⊕ · · · ⊕ c0 ⊕ · · ·)2 2

and their duals fail to have a unique unconditional basis, up to equivalence and permutation. n P ROOF. We shall treat here only the case of the dual direct sum X = ( ∞ n=1 ⊕1 )2 ; the other cases can be easily derived from it without too much difﬁculty. In order to exhibit a normalized unconditional basis of X which is not permutatively equivalent to the unit vector basis of X, we ﬁrst ﬁx n and let Fi = {Ai,j }nj=1 ; 1 i n, the independent partitions of the interval [0, 1] into sets of measure equal to 1/n (i.e., for A ∈ Fi , B ∈ Fj and i = j, μ(A ∩ B) = μ(A)μ(B) = 1/n2 ). With {ei }ni=1 denoting the unit vector basis of the space n1 , consider the vector valued functions fi,j (t) = n1/2 χAi,j (t)ei ;

1 i, j n,

which form a normalized 1-unconditional basis in a subspace Yn of the space L2 ([0, 1], n1 ) since n 2 1/2 n n 1 1/2 ai,j fi,j = n ai,j χAi,j (t) dt , 0 i,j =1

i=1 j =1

for any choice of {ai,j }ni,j =1 . Indeed, for each 1 i n, the functions n ai,j χAi,j (t) j =1

1648

L. Tzafriri

and n |ai,j |χAi,j (t) j =1

are both Fi measurable and have the same independence of Fi , distribution. Hence, the 1 i n, ensures that that the functions | nj=1 ai,j χAi,j (t)| and nj=1 |ai,j |χAi,j (t) have the same norm in the space L2 ([0, 1], n1 ). Now notice that if Ei denotes the conditional expectation operator associated to the partition Fi , i.e., the operator deﬁned by Ei ϕ =

ϕ(t) dt χAi,j ,

n Ai,j

j =1

for ϕ ∈ L1 (0, 1), then one can deﬁne a projection Pn from L2 ([0, 1], n1 ) onto Yn by setting Pn =

n (Ei ϕi )ei , i=1

whenever f (t) = ni=1 ϕi (t)ei is an element of L2 ([0, 1], n1 ). Then, by using the independence of the partitions {Fi }ni=1 , one gets that Pn f = 2

1 n 0

=

2 |Ei ϕi |

dt

i=1

n

1

|Ei ϕi |2 dt +

i=1 0

=

n

n i,j =1 i =j

1

|Ei ϕi |2 dt +

i=1 0

1

|Ei ϕi ||Ej ϕj | dt

0

n i,j =1 i=j

1 0

1

|Ei ϕi | dt

|Ej ϕj | dt ,

0

for any f as above. Hence, Pn f 2 2

n

1

|ϕi |2 dt = 2f 2 ,

i=1 0

√ √ i.e., Pn 2, which shows that Yn is 2-complemented in L2 ([0, 1], n1 ), or even in the subspace (n1 ⊕ n1 ⊕ · · · ⊕ n1 )2 (nn factors) of L2 ([0, 1], n1 ). It follows that the space Y =( ∞ ⊕Y n )2 is isomorphic to a complemented n=1 subspace of X. On the other hand, the block basis yi = n−1/2 nj=1 fi,j ; 1 i n, of {fi,j }ni,j =1 is 1-equivalent to the unit vector basis of n1 , and thus its span is complemented in Yn , since

Uniqueness of structure in Banach spaces

1649

1 is block injective, i.e., 1-complemented, whenever it is embedded as a block basis of a 1-unconditional basis. Consequently, Y contains a 1-complemented copy of X and thus, by the decomposition method, X ≈ Y . The proof will be completed once we show that the natural basis of Y is not equivalent to a permutation of the unit vector basis in X. To this end, notice that the unit vector basis of X has the property that any of its subsets of ﬁnite codimension contains, for each n, a subsequence 1-equivalent to the unit vector basis of n1 . The natural basis of Y has, however, a diametrically opposite behavior. Indeed, for each ε > 0 and each integer k, there exists an integer n = n(ε, k) such that any subset of {fi,j }ni,j =1 of cardinality k is (1 + ε)-equivalent to the unit vector basis of k2 and, moreover, this property remains valid in Y provided that, for any such k, we eliminate a suitable ﬁnite set of vector from the basis of Y . Another quite well-known space for which the question of uniqueness, up to a permutation, could be settled is the so-called Tsirelson space, introduced in [54] (see also [20]). This space is the completion of the space of sequences x ∈ c0,0 under the minimal norm · T satisfying the conditions: (i)

xT x0 ,

for all x ∈ c00 , and (ii)

1 xi = xi T , 2 i=1

T

i=1

whenever support x1 < support x2 < · · · < support x ; = 1, 2, . . . . The fact that such a norm exists is proved in [20]. T HEOREM 3.6. Every complemented subspace of T with an unconditional basis has, up to permutation and equivalence, a unique unconditional basis. Actually, this result was preceded by a theorem proved in [6] which asserts that the 2-convexiﬁcation T (2) of T , as well as its complemented subspaces with an unconditional basis, have a unique unconditional basis, up to equivalence and permutation (recall that 1/2 T (2) is obtained from T by deﬁning the norm in it as follows: xT (2) = |x|2 T , for vectors x having the property that the square of their absolute value belongs to T ). The proof is quite difﬁcult. The proof for T (i.e., of Theorem 3.6), due to Casazza and Kalton [9], is the byproduct of a more general study of the uniqueness property in spaces which do not contain uniformly complemented copies of n2 , for all n. For 1 < p = 2, the p-convexiﬁcation T (p) of T does not have a unique unconditional basis, up to permutation and equivalence, since, as was pointed out by Kalton, T (p) can be represented a Tsirelson sum of np ’s and, in this sum, the factor np has an unconditional basis containing among its vectors an k2 with k ∼ log n. The Banach spaces with a unique unconditional basis, up to permutation and equivalence, which were considered so far in this section, have the additional property that also

1650

L. Tzafriri

their complemented subspaces with an unconditional basis share the uniqueness feature. For such spaces one can introduce the notion of genus (cf. [6]). A Banach space with an unconditional basis is said to be of genus n if in all its complemented subspaces with an unconditional basis, the normalized unconditional basis is unique, up to equivalence and permutation, and there are exactly n different isomorphic types of complemented subspaces with an unconditional basis. The spaces 1 , 2 and c0 are clearly of genus 1. It turns out (cf. [6]) that these three spaces are the only ones of genus 1. T HEOREM 3.7. 1 , 2 and c0 are the only spaces with a (unique) unconditional basis of genus 1. The idea of the proof from [6] is the following: if X is a space with a normalized uncon∞ ditional basis {xn }∞ n=1 of genus 1 then every inﬁnite subsequence of {xn }n=1 is equivalent . Then, by using a sort of compactiﬁcation argument involving to a permutation of {xn }∞ n=1 is, up to permutation, a subspreading models and ultraproducts, one can show that {xn }∞ n=1 itself is subsymmetric. symmetric basis. Therefore, one can assume w.l.o.g. that {xn }∞ n=1 ∞ Since {xn }∞ n=1 is subsymmetric any block basis with constant coefﬁcients of {xn }n=1 spans a complemented subspace U of X. Moreover, by a slightly more complicated argument than that used in the case when {xn }∞ n=1 is symmetric, it can be easily shown that X ≈ X ⊕ U . Hence, any normalized block basis with constant coefﬁcients of {xn }∞ n=1 is , for some permutation π of the integers. Then, by a simple modequivalent to {xπ(n) }∞ n=1 iﬁcation of Zippin’s characterization of perfectly homogeneous bases from [58], one concludes that {xn }∞ n=1 is equivalent to the unit vector basis in c0 or p , for some p 1. The cases when 1 < p = 2 can be easily dismissed, as shown ∞ before. n n Theorems 3.3 and 3.4 above show that the sums ( ∞ n=1 ⊕2 )0 , ( n=1 ⊕1 )0 and their duals are spaces of genus 2. However, it is not known if there exist other spaces of genus 2. The spaces 1 ⊕ 2 , 1 ⊕ c0 and 2 ⊕ c0 are of genus 3 but, again, we do not have a complete characterization of this class. It is quite possible that the class of spaces of ﬁnite genus coincides with that of spaces which are obtained from Hilbert space by taking repeated ﬁnite or inﬁnite direct sums in the sense of c0 or 1 . Casazza and Lammers [12] obtained many results on spaces of ﬁnite genus, e.g., that the unconditional basis contains a subsequence equivalent to the unit vector basis in either 1 , or 2 or c0 . There is a feeling that this class will be well understood once spaces of genus 2 are characterized. The Tsirelson space T and its 2-convexiﬁcation are of inﬁnite genus. The question whether the uniqueness of the unconditional basis, up to permutation and equivalence, is a hereditary property has a negative answer. More precisely, Casazza and Kalton [9] constructed the ﬁrst example of a Banach space with a unique unconditional basis, up to equivalence and permutation, which has complemented subspaces with an unconditional basis lacking the uniqueness property. Their starting point in the Orlicz sequence space M , where M(t) = t/(1 + |log t|), for t 0. In this, as in any other Orlicz block basis with constant coefﬁcients of the form qj+1 space, a normalized qj+1 e )/ e uj = ( n=q n n=q +1 n ; j = 1, 2, . . . , generates a so-called modular space j−1 +1 qj+1j−1 M [sj ], where sj = 1/ n=qj−1 +1 en , for all j . This subspace, which is clearly comple-

Uniqueness of structure in Banach spaces

1651

mented in M , can be described as the space of all sequences a = (a1 , a2 , . . . , aj , . . .) so that ∞ j =1 Msj (|aj |) < ∞, where Ms (t) = M(st)/M(s). It can be easily veriﬁed that if sj → 0 very fast, as j → ∞, then M [sj ] is isomorphic to 1 , and, on the other hand, if infj sj > 0 then M [sj ] ≈ M . By manipulating between these diametrically opposite situations, one can select a normalized block basis {uj }∞ j =1 of the unit vector basis in M which is permutatively equivalent to its square but M [sj ] is not isomorphic to either 1 or M itself. Casazza and Kalton have shown in [9] that in this case {uj }∞ j =1 , is, up to permutation and equivalence, the unique unconditional basis of M [sj ] and, moreover, M [sj ] contains complemented subspaces with a non-unique unconditional basis. The fact that direct sums in the sense of c0 of spaces such as 2 or 1 , which do have a unique unconditional basis, have also the uniqueness property, led to the question (stated explicitly in [6]) whether, whenever a space X has unique unconditional basis, up to permutation, then (X ⊕ X ⊕ · · · ⊕ X ⊕ · · ·)0 also has a unique unconditional basis. It turns out that the answer to this question is negative: Casazza and Kalton [9] have proved that direct sums of T or T (2) in the c0 -sense do not have a unique unconditional basis, up to permutation, in spite of the fact mentioned above that both T and T (2) have this property. 4. Uniqueness in ﬁnite-dimensional spaces Since any two bases of a ﬁnite-dimensional space are always equivalent the question of uniqueness of the basis makes no sense in the framework of a single ﬁnite-dimensional Banach space but rather in that of families of such spaces. As in the case of inﬁnitedimensional spaces, the most interesting results are obtained for families of spaces with an unconditional or symmetric basis. D EFINITION 4.1. Let F be a family of ﬁnite-dimensional spaces each of which has a normalized 1-unconditional basis. We say that the members of F have a unique unconditional basis, up to equivalence (and permutation), if there exists a function ψ : [1, ∞) → [1, ∞) such that, whenever a space E ∈ F has another normalized unconditional basis {ej }nn=1 whose unconditional constant is K, then {ej }nj=1 is ψ(K)-equivalent to (a permutation of) the given 1-unconditional basis of E. By replacing in the above deﬁnition the word “unconditional” with “symmetric”, one deﬁnes the notion of uniqueness of the symmetric basis for the elements of F . Typical families studied in this section are Fp = np ; n = 1, 2, . . . ; 1 p ∞. The same argument, involving the parallelogram identity, which was used in the case of 2 , shows that the members of F2 have a unique unconditional basis, up to equivalence. Furthermore, by using the same argument involving 1-summing and 2-summing operators, as in the case of 1 and c0 , one can show that also F1 and F∞ have the same uniqueness property as F2 . P ROPOSITION 4.2. The members of the families F1 , F2 and F∞ have a unique unconditional basis, up to equivalence.

1652

L. Tzafriri

The analogy with the inﬁnite-dimensional case might lead one to believe that these three families above are the only ones having, up to equivalence, a unique unconditional basis. It turns out that this fact is false. In order to produce examples of other families whose member have a unique unconditional basis, up to equivalence, one needs a result of Dor and Starbird [14] asserting that, for p > 2, a normalized unconditional basis of p is either equivalent to the unit vector basis of p or it admits, for each n, a block basis which is 2-equivalent to the unit vector basis of n2 . By using this fact together with the known assertion that Hilbert space is uniformly complemented in Lp -spaces, for p > 2 (cf. [48] or [41]), it is deduced in [27, p. 48] that: P ROPOSITION 4.3. There exist constants C < ∞ and M < ∞ such that every normalized K-unconditional basis of p ; p > 1, with √ # K C max p, p/(p − 1) , for some constant C < ∞, is M-equivalent to the unit vector basis of p . We omit the details of the proof but just mention that the bound for√K, appearing in the right-hand side of the above inequality, is actually equal to γp (2 )/ 2, where γp (2 ) denotes as usual the factorization constant of 2 through Lp -spaces. An immediate consequence of Proposition 4.3 is the fact that, for each sequence pn → ∞, the members of the family F = npn ; n = 1, 2, . . . have a unique unconditional basis, up to equivalence. We pass now to some questions concerning the uniqueness of the symmetric basis. We begin with subspaces of Lp . Among the ﬁnite-dimensional subspaces of Lp with a symmetric basis one can ﬁnd the families Fp and F2 . Other interesting subspaces of Lp are (p) the “diagonals” of np ⊕ n2 generated by vectors of the form {ej + wj ej(2)}nj=1 , where (p)

{ej }nj=1 and {ej(2)}nj=1 denote the unit vectors in np , respectively n2 , and {wj }nj=1 is an arbitrary sequence of scalars. Spaces of this type were studies by Rosenthal [51] who coined for them the name Xp -space. If wj = w; j = 1, 2, . . . , n, then obviously we deal with symmetric Xp -space. It turns out that every symmetric basic sequence in Lp ; p > 2, is equivalent to a symmetric Xp -space. This characterization has been proved in the Memoir [27, p. 34]. T HEOREM 4.4. For every p > 2 and K 1, one can ﬁnd a constant D = D(p, K) < ∞ so that any normalized basic sequence {xj }nj=1 in Lp (0, 1), whose symmetry constant is (p)

K, is D-equivalent to the symmetric Xp -space generated by {ej + wej(2)}nj=1 , where n √ w= xj / n. j =1

Uniqueness of structure in Banach spaces

1653

P ROOF. In the ﬁrst step of the proof, the symmetric basic sequence in Lp is replaced by a sequence of so-called “symmetrically exchangeable” random variables, i.e., a sequence of functions in Lp whose joint distribution in Rn remains invariant under permutation and change of sign. In order to describe this construction, let {xj }nj=1 be a normalized K-symmetric basic sequence in Lp (0, 1); p > 2, and Hn the family of all distinct n!2n pairs {π, (εj )nj=1 }, where π is a permutation of {1, 2, . . . , n} and εj = ±1, for all 1 j n. The elements of Hn can be put in a one-to-one correspondence with unit subintervals of [0, n!2n ] of the form [k, k + 1], with k being an integer. If a pair {π, (εj )nj=1 } is in correspondence with the subinterval I of [0, n!2n ] then we deﬁne fj (t) = εj xπ(j )(t);

t ∈ I ; 1 j n.

In order to make the functions {fj }nj=1 , which are deﬁned on the interval [0, n!2n ], into an exchangeable sequence we compress the interval [0, n!2n ] into [0, 1] in the obvious linear way thus obtaining a new sequence {gj }nj=1 which is symmetrically exchangeable and K-equivalent to the original sequence {xj }nj=1 . It is easily seen that there is no loss of generality in assuming that each point of [0, 1] belongs to the support of at least one of the functions {xj }nj=1 . Then, by a corresponding change of density, one can make the expression ( nj=1 |xj |2 )1/2 into a constant C. This n however implies that also the square function ( j =1 |gj |2 )1/2 is a constant equal to C = ( nj=1 |xj |2 )1/2 . Since {gj }nj=1 is a 1-unconditional basic sequence in Lp (0, 1) and the space Lp (0, 1) is of cotype p, for p > 2, it follows that n n 1/p p aj gj |aj | , j =1

j =1

for any choice if {aj }nj=1 . On the other hand, since {gj }nj=1 is an orthogonal sequence in √ L2 (0, 1) with gj 2 = C/ n, for all 1 j n, we get that n n n n 1/2 1/2 C 2 2 2 aj gj aj gj = |aj | gj 2 =√ |aj | , n j =1

j =1

j =1

2

j =1

for all 1 j n. This proves one part of 4.3 since n n aj xj K −1 aj gj j =1

j =1

K

−1

max

n j =1

1/p |aj |

p

n 1/2 C 2 , , √ |aj | n j =1

1654

L. Tzafriri

for all {aj }nj=1 , and, by Khinchine’s inequality, n xj K j =1

n p 1/p εj xj dε j =1

p 1/p KBp C. = K εj xj dε

The opposite inequality is more difﬁcult and requires some ideas of Rosenthal [51] and a certain averaging procedure. We omit this argument which is described in detail in [27]. The signiﬁcance of Theorem 4.4 becomes √ clear only after we fully understand the meaning of the expression w = nj=1 xj / n. P ROPOSITION 4.5. For every K 1, there exists a constant M = M(K) such that, whenever {xj }nj=1 is a normalized K-symmetric basis in a ﬁnite-dimensional subspace X of L n2 ) is M-equivalent to the expression p (0, 1), then d(X, √ the Banach–Mazur distance √ n n j =1 xj / n if 1 p 2 and to n/ j =1 xj if p > 2. P ROOF. Suppose that p > 2. It follows from 4.4 that n n n 1/2 j =1 xj −1 2 aj xj D(K, p) √ |aj | , n j =1

j =1

for any choice of {aj }nj=1 . Furthermore, by using Khinchine’s inequality in Lp (0, 1) and the 2-convexity of Lp (0, 1), for p > 2, we also get that n aj xj K j =1

p 1/p n aj εj xj dε = K j =1

p 1/p n aj εj xj dε j =1

n 1/2 n 1/2 |aj |2 |xj |2 |αj |2 , KBp KBp j =1

j =1

again for any choice of {aj }nj=1 . Combining these two inequalities, we conclude that √

n . d X, n2 KBp D(K) n xj j =1

Next notice that, for every sequence {aj }nj=1 of scalars, we also get n n 1 max |aj | aj xj K xj max |aj |, 1j n 2K 1j n j =1

j =1

Uniqueness of structure in Banach spaces

1655

i.e., n

n 2 d X, ∞ 2K xj . j =1

Hence,

n

√ n d X, n2 d X, n∞ 2K 2 xj d X, n2 , j =1

i.e.,

√

−1 n n , d X, n2 2K 2 xj j =1

which completes the proof for p > 2. The case 1 p 2 is treated in a similar manner. Proposition 4.5 shows that, for a given ﬁnite-dimensional subspace X of Lp (0, 1); 1 p ∞, with a symmetric basis, the expression nj=1 xj is, up to a constant depending only on the symmetry constant of {xj }nj=1 , an invariant of the space X rather than that of the particular symmetric basis {xj }nj=1 . This fact together with Theorem 4.3 imply the following consequence from [27, p. 39]. C OROLLARY 4.6. For p > 2, let Gp denote the family of all ﬁnite-dimensional subspaces of Lp (0, 1) which have a 1-symmetric basis. Then the members of Gp have, for any ﬁxed p, a unique symmetric basis, up to equivalence. A trivial duality argument, together with 4.2, shows that, for 1 p ∞, each member of the family Fp has, up to equivalence, a unique symmetric basis. This means that, for every 1 p ∞, there is a function ψp (K) which corresponds to the family Fp by Deﬁnition 4.1. It turns out that these functions can be selected as not to depend on p and the following result from [27, p. 39] is true. T HEOREM 4.7. The members of the family symmetric basis.

1p∞ Fp

have, up to equivalence, a unique

We omit the proof which is given in detail in the Memoir [27]. We already see that the class of families whose members have, up to equivalence, a unique symmetric basis is quite large. In an attempt to discover the largest family of space with a unique symmetric basis, Schütt studied in [53] the family Dα of all ﬁnitedimensional Banach spaces X with a normalized 1-symmetric basis {xj }nj=1 which satisfy the condition

d X, n2 nα , and proved the following beautiful result which we quote here without giving its proof.

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T HEOREM 4.8. For any α > 0, each member of the family Dα has, up to equivalence, a unique symmetric basis. The abundance of families of ﬁnite-dimensional spaces with a unique symmetric basis leads naturally to the question, which was raised in the Memoir [27], whether this is not always the case. Theorem 4.8, mentioned above, shows that the construction of a counterexample is not an easy matter. Eventually, Gowers produced in [23] ingenious examples of ﬁnite-dimensional normed spaces with two (asymptotically) non-equivalent symmetric bases. The precise statement is as follows. T HEOREM 4.9. For each integer k, there exists a Banach space of dimension n = 2k with two 2-symmetric normalized symmetric bases {ei }ni=i and {ei }ni=1 whose constant of equivalence is at least exp(log log n/8 log log log n). We shall describe here only the main ideas of Gowers’ construction; the details can be found in the paper [23]. Fix n and let A : Rn → Rn be a linear map deﬁned by an orthogonal matrix which will be speciﬁed later. Let {ei }ni=i denote the unit vector basis of Rn and put ei = Aei , for all 1 i n. These will be the two bases under consideration. The next step is to deﬁne a norm · on Rn so that both {ei }ni=i and {ei }ni=1 become 2-symmetric bases. To this end, let Ω be the group of symmetries associated to the ﬁrst basis {ei }ni=1 , i.e., of the linear maps w of the form w

n

a i ei =

i=1

n

εi ai eπ(i) ,

i=1

where εi = ±1, for all 1 i n, and π is a permutation of the integers {1, 2, . . ., n}. To the second basis {ei }ni=1 , we associate the group Ω = {AwA−1 ; w ∈ Ω}. Then put X0 = {εi ei ; εi = ±1, 1 i n} and deﬁne sets {Xj }∞ j =1 by induction, as follows: if j 1 is odd then Xj = w x; x ∈ Xj −1 , w ∈ Ω , and if j 1 is even, then Xj = {wx; x ∈ Xj −1 , w ∈ Ω}. n Once the sets {Xj }∞ j =0 have been introduced, we deﬁne the norm of a vector x ∈ R by setting

x = max 2−j x, xj ; xj ∈ Xj , j = 0, 1, 2, . . . .

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The fact that · is indeed a norm on Rn is trivial since the above maximum is always attained on a ﬁnite set of indices depending of course on the vector x under consideration. In order to prove that {ei }ni=1 is a 2-symmetric basis, ﬁx a vector x = ni=1 ai ei ∈ Rn and assume that x = 2−j |x, xj |, for some integer j and some xj ∈ Xj . If j is odd then, for any w ∈ Ω, wxj ∈ Xj +1 so that wx 2−(j +1)wx, wxj = 2−(j +1) x, xj = 2−1 x. On the other hand, if j is even then xj = wx ˜ j −1 , for some w˜ ∈ Ω and xj −1 ∈ Xj −1 . Hence, ˜ j −1 ∈ Xj so that for any w ∈ Ω, wxj = (ww)x wx 2−j wx, wxj = 2−j x, xj = x, which proves that {ei }ni=1 is indeed 2-symmetric. The proof of the fact that {ei }n2=1 is also 2-symmetric is done in exactly the same way. The above construction is independent of the particular choice of the orthogonal matrix A. The idea now is to select such a matrix A so that e1 = Ae1 is as small as possible and it turns out that one can construct an orthogonal matrix A for which the corresponding vector e1 satisﬁes e exp(− log log n/8 log log log n). 1 This will sufﬁce since it is not too difﬁcult to show that {ei /ei }ni=1 and {ei /ei }nu=1 are at best e1 -equivalent; this fact follows by comparing the expectations e11 E ni=1 gi ei 1 and e1 E ni=1 gi ei , where {gi }ni=1 is a sequence of independent identically distributed 1 random variables. The construction of the orthogonal matrix A so that e1 = Ae1 is “small” is done by induction on its size. For k = 0 and therefore n = 20 = 1, we let A 0 = (1) while, for k > 0, A k is deﬁned by A k

=

A k−1 Ik−1

Ik−1 −A k−1

,

where Ik−1 denotes the (2k−1 × 2k−1 )-identity matrix. Once the (2k × 2k )-matrix A k is deﬁned, we put Ak = k −1/2 A k . It is easily seen that the matrices {Ak }∞ k=0 are not only orthogonal but also symmetric. The most difﬁcult part of the proof is to show that, for n = 2k , the matrix A = Ak has the property that e1 exp(− log log n/8 log log log n). This argument is quite technical and it relies on a lemma of Harper [24], Bernstein [3], Hart [25] and Lindsey [40] which gives an estimate from below for the number of edges joining vertices of the k-cube of a given cardinality r. We omit these details which, as we mentioned before, can be found in [23]. We conclude this section with some remarks on two notions of uniqueness which are in the spirit of the so-called “proportional” theory of ﬁnite-dimensional spaces. The main deﬁnition introduced in [11] is the following.

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D EFINITION 4.10. Let F be a family of ﬁnite-dimensional spaces each of which has a normalized 1-unconditional basis. We say that the members of F have an almost (somewhat) unique unconditional basis provided there exists a function ϕ(K, λ), deﬁned for all K 1 and 0 < λ < 1, such that, whenever X ∈ F with the given normalized 1-unconditional basis {xi }ni=1 has also another normalized K-unconditional basis {yi }ni=1 then, for any (some) 0 < λ < 1, there exist a subset σ ⊂ {1, 2, . . . , n} and a one-to-one function π : σ → {1, 2, . . . , n} so that {xi }i∈σ is ϕ(K, λ)-equivalent to {yπ(i)}i∈σ . These notions are obviously an extension of the notion of unique unconditional basis, up to equivalence and permutation, introduced above. A thorough study of almost and somewhat uniqueness of unconditional basis is made in the paper [11]. We quote here, without proof, the following result from [11], which is clearly a generalization of Theorem 4.8. T HEOREM 4.11. For any α > 0, each member of the family Dα has an almost unique unconditional basis.

5. Uniqueness of rearrangement invariant structures While in the preceding sections we focused on the uniqueness question only in the setting of sequence spaces, in the present one we pass to a study of similar problems in the framework of rearrangement invariant (r.i.) function spaces on a non-atomic measure space. The main requirement imposed on an r.i. function space X on a ﬁnite or σ -ﬁnite nonatomic measure space (Ω, Σ, μ) is that, for any automorphism τ of Ω and every measurable function f ∈ X, the function f (τ −1 ) also belongs to X and has the same norm as f . If the measure space (Ω, Σ, μ) is assumed to be non-atomic and separable (i.e., Σ endowed with the usual metric ρ(τ, η) = μ(σ Δη); σ, η ∈ Σ, is a separable metric space) then it is well known that (Ω, Σ, μ) is isomorphic to a ﬁnite or inﬁnite interval endowed with the usual Lebesgue measure. Hence, in principle, we can restrict our attention to the canonical cases Ω = [0, 1] and Ω = [0, ∞), both endowed with the Lebesgue measure. In the Basic Concepts article such spaces are called symmetric lattices. In the case when a function space X is invariant with respect to the automorphisms of Ω then the same is true for X , the subspace of the dual X∗ of X which consists of “integrals”, i.e., of functionals of the form xg∗ (t) =

fg dμ,

f ∈ X.

Ω

In most of the interesting cases that appear in analysis, X is a norming subspace of X∗ . This occurs if and only if 0 fn (w) ↑ f (w) a.e. on Ω implies that limn→∞ fn = f . The proof of this simple assertion can be found in [38, 1.b.18]. For convenience, we shall assume here that this is always the case. The formal deﬁnition of the notion of r.i. function space, which will be used in the sequel, is the following.

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D EFINITION 5.1. An r.i. function space X on the interval Ω = [0, 1] or on the interval Ω = [0, ∞) is a Banach space of classes of equivalence of measurable functions on Ω such that: (i) For any automorphism τ of Ω, a function f ∈ X if and only if f (τ −1 ) ∈ X, and if this is the case then f (τ −1 ) = f . (ii) X is a norming subspace of the dual X∗ of X and thus X is order isometric to a subspace of X

. As a subspace of X

, the space X is either minimal (i.e., X is the closure of the simple integrable functions on Ω) or it is maximal (i.e., X = X

). (iii) If Ω = [0, 1] then L∞ (0, 1) ⊂ X ⊂ L1 (0, 1), with the inclusion maps being of norm one, i.e., f 1 f X f ∞ , for all f ∈ L∞ (0, 1). (iii ) If Ω = [0, ∞) then L1 (0, ∞) ∩ L∞ (0, ∞) ⊂ X ⊂ L1 (0, ∞) + L∞ (0, ∞), again with the inclusion maps being of norm one. Recall that the norm of a function f in the space L1 (0, ∞) ∩ L∞ (0, ∞) is deﬁned as

f = max f 1 , f ∞ . The space L1 (0, ∞) + L∞ (0, ∞) is often used in interpolation theory and the norm of a function f in this space is usually deﬁned by the formula f = inf g1 + h∞ ; f = g + h , the inﬁmum being taken over all decompositions f = g + h, with g ∈ L1 (0, ∞) and h ∈ L∞ (0, ∞). It is easily veriﬁed that if Y = L1 (0, ∞) + L∞ (0, ∞) then Y = L1 (0, ∞) ∩ L∞ (0, ∞). The norm in the space Y can be alternatively deﬁned with the aid of the notion of decreasing rearrangement of a function f on either Ω = [0, 1] or on Ω = [0, ∞). The decreasing rearrangement f ∗ of a function f 0 is deﬁned as the right continuous inverse of the distribution function df of f , which is deﬁned by df (t) = μ w ∈ Ω; f (w) > t . In other words, f ∗ (x) = inf t > 0; df (t) x ;

0 x < μ(Ω).

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If f is not 0 then f ∗ is deﬁned as the decreasing rearrangement of the absolute value |f | of f . It turns out that the norm of a function f ∈ Y = L1 (0, ∞) + L∞ (0, ∞) is equal to

1

f ∗ (x) dx.

0

Indeed, for any decomposition f = g + h, as above, and any subset σ ⊂ [0, ∞), we have that σ

f (t) dt g1 + h∞ μ(σ ).

Hence,

1

f ∗ (x) dx = sup

0

f (t) dt; μ(σ ) = 1 f .

σ

Conversely, ﬁx f ∈ Y and put λ = f ∗ − f ∗ χ[0,1] ∞ . Then

f = f ∗ f ∗ − min λ, f ∗ 1 + min(λ, f )∞ 1

= f ∗ − λ χ[0,1] 1 + λ = f ∗ (x) dx. 0

As in the case of spaces with a symmetric basis, the question of uniqueness (this time of the r.i. structure) has been studied most extensively in Lp -spaces. A result on r.i. function spaces on [0, 1], which is quite useful in the study of uniqueness in Lp -spaces, was proved in the Memoir [27, p. 41]. T HEOREM 5.2. An r.i. function space X on [0, 1], which is isomorphic to a subspace of Lp (0, 1); p > 2, coincides with either Lp (0, 1) or L2 (0, 1), up to an equivalent renorming. P ROOF. Let X be an r.i. function space on [0, 1] and let T be an isomorphism from X into Lp (0, 1); p > 2. For every n and 1 i 2n , denote by ϕn,i the characteristic funcn tion of the interval [(i − 1)/2n , i/2n ). Since, for every n, the sequence {T ϕn,i }2i=1 is a K-symmetric basic sequence in Lp (0, 1) with K T · T −1 , one can use Theorem 4.4 from the previous section and conclude the existence of a constant D < ∞, depending only n on p and on T , so that, for every choice of scalars {ai }2i=1 , we have 2n 2n 2n 1/p 1/2 ϕn,i D p 2 ai |ai | ,w |ai | , ∼ max ϕn,i X i=1

X

i=1

i=1

Uniqueness of structure in Banach spaces

1661

where w=

ϕn,i i=1 ϕn,i X X

2n

√ 2n

=

1

√ . ϕn,1 X 2n

Hence, for any simple function f over the ﬁeld generated by the intervals [(i − 1)/2n , i/2n ); 1 i 2n , we have that D f X ∼ max ϕn,1 X 2n/p f p , f 2 . Taking f ≡ 1 we get that the sequence {ϕn,1 X 2n/p }∞ n=1 is bounded by D and thus, with α = lim inf ϕn,1 X 2n/p , n→∞

one concludes that, D f X ∼ max αf p , f 2 , for any simple function f over the dyadic intervals in [0, 1]. If α = 0 then obviously X = L2 (0, 1) and if α > 0 then, since p > 2, X = Lp (0, 1), both equalities up to an equivalent renorming. Theorem 5.2 has been generalized in [26] to a quite large class of pairs X and Y , where Y is an r.i. function space on [0, ∞) and X is a non-atomic Banach lattice isomorphic to a subspace of Y . Under different conditions on Y , stated mostly in terms of p-convexity and q-concavity-notions which are described in the Basic Concepts article – it is shown in [26] that X is isomorphic to a sublattice of Y . For instance, this is the case when Y is p-convex and q-concave, for some p > 2 and q < ∞, and X is r-convex, for some r > 2. The same type of assumptions imply that if Y is an r.i. function space on [0, 1] then X contains a non-trivial band lattice isomorphic to a sublattice of Y . The paper [26] contains also some results on complemented spaces. For instance, if Y is a separable r.i. function space on either [0, 1] or [0, ∞), which contains no 2 as a complemented sublattice, and X is a p-convex Banach lattice, for some p > 2, which is isomorphic to a complemented subspace of Y then X is even lattice-isomorphic to a complemented sublattice of Y . In exactly the same manner as in the proof of Theorem 5.2, one can prove the following version for [0, ∞) (cf. [27, p. 43]). T HEOREM 5.3. An r.i. function space X on [0, ∞), which is isomorphic to a subspace of Lp (0, ∞); p > 2, coincides with one of the spaces Lp (0, ∞), L2 (0, ∞) or L2 (0, ∞) ∩ Lp (0, ∞), up to an equivalent renorming. The norm of a function f ∈ L2 (0, ∞) ∩ Lp (0, ∞) is of course deﬁned by f = max(f 2 , f p ).

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Theorem 5.2 above implies the uniqueness of the r.i. structure on [0, 1] of Lp (0, 1), for p > 2, and thus, by duality, also of 1 < p < 2. The uniqueness of the r.i. structure of L2 (0, 1) is quite trivial, in view of 1.1 above. The uniqueness of the r.i. structure on [0, 1] of L∞ (0, 1) can be easily reduced to that of L1 (0, 1). In order to prove the uniqueness of the r.i. structure on [0, 1] of the space L1 (0, 1), we need the fact that if X is an r.i. function space on [0, 1], which is isomorphic to L1 (0, 1), then, for every n, the sequence of characteristic functions ϕn,i = χ[(i−1)/2n ,i/2n ) ; 1 i 2n , forms a 1-unconditional basic sequence in X whose span is 1-complemented in X, by the conditional expectation operator relative to the ﬁeld generated by the intervals [(i − 1)/2n , i/2n ); 1 i 2n (i.e., by the operator deﬁned by n i/2n En f = 2n 2i=1 ( (i−1)/2n f dx)ϕn,i ). Then, the same argument, as the one used in Secn

tion 1 to prove that 1 has a unique unconditional basis, shows that {ϕn,i /ϕn,i X }2i=1 is n equivalent to the unit vector basis of 21 , with a constant of equivalence independent of n. We summarize the above observations in the following theorem which, again, has been proved in [27]. T HEOREM 5.4. The space Lp (0, 1) has a unique structure as an r.i. function space on [0, 1], for any value of 1 p ∞. R EMARK . In the case p = 1, Kalton [29] proved a much stronger result: an r.i. function space on [0, 1] which contains a copy of L1 (0, 1), already coincides with L1 (0, 1), up to an equivalent norm, provided it does not contain uniformly isomorphic copies of n∞ . Contrary to an initial belief, the space Lp (0, ∞) does not have a unique structure as an r.i. function space on [0, ∞), unless p = 1, 2 or ∞. In fact, the following result can be easily deduced from Theorem 5.3. T HEOREM 5.5. The space Lp (0, ∞); 1 2,

Lp (0, ∞)

and L2 (0, ∞) + Lp (0, ∞),

if 1 2, the spaces Lp (0, ∞) and Z = L2 (0, ∞)∩Lp (0, ∞) are isomorphic. To this end, notice that the restriction Z|[0,1] is isomorphic to Lp (0, 1) and thus Z contains a complemented subspace isomorphic to n Lp (0, ∞). Conversely, for any n and m, the sequence {χ[(i−1)/2n,i/2n ) }2i=1 spans in Z a symmetric Xp -space, and thus its span embeds complementably in Lp (0, ∞), in a uniform manner. Hence, by a compactness argument using, for instance, ultraproducts, one can show that Z is isomorphic to a complemented subspace of Lp (0, ∞). Then the decomposition method shows that Z ≈ Lp (0, ∞). Many other classes of spaces which admit a unique representation as an r.i. function space on [0, 1] were exhibited in the Memoirs [27] and [29]. We shall state some of these results without proof.

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T HEOREM 5.6. An r.i. function space X on [0, 1], which is q-concave for some q < 2, has unique structure as an r.i. function space on [0, 1]. The other result that we want to state below involves the notion of the Haar basis. Recall k that the Haar system {χn }∞ n=1 on [0, 1] is deﬁned by χ1 (t) ≡ 1 and, for = 1, 2, . . . , 2 and k = 0, 1, . . . , by ⎧

⎪ if t ∈ (2 − 2)/2k+1, (2 − 1)/2k+1 , ⎨1

χ2k + (t) = −1 if t ∈ (2 − 1) 2k+1 , 2/2k+1 , ⎪ ⎩ 0 otherwise. It is an immediate consequence of basic interpolation theorems that the Haar system forms a (monotone) basis in any separable r.i. function space on [0, 1]. We now present a result on uniqueness which was originally proved in [27] under some additional assumptions and whose deﬁnitive form, stated below, is due to Kalton [29]. T HEOREM 5.7. Let X be a separable r.i. function space on [0, 1] such that the Haar basis of X is not equivalent to a sequence of mutually disjoint functions in X whose linear span is complemented in X. Then X has unique structure as an r.i. function space on [0, 1]. For using interpolation between two Lp -spaces other than L1 and L∞ it is useful to consider the so-called Boyd indices. In order to deﬁne these indices for an r.i. function space X on [0, 1], we need the dilation operator Ds restricted to [0, 1], which is deﬁned for 0 < s < ∞ and f on [0, 1], by f (t/s), 0 t min(1, s), (Ds f )(t) = 0, s < t 1 (in the case s < 1). The operator Ds is a dilation by the ratio s : 1 in the positive direction of the t-axis followed by restriction to [0, 1]. It is easily veriﬁed that, for every choice of r and s, and every r.i. function space X on [0, 1], Drs X Dr X Ds X , which eventually ensures that the following limits exist: pX = lim

log s log s = sup log Ds X s>1 log Ds X

qX = lim

log s log s = sup . log Ds X 0<s<1 log Ds X

s→∞

and s→0+

The importance of these indices stems from the fact, proved by Boyd [7], that, whenever 1 p < pX and qX < q ∞ for some r.i. function space X on [0, 1], then every linear operator T , which is bounded on both Lp (0, 1) and Lq (0, 1), is also bounded on X.

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A result due to Paley [46], proved in detail, e.g., in [38, p. 155] (cf. also the article of Burkholder [8] and the article of Figiel and Wojtaszczyk [21]), asserts that the Haar system forms an unconditional basis in every Lp (0, 1)-space, if 1 < p < ∞. It follows immediately that the Haar system is also unconditional in every separable r.i. function space X on [0, 1], whose Boyd indices are non-trivial, i.e., 1 < pX and qX < ∞. In the case that one of the Boyd indices is trivial, Kalton [29] proved the following uniqueness result. T HEOREM 5.8. Let X be a separable r.i. function space on [0, 1] such that either pX = 1 or qX = ∞. Then X has unique structure as an r.i. function space on [0, 1]. Finally, we mention the following result which was proved in [27, p. 169], in the reﬂexive case, and follows from 5.8 in the non-reﬂexive one. T HEOREM 5.9. A separable Orlicz function space on [0, 1] has unique structure as an r.i. function space on [0, 1]. It turns out that there are also many r.i. function spaces on [0, 1] which fail to have a unique structure as an r.i. function space on [0, 1]. The construction of such examples has something in common with that of a space with “many” mutually non-equivalent symmetric bases, presented in Section 2. We start, as in Section 2, with the space U1 deﬁned there, whose normalized unconditional basis {un }∞ n=1 is universal for all normalized unconditional basic sequences in the sense that each such sequence is equivalent to a subsequence of {un }∞ n=1 . We could proceed as in Section 2 and interpolate U1 between two distinct Lp -spaces but the r.i. space constructed in this way will not be isomorphic to U1 . In order to avoid this problem, we shall ﬁrst p-convexify and 2-concavify the space U1 , for some 1 < p < 2, thus obtaining a new space whose unconditional basis is universal for all p-convex and 2-concave normalized unconditional basic sequences. To this end, with 1 < p < 2 and q = p∗ = p/(p − 1), we introduce a new norm ||| · ||| on U1 by setting for u ∈ U1 , |||u||| = sup

n (k) q v k=1

1/q

; u=

n (k) q v

1/q ,

k=1

where the supremum is taken over all decompositions of u, as above. It is easily checked that {un }∞ n=1 is in (U1 , ||| · |||) q-concave and universal for all q-concave normalized unconditional basic sequences. Hence, the biorthogonal functionals form in the dual V of (U1 , ||| · |||) a p-convex unconditional basis which is universal for all p-convex normalized unconditional basic sequences. Repeating this procedure, this time with 2 instead of q, (p) we eventually get a space Wp , with a normalized unconditional basis {wn }∞ n=1 , which is p-convex and 2-concave and, moreover, each p-convex and 2-concave normalized uncon(p) ditional sequence is equivalent to a subsequence of {wn }∞ n=1 . Next, with p r < 2, we interpolate Wp between Lr (0, 1) and L2 (0, 1) thus obtaining a p-convex and 2-concave r.i. function space Yp,r on [0, 1]. The Boyd indices of this space

Uniqueness of structure in Banach spaces

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are non-trivial and therefore the Haar system in Yp,r is unconditional and this unconditional structure is p-convex and 2-concave. Hence, Yp,r is isomorphic to a complemented subspace of Wp . The interpolation procedure discussed above is done by choosing a sequence {nk }∞ k=1 and by deﬁning f k = inf nk gr + n−1 k h2 ; f = g + h and f Yp,r

∞ (p) = f k wk =1

,

Wp

for any f ∈ Lr (0, 1). One can easily show that if {nk }∞ k=1 tends “fast” to ∞ then the corresponding space Yp,r contains a complemented subspace isomorphic to Wp . Then, by using the decomposition method, we conclude that Yp,r ≈ Wp , for any choice of p < r < 2. Finally, it is not very hard to verify that, for different values of p < r < 2, we obtain mutually non-equivalent r.i. function spaces Yp,r on [0, 1], which are all isomorphic to Wp . While the class of r.i. function spaces on a non-atomic measure space can be considered as a continuous variant of the class of spaces with a symmetric basis, non-atomic Banach lattices are the continuous analogue of spaces with an unconditional basis. The ﬁrst result on uniqueness of structure for non-atomic Banach lattices was proved by Abramovich and Wojtaszczyk [1] who showed that L1 and L2 have unique structure as non-atomic Banach lattices. It turns out that, contrary to the case of spaces with unique unconditional basis, there exist other spaces with unique non-atomic structure. For example, the Orlicz space LM (0, 1), with M(t) ∼ t (log t)α , for large t and 0 < α < 1/2, is such an example (cf. [29]). In particular, this is an example of an r.i. function space on [0, 1] which is not isomorphic to an r.i. function space on [0, ∞].

6. Uniqueness of bases in non-Banach spaces As we have seen in the previous sections, it is quite rare for an unconditional basis in a Banach space to be unique, even up to a permutation. It turns out that in spaces other than Banach spaces one can ﬁnd relatively many unconditional bases having the uniqueness property and, moreover, some of these uniqueness results have quite interesting applications. One class of spaces with a rich structure is that of so-called quasi-Banach spaces. A quasi-Banach space is a vector space X endowed with a quasi-norm · which satisﬁes the usual axioms of the norm except that the triangle inequality is replaced by the inequality

x + y C x + y ,

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for all x, y ∈ X and some ﬁxed constant C 1. A survey of the theory of these spaces is [31] or the article of Kalton in this Handbook. In the class of quasi-Banach spaces the uniqueness of unconditional bases, up to equivalence, is quite a common occurrence. For instance, it was shown by Kalton [28] that the spaces p have, up to equivalence, a unique unconditional basis, for every value of 0 < p < 1. As we have seen in the previous sections, this result is false for 1 < p = 2, even up to a permutation. In the same paper [28], the author also exhibits a larger class of non-locally convex Orlicz sequence spaces with a unique unconditional basis, up to equivalence. One can consider the problem of uniqueness for the so-called non-locally convex Lorentz sequence spaces d(w, p), where p > 0 and w = (wn )∞ n=1 is a non-summable monotone decreasing sequence of positive numbers and the quasi-norm xw,p of an element x ∈ d(w, p) is deﬁned by xw,p = sup π

∞

1/p |xπ(n)| wn p

< ∞.

n=1

In this deﬁnition, the supremum is taken over all permutation π of the integers. The uniqueness of the basis in d(w, p) was studied in [45] and further in [30]. One result in this direction, which is proved in [30] and provides the solution to an open question raised in [45], asserts that the space d(w, p) has, up to equivalence, a unique unconditional basis, whenever 0 < p < 1 and the sequence of weights w satisﬁes the condition 1 (w1 + · · · + wn )1/p = ∞. n→∞ n lim

Another remarkable result proved in [30] states that also the spaces p (q ) have a unique unconditional basis, up to equivalence and permutation, as long as 0 < p, q < 1. The same is true for the spaces c0 (p ), as long as 0 < p < 1 (cf. [33]). Perhaps, the most interesting class of quasi-Banach spaces is that of the Hardy spaces Hp (T), 0 < p < 1, where T denotes the circle, and their m-dimensional version Hp (Tm ), where Tm denotes the m-dimensional torus. The fact that these quasi-Banach spaces have an unconditional basis is not obvious. For m = 1, this follows from the fact proved in [55], that Hp (T) is isomorphic to the dyadic Hardy space Hp (0, 1), for all 0 < p < 1. The usual Haar basis {hn }∞ p (0, 1), 0 < p < ∞, since the dyadic n=1 forms an unconditional basis in H Hp (0, 1) consists of all distribution of the form f = ∞ n=1 an hn for which the expression f p =

∞ 1 0

p/2 1/p |an hn |

2

< ∞.

n=1

For higher dimensions, one veriﬁes that the m-times tensor product of the Haar basis is an unconditional basis in the dyadic Hardy space Hp (0, 1)m , for every value of 0 < p < 1, and that this space is isomorphic to the usual space Hp (Tn ) (cf. [55]). A remarkable result from [56] asserts that every normalized unconditional basis in Hp (T); 0 < p < 1, is equivalent to the Haar basis in some order, i.e., the space Hp (T) has, up to equivalence and permutation, a unique unconditional basis.

Uniqueness of structure in Banach spaces

1667

A nice application of the uniqueness result in non-locally convex Hardy spaces is the fact (proved ﬁrst by Kalton, Leranoz and Wojtaszczyk [30]) that, for 0 < p < 1, the spaces Hp (Tm ), m = 1, 2, . . . , are mutually non-isomorphic. If, for example, Hp (Tm ) were isomorphic to Hp (T), for some 0 < p < 1 and some integer m, then the corresponding normalized unconditional bases of these two spaces would be equivalent, up to a permutation. That this is false is checked directly. One should point out that the result asserting that the spaces H1 (Tm ), m = 1, 2, . . . , are mutually non-isomorphic was proved before by Bourgain ([4,5]). Uniqueness of bases can be also studied in the more general context of locally convex spaces (l.c.s.), i.e., vector spaces which admit a fundamental system of convex neighborhoods of 0. The topology of such a space can be deﬁned with the aid of a family of semi-norms. Of particular interest are the Fréchet spaces (recall that these are l.c.s. whose topology is both metrizable and complete). The main difﬁculty faced in studying uniqueness in l.c.s. is the fact that the notion of “normalized” basis does not make any sense in these spaces. The only Fréchet spaces which have a unique unconditional basis are ω, the space of all scalar sequences, and its dual ω∗ , the space of all scalar sequences which are eventually zero. The uniqueness assertion for ω and ω∗ is due to Köthe and Töplitz [32]. This is the result that in some sense initiated the research on uniqueness of bases. That ω and ω∗ are the only spaces with this uniqueness property is due to Dragilev (cf. [17]). Less restrictive is the notion of quasi-equivalence. Two unconditional bases {xn }∞ n=1 and {yn }∞ n=1 in l.c.s. X, respectively Y , are said to be quasi-equivalent if there exists a ∞ permutation π of the integers and a sequence λ = (λn )∞ n=1 of scalars such that {λn xπ(n) }n=1 and {yn }∞ are equivalent, i.e., there exists an isomorphism T from X onto Y so that n=1 yn = T (λn xπ(n)), for all n. A remarkable result, also due to Dragilev [15], asserts that the Fréchet space A(D) of all the functions which are analytic on the open disk D (endowed with the topology of uniform convergence on each compact subset of the disk) has the property that all its unconditional bases are quasi-equivalent. This fact is also true in the class of all nuclear power series space (cf. [43]). A systematic study of quasi-equivalence in l.c.s. has been carried out by Dragilev [16], Dubinsky [18], Mityagin [44] and recently by Zahariuta [57].

References [1] Y.A. Abramovich and P. Wojtaszczyk, On the uniqueness of order in the spaces p and Lp [0, 1], Mat. Zametki 18 (1975), 313–325 (Russian). [2] K.I. Babenko, On conjugate functions, Dokl. Akad. Nauk SSSR 62 (1948), 157–160 (Russian). [3] A.J. Bernstein, Maximally connected arrays on the n-cube, Discrete Math. 15 (1967), 1485–1489. [4] J. Bourgain, The non-isomorphism of H 1 -spaces in one and several variables, J. Funct. Anal. 46 (1982), 45–47. [5] J. Bourgain, The non-isomorphism of H 1 -spaces in a different number of variables, Bull. Soc. Math. Belg. Sér. B 35 (1983), 127–136. [6] J. Bourgain, P.G. Casazza, J. Lindenstrauss and L. Tzafriri, Banach Spaces with a Unique Unconditional Basis, up to Permutation, Mem. Amer. Math. Soc. 322 (1985).

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[7] D.B. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245–1254. [8] D.L. Burkholder, Martingales and singular integrals in Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 233–269. [9] P.G. Casazza and N.J. Kalton, Uniqueness of unconditional basis in Banach spaces, Israel J. Math. 103 (1998), 141–176. [10] P.G. Casazza and N.J. Kalton, Uniqueness of unconditional bases in c0 products, Preprint. [11] P.G. Casazza, N.J. Kalton and L. Tzafriri, Uniqueness of unconditional and symmetric structions in ﬁnite dimensional spaces, Illinois J. Math. 34 (1990), 793–836. [12] P.G. Casazza and M.C. Lammers, Genus N Banach spaces, Preprint. [13] W.J. Davis, Embedding spaces with unconditional bases, Israel J. Math. 20 (1975), 189–191. [14] L.E. Dor and T. Starbird, Projections of Lp onto subspaces spanned by independent random variables, Compositio Math. 39 (1979), 141–175. [15] M.M. Dragilev, Canonical forms of a basis in the space of analytic function spaces, Uspekhi Mat. Nauk 15 (1960), 181–188. [16] M.M. Dragilev, On regular bases in nuclear spaces, Mat. Sbornik 68 (1965), 153–173 (Russian). [17] M.M. Dragilev, Topological vector spaces with equivalent bases, Mat. Zametki 28 (1980), 947–951. [18] E. Dubinsky, The Structure of Nuclear Fréchet Spaces, Lecture Notes in Math. 720, Springer (1979). [19] I.S. Èdelšte˘ın and P. Wojtaszczyk, On projections and unconditional bases in direct sums of Banach spaces, Studia Math. 56 (1976), 263–276. [20] T. Figiel and W.B. Johnson, A uniformly convex Banach space which contains no p , Compositio Math. 29 (1974), 179–190. [21] T. Figiel and P. Wojtaszczyk, Special bases in functional spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 561–597. [22] J. Garnett, Bounded Analytic Functions, Academic Press, Orlando (1981). [23] W.T. Gowers, A ﬁnite-dimensional Banach space with two non-equivalent symmetric bases, Israel J. Math. 87 (1994), 143–151. [24] L.H. Harper, Optimal assignments to number of vertices, SIAM J. Appl. Math. 12 (1964), 131–135. [25] S. Hart, A note on the edges of the n-cube, Discrete Math. 14 (1976), 157–163. [26] F. Hernández and N.J. Kalton, Subspaces of rearrangement invariant spaces, Canad. J. Math. 48 (1996), 794–833. [27] W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric Structures in Banach Spaces, Mem. Amer. Math. Soc. 217 (1979). [28] N.J. Kalton, Orlicz sequence spaces without local convexity, Math. Proc. Cambridge Philos. Soc. 81 (1977), 253–278. [29] N.J. Kalton, Lattice Structures on Banach Spaces, Mem. Amer. Math. Soc. 493 (1993). [30] N.J. Kalton, C. Leranoz and P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces, Israel J. Math. 72 (1990), 299–311. [31] N.J. Kalton, N.T. Peck and J.W. Roberts, An F -space Sample, London Math. Soc. Lecture Notes Ser. 89, Cambridge Univ. Press (1984). [32] G. Köthe and O. Töplitz, Theorie der halbﬁniten unendlichen Matrizen, J. Reine Angew. Math. 165 (1931), 116–127. [33] C. Leranoz, Uniqueness of unconditional bases of c0 (p ), 0 < p < 1, Studia Math. 102 (1992), 193–207. [34] J. Lindenstrauss, A remark on symmetric bases, Israel J. Math. 13 (1972), 317–320. [35] J. Lindenstrauss and A. Pelczynski, Absolutely summing operators in Lp spaces and their applications, Studia Math. 29 (1968), 275–326. [36] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces III, Israel J. Math. 14 (1973), 368–389. [37] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer, Berlin (1977). [38] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer, Berlin (1979). [39] J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, J. Funct. Anal. 3 (1969), 115–125. [40] J.H. Lindsey, Assignments of numbers to vertices, Amer. Math. Monthly 71 (1964), 508–516. [41] B. Maurey, Théoremes de factorisation pour les operateures a valeurs dans un espace Lp , Asterisque 11 (1974).

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[42] C.A. McCarthy and J. Schwartz, On the norm of a ﬁnite Boolean algebra of projections and applications to theorems of Kreiss and Morton, Comm. Pure Appl. Math. 18 (1965), 191–201. [43] B.S. Mityagin, Nuclear Riesz scales, Dokl. Akad. Nauk SSSR 137 (1961), 519–522. [44] B.S. Mityagin, Approximative dimension and bases in nuclear spaces, Uspekhi Mat. Nauk 16 (1961), 63– 132 (Russian). [45] M. Nawrocki and A. Ortynski, The Mackey topology and complemented subspaces of Lorentz sequence spaces d(w, p) for 0 < p < 1, Trans. Amer. Math. Soc. 287 (1985), 713–722. [46] R.E. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241–264. [47] A. Pelczynski, Universal bases, Studia Math. 32 (1969), 247–268. [48] A. Pelczynski and H.P. Rosenthal, Localization techniques in Lp -spaces, Studia Math. 52 (1975), 263–289. [49] A. Pelczynski and I. Singer, On non-equivalent bases and conditional bases in Banach spaces, Studia Math. 25 (1964), 5–25. [50] C.J. Read, A Banach space with up to equivalence, precisely two symmetric bases, Israel J. Math. 40 (1981), 33–53. [51] H.P. Rosenthal, On the subspaces of Lp (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303. [52] G. Schechtman, On Pelczynski’s paper, “Universal bases”, Israel J. Math. 22 (1975), 181–184. [53] C. Schütt, On the uniqueness of symmetric bases in ﬁnite-dimensional Banach spaces, Israel J. Math. 40 (1981), 97–117. [54] B.S. Tsirelson, Not every Banach space contains p or c0 , Functional Anal. Appl. 8 (1974), 138–141 (translated from Russian). [55] P. Wojtaszczyk, Hp -spaces, p 1, and spline systems, Studia Math. 77 (1984), 289–320. [56] P. Wojtaszczyk, Uniqueness of unconditional bases in quasi-Banach spaces with applications to Hardy spaces II, Israel J. Math. 97 (1997), 253–280. [57] V.P. Zahariuta, On Isomorphic Classiﬁcation of F -spaces, Lecture Notes in Math. 1043, Springer (1983). [58] M. Zippin, On perfectly homogeneous bases in Banach spaces, Israel J. Math. 4 (1966), 265–272.

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CHAPTER 39

Spaces of Analytic Functions with Integral Norm P. Wojtaszczyk Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland E-mail: [email protected]

Contents 1. 2. 3. 4.

Notation . . . . . . . . . . . Bergman spaces . . . . . . . Hardy spaces . . . . . . . . Special operators . . . . . . 4.1. Coefﬁcient multipliers 4.2. Composition operators 5. Isomorphisms of H1 . . . . 6. Isomorphic structure of H1 . 7. Isometric questions . . . . . References . . . . . . . . . . .

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. 1673 . 1673 . 1677 . 1680 . 1680 . 1683 . 1684 . 1696 . 1699 . 1700

Abstract We discuss properties of Banach spaces of analytic functions with integral type norms, in particular Hardy spaces and Bergman spaces. We present results about the isomorphic structure of those spaces, subspaces and complemented subspaces, special operators like multipliers and composition operators. The basic results about the real variable theory of H1 spaces are discussed and the connections with complex theory are explained. Isometric questions are also presented.

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2 Edited by William B. Johnson and Joram Lindenstrauss © 2003 Elsevier Science B.V. All rights reserved 1671

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Spaces of analytic functions with integral norm

1673

1. Notation In addition to the standard notations of this Handbook as explained in [21] we will also use the following notations: D = z ∈ C: |z| < 1 , Bn = (z1 , z2 , . . . , zn ) ∈ C : n

n

|zj | < 1 , 2

j =1

Sn = (z1 , z2 , . . . , zn ) ∈ C : n

n

|zj | = 1 , 2

j =1

C+ = {z ∈ C: 'z > 0}. Obviously Dn ⊂ Cn will denote the n-fold Cartesian product of D and Tn ⊂ Cn will denote the n-fold Cartesian product of T. By ν we will denote the natural volume Lebesgue measure on any subset of Cn . On the sets D, Dn or Bn this measure is normalized so the measure of the whole set equals 1. The natural arc length probability measure of T will be denoted by λ and the corresponding product measure on Tn by λn . The natural rotation invariant probability measure on Sn will be denoted by σn . For any open subset D ⊂ Cn by H(D) we will mean the space of all holomorphic functions on D. 2. Bergman spaces Let D ⊂ Cn be an open set. For 0 0 be such that z + rBn ⊂ D. The following mean value formula for f ∈ H(D) easily follows from the Cauchy formula (or from expansion of f into power series) f (w) dν(w). (2) f (z) = r −2n z+rBn

From (2) follows immediately that for p 1

f (z) Cf p dist z, Cn \ D −2n/p .

(3)

From (3) we infer easily that a sequence Cauchy in the norm of Bp (D) is almost uniformly convergent in D, so we get

1674

P. Wojtaszczyk

T HEOREM 1. The space Bp (D) is a closed subspace of Lp (D, ν), in particular it is a Banach space. From (3) and Montel’s theorem we also infer that for each compact subset K ⊂ D the restriction f → f |K is a compact operator from Bp (D) into C(K). This implies C OROLLARY 2. The space Bp (D) is isomorphic to a subspace of p . Tosee it simply take a sequence ∅ = K0 ⊂ K1 ⊂ K2 ⊂ · · · of compact subsets of D such that ∞ j =1 Kj = D, and consider the isometric embedding Bp (D) −→

∞

Lp (Kj +1 \ Kj )

j =0

p

deﬁned as f → (f | Kj +1 \ Kj )∞ j =0 . Since each restriction is a compact operator we can ﬁnd a ﬁnite partition of Kj +1 \ Kj such that for the averaging projection Pj (with respect to this partition) we have for f ∈ Bp (D) 1/p

Kj+1 \Kj

|f − Pj f |

p

10−1 f p .

This gives that f → (Pj (f | Kj +1 \ Kj ))∞ j =0 ∈ ( ding into a space isometric to p .

∞

j =0 Im Pj )p

is an isomorphic embed-

R EMARK 3. An obvious modiﬁcation of the above proof gives that Bp (D) is almost isometric to a subspace of p . It follows easily from [13] that for p = 2k with k = 2, 3, . . . it is not isometric to a subspace of p . For p = 2k it is unknown. From (3) we see that for z ∈ D the value at z is a continuous linear functional on Bp (D). Since B2 (D) is a Hilbert space, from the Riesz theorem we get that for each z ∈ D there exists a function Kz (w) ∈ B2 (D) such that the orthogonal projection P from L2 (D, ν) onto B2 (D) is given by the formula f (w)Kz (w) dν(w). (4) (Pf )(z) = D

This projection is called the Bergman projection and the function of two variables KD (z, w) = Kz (w) is called the Bergman kernel. The way to construct the Bergman kernel of a given domain is to take an orthonormal basis (ϕn )∞ n=0 in B2 (D) and observe ∞ (w) = n=0 ϕn (w)ϕn (z). Taking as an orthonormal basis in B2 (D)√the system that Kz√ ϕ(z) = n + 1zn we get the Bergman kernel for D and taking the tensors of n + 1zn as a basis for B2 (Dn ) we get the Bergman kernel for the polydisc. The ball is treated similarly, although technical details are slightly more complicated (cf. [41], 3.1). In general it is quite difﬁcult to calculate the Bergman kernel explicitly but in the above-mentioned most important cases we get:

Spaces of analytic functions with integral norm

1675

• the Bergman kernel for the unit ball Bn is (1 z, w)−n−1 , *− n n • the Bergman kernel for the polydisc D is i=1 (1 − zi w¯ i )−2 . Note that the Bergman kernel for the polydisc is the product of Bergman kernels for the discs. It is a general fact for product domains – this follows from the above argument. Using the above explicit representations we obtain (cf. [41]) T HEOREM 4. Let D be either Bn or Dn and let 1 < p < ∞. The Bergman projection is (extends to) a bounded projection from Lp (D, ν) onto Bp (D). This is an optimal result. Since H∞ is not complemented in any L∞ space, the Bergman projection can not be bounded for p = ∞ (cf. [16]). Since it is orthogonal it cannot be bounded for p = 1 either. There are however (non-orthogonal) projections which are bounded for 1 p < ∞. We have the following T HEOREM 5 (Forelli–Rudin, cf. [41], 7.1). Let P1 f (z) = n

Bn

f (w)(1 − |w|2 ) dν(w). (1 − z, w)n+2

(5)

The operator P1 is a bounded projection from Lp (Bn , ν) onto Bp (Bn ) for 1 p < ∞. Using those projections (for n = 1) coordinatewise we can get analogous result for polydiscs. When we compare this with Corollary 2 and use the classical result of Pełczy´nski we get T HEOREM 6 ([29]). Let D be either Bn or Dn and let 1 p < ∞. The space Bp (D) is isomorphic to p . Note also that Theorems 4 and 5 allow us to describe (isomorphically) duals Bp (D)∗ . For 1 < p < ∞ we can identify (using the natural Hilbert space pairing) Bp (D)∗ with Bq (D), p−1 + q −1 = 1. Theorem 5 allows us to identify (for 1 p < ∞) the space Bp (D)∗ with Im P1∗ ⊂ Lq (D, ν), i.e., with the space of those f (z) analytic in D that |f (z)|(1 − |z|2 ) ∈ Lq (D, ν). P ROBLEM 7. This theorem settles the isomorphic structure of Bergman spaces on balls and polydiscs, however many isometric questions remain open. For example, we do not know the Banach–Mazur distance between Bp (D) and p (except for p = 2). Clearly it goes to inﬁnity as p → ∞ but the exact values or good estimates are unknown. Also we do not know basis constants or unconditional basis constants. It seems likely that (for p = 2) there is no monotone basis (or even basic sequence) in Bp (D) nor even a ﬁnite-dimensional (but not one-dimensional) norm one projection. Also the description of extreme points (and more reﬁned extreme structure) of the unit ball in B1 (D) is unknown. The same questions make sense for other Bergman spaces. The problem of boundedness in Lp of Bergman projections and analogues of Riesz projections (Szegö projections) for other domains is very delicate and not solved in general.

1676

P. Wojtaszczyk

We refer the interested reader to [4] and the references quoted there. Let us simply point out that there are very regular domains D ⊂ Cn for which the Bergman projection is bounded only in some interval around 2, not in (1, ∞) as we have seen in our examples. Also there are domains for which Szegö projections are bounded only when p = 2. The question if Bp (D) is always isomorphic to p seems to be open. The feature that greatly simpliﬁes the study of Taylor expansions of functions in Bergman spaces isn that dyadic blocks sum absolutely. To make it more precise let us take f (z) = ∞ n=0 an z ∈ Bp (D). Writing the integral in polar coordinates as f Bp =

1 π

1 0

iθ p f r e dθ r dr

2π

1/p

0

we see that every multiplier Λ = (λn ) which acts on Hp (D) acts also on Bp (D). If for p 1 we take multipliers εn Λn discussed after Theorem 18 or for p > 1 we take as Λn the sequence χ[2n ,2n+1 ] we infer (either from Theorem 18 or from the Littlewood–Paley n theory) that the series ∞ n=0 Λ f converges to f unconditionally in Bp (D), and so > L =:

D

∞ n Λ f (z)2

∼ f Bp .

dν(z)

n=0

Now we have (remember that Λn f = Lp

?1/p

p/2

∞

2n+2

2n−1 γn z

n)

1−2−n−1 2π

Λn (f ) r eiθ p dθ r dr

−n 0 n=0 1−2 ∞ 2π −n n

c

p Λ (f ) eiθ dθ.

2

On the other hand p f Bp

1 0

0

0

1

∞ 2π

>

1

>

0 ∞ n=0

p

Λ (f ) r e dθ r dr

n

Λn (f ) r eiθ p dθ

0

r

2n−1

Λ (f )Hp

n=0 ∞ 2n−1 n/p q r 2

p n Λ (f )Hp ·

−n

r dr

r dr

?p/q > ·

1 ∞ 0

1/p ?p

?p

n

n=0

2

iθ

n=0 ∞ 2π

n=0 > ∞ 1

0

=

(6)

0

n=0

n=0

∞

2

? p n Λ (f )Hp r dr

−n

n=0

r

q/p

q2n−1 nq/p

2

r dr.

(7)

Spaces of analytic functions with integral norm

1677

A relatively straightforward calculation shows that the last integral is ﬁnite, so from (6) and (7) we obtain f Bp ∼

∞

2

p n Λ (f )Hp

−n

1/p (8)

.

n=0

The details of the above calculation can be found (in much greater generality) in [32] or [53], Section 4. The relation (8) allows an explicite construction of an unconditional basis in Bp (D) equivalent to the unit vector basis in p , 1 p < ∞ (cf. [32,53]). For 1 < p < ∞ the following simple system works after we normalise it properly: we deﬁne n −1 k 0 = 1 and for z for n = 0, 1, 2, . . . . The desired system consists of f−1 fn (z) = 2k=0 n −n k 2 −2πk2 n n = 0, 1, 2, . . . fn (z) = z fn (e z) with k = 0, 1, . . . , 2 − 1.

3. Hardy spaces The general theory of Hardy spaces is presented, from various points of view in many books, e.g., [18,26,41,56,14]. In what follows we will present only those facts from the general theory that have big impact on Banach space properties of those spaces. Let us start with the most important case, of Hardy spaces on the unit disc D. For f ∈ H(D) and every r, 0 < r < 1, and p, 0 < p < ∞, we deﬁne Mp (f ; r) =

1 2π

2π

iθ p f r e dθ

1/p .

(9)

0

For each f and p the function Mp (f ; r) is an increasing function of r and we denote Hp (D) = f ∈ H(D): sup Mp (f ; r) < ∞ .

(10)

r<1

The Fatou theorem asserts that we can identify an analytic function f ∈ Hp (D) with its “boundary values” f ∗ on T. More precisely it says n iθ T HEOREM 8. Suppose that f (z) = ∞ n=0 an z ∈ Hp (D). Then the limit limr→1− f (r e ) ∗ iθ ∗ = f (e ) exists almost everywhere and in Lp (T) and we have supr<1 Mp (f ; r) = f p . einθ . Conversely, for p 1 if a If p 1 the function f ∗ has the Fourier series ∞ n=0 an inθ function g ∈ Lp (T) has the Fourier series of the form ∞ n=0 an e , then the analytic ∞ n ∗ function f (z) = n=0 an z ∈ Hp (D) and f = g. We also have

1 f r eiθ = 2π where P (r, θ ) =

2π

P (r, θ − t)g eit dt,

0

1−r 2 1−2r cos θ+r 2

is the Poisson kernel for the disc.

(11)

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P. Wojtaszczyk

Generally we will treat the function as extended by its boundary values and so we will omit the ∗ in the sequel. Note that for p > 1 the limit limr→1− f (r eiθ ) is easily seen to exists weakly in Lp (T), and so is a function from Lp (T). Then we can use the properties of the Poisson kernel to get the norm convergence. For p = 1 there is a serious problem, because w∗ -limr→1− f (r eiθ ) is in general a measure. That it is actually a function follows from the following important T HEOREM 9 (Riesz). Let μ be a Borel measure on T such that T einθ dμ(θ ) = 0 for n = 1, 2, . . . . Then μ is absolutely continuous with respect to the Lebesgue measure. There are many beautiful proofs of this theorem, three can be found in [26]. As an important corollary from Theorem 8 let us note that spaces Hp (D) are actually subspaces of Lp (T). This allows us to consider the natural orthogonal projection R from (T) onto H2 (D), called projection. In terms of the Fourier series this is simply L2 ∞the Riesz inθ ) = inθ . One can give a representation of this projection as a e e a a R( ∞ n=−∞ n n=0 n Cauchy integral. onto

T HEOREM 10. The Riesz projection R extends to a bounded projection R : Lp (T) −→ Hp (D) for 1 < p < ∞. The norm of this projection satisﬁes Rp ∼ for p = 1 nor for p = ∞.

p2 p−1 . R

is not bounded

Let us note two corollaries of this theorem. C OROLLARY 11 (Boas [7]). For 1 < p < ∞ the (D) is isomorphic p (T) ∞ to Li2nθ space Hpinθ e e a ) = a + a + and the isomorphism B can be given as B( ∞ 0 n=−∞ n n=1 n ∞ i(2n−1)θ . n=1 a−n e C OROLLARY 12. The system (zn )∞ n=0 in the natural order is a Schauder basis in Hp (D) for 1 < p < ∞. Factorization on D. factorization

Any function f ∈ Hp (D), 0 < p ∞ admits the so-called canonical

f (z) = I (z) · F (z) = B(z) · S(z) · F (z), where (1) The function I (z) is inner, i.e., |I (z)| 1 for z ∈ D and |I (eit )| = 1 a.e. (2) The function F (z) is outer. For any function ϕ(t) 0 deﬁned on T such that ϕ ∈ Lp (T) and ln ϕ ∈ L1 (T) we can deﬁne an outer function in Hp by the formula Fϕ (z) = exp

1 2π

2π 0

eit +z log ϕ(t) dt . eit −z

The function Fϕ is in Hp (D) and |Fϕ (eit )| = ϕ(t). Moreover Fϕ has no zero in D. The function F (z) equals F|f (eit )| .

Spaces of analytic functions with integral norm

1679

(3) The function B(z) is a Blaschke product ∞ $ |an | an − z B(z) = , an 1 − a¯ n z n=1

where an ’s are all zeros of f (z) in D counted with multiplicities. It is an inner function. (4) The function S(z) is a singular inner function, i.e., an inner function with no zero in D. We start this factorisation by considering B(z). First by a direct calculation we check that the product deﬁning B(z) converges almost uniformly if and only if (1 − |an|) < ∞. Then it follows from Jensen’s inequality that for a function from Hp (D) the sequence of its zeros (counted with multiplicities) gives the convergent Blaschke product. We deﬁne the outer function F (z) as (ii) and the rest is S(z). For an inner function I (z) the space I · Hp = {I · f : f ∈ Hp (D)} is a closed subspace of Hp (D) isometric to Hp (D). Such a subspace is invariant under multiplication by z, i.e., z · I · Hp ⊂ I · Hp and Buerling’s theorem asserts that for 1 p < ∞ those are all such subspaces (= {0}). For a function f ∈ H(Dn ) and r ∈ [0, 1) we deﬁne Mp (f, r) =

Tn

f (rz)p dλn (z)

1/p .

(12)

Analogously for f ∈ H(Bn ) and r ∈ [0, 1) we deﬁne Mp (f, r) =

Bn

f (rz)p dσn (z)

1/p .

(13)

By Hp (D) where D is any one of the above sets we denote the set of all functions f ∈ H(D) such that supr Mp (f, r) = f p < ∞. It is known that in all the above cases + Mp (f, r) is an increasing function of r. We can also deﬁne ∞ Hp on C . pThe function Mp (f, r) is then deﬁned for all r > 0 by Mp (f, r) = ( −∞ |f (x + ir)| dx)1/p and Hp (C+ ) is the set of all f ∈ H(C+ ) such that supr Mp (f, r) = f p < ∞. For such f ’s the function Mp (f, r) is a decreasing function of r. Like in the case of the disc we can deﬁne boundary values of functions in Hp and we get the conclusion that all Hp spaces deﬁned above are subspaces of Lp . The orthogonal projection from L2 onto H2 (after the identiﬁcation via boundary values) is called a Riesz projection. Like in D it is bounded from Lp onto Hp for 1 < p < ∞ (and unbounded for p = 1 and p = ∞). For Dn we apply the one-dimensional result coordinatewise; for Bd it is more difﬁcult (cf. [41], 6.3). There is a natural equivalence between Hp (D) and Hp (C+ ) which is a reﬂection of the fact that those domains are conformally equivalent in a very nice way. Let us ﬁx the map ϕ(z) = (i − z)/(i + z) which maps C+ conformally onto D. For a function f holomorphic

1680

P. Wojtaszczyk

in some neighbourhood of D we see that f ◦ ϕ is holomorphic in some neighbourhood of C+ . Moreover we have p ∞ ∞ 1 f i − x f ◦ ϕ(x)p 1 dx = dx. 2 1+x i+x 1 + x2 −∞ −∞ π Substituting eiθ = (i − x)/(i + x) we see that the above integral equals −π |f (eiθ )|p 21 dθ . This means that we have an isometry between Hp (D) and a weighted Hp space on C+ with respect to the measure 1/(1 + x 2 ) dx. A natural way to deal with this weight is to apply an appropriate multiplication operator. The results are summarised in the following T HEOREM 13. The map Ip deﬁned on Hp (D) as

(Ip f )(z) =

1 π 1/p

(f ◦ ϕ)(z) · (1 − iz)

−2/p

i−z = 1/p f (1 − iz)−2/p i+z π 1

(14)

is an isometry from Hp (D) onto Hp (C+ ) for 0 < p ∞. Note ﬁrst that (1 − iz)−2/p is a well deﬁned analytic function on C+ . Computing the norm we get i − x p −2 f i + x |1 − ix| dx −∞ π iθ p 1 ∞ i − x p 1 1 f e dθ. f = dx = 2 π −∞ i+x 1+x 2π −π

Ip (f )p = 1 p π

∞

Thus (some technicalities aside) we see that Ip is an isometry from Hp (D) into Hp (C+ ). To see that it is onto it sufﬁces to solve (14) for f . Note that the above theorem allows us to carry many properties (in particular the canonical factorisation) from H1 (D) to H1 (C+ ). Using this substitution ∞ we can also compute the Poisson kernel for C+ and check that if f ∈ H1 (C+ ) then −∞ f (t) dt = 0. 4. Special operators The fact that our spaces consist of analytic functions and that we have natural projections associated with them allows us to consider several natural classes of operators.

4.1. Coefﬁcient multipliers Coefﬁcient multipliers are most naturally considered on or on Dn . For simplicity we will D ∞ consider only functions on D. Then a function f (z) = k=0 ak zk analytic in D is identiﬁed

Spaces of analytic functions with integral norm

1681

∞ ∞ with a sequence of coefﬁcients (ak )∞ k=0 and we consider an operator (ak )k=0 → (λk ak )k=0 where the range space is either some sequence space or a space of analytic functions on D.

Hardy spaces. We will consider only coefﬁcient multipliers on H1 (D) because for p > 1 those are the same as Fourier multipliers on Lp (T) (this follows from Corollary 11) and thus are more properly treated in the theory of Fourier series. Moreover multipliers of H1 (D) have important Banach space consequences. T HEOREM 14 (Fefferman). A multiplier Λ = (λn )∞ n=0 maps H1 (D) into 1 if and only if sup

∞

(k+1)m

m1 k=1

2 |λn |

< ∞.

(15)

n=km+1

This theorem was obtained by Fefferman but never published. A proof of an analogous result on Rn was published in [47]. In Section 5 we will give a proof which follows unpublished argument from [49]. C OROLLARY 15 (Hardy). If f =

∞

n=0 an z

n

∈ H1 (D) then

∞ |an | πf 1 . n+1

(16)

n=0

Clearly the constant π in the above inequality does not follow from Theorem 14. There is also a full description of multipliers from H1 (D) into 2 (cf. [14, 6.4]), namely T HEOREM 16. A multiplier Λ = (λn )∞ n=0 maps H1 (D) into 2 (or equivalently into H2 (D)) if and only if sup N −2

N (n + 1)2 |λn |2 < ∞.

N1

n=0

C OROLLARY 17 (Paley). If f =

∞

∞

n=0 an z

(17) n

∈ H1 (D) then

1/2 |a2n |

2

Cf 1 .

(18)

n=1

From this corollary in particular follows the following well known version of Khinchin’s inequality for lacunary trigonometric series: ∞ ∞ ∞ i2n θ i2n θ i2n θ C an e an e an e , n=0

2

n=0

1

n=0

2

(19)

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P. Wojtaszczyk

n so we infer that the projection P (called Paley’s projection) deﬁned as P ( ∞ n=0 an z ) = ∞ k 2 k=0 a2k z is a bounded projection on H1 (D) and its range is isomorphic with 2 . This in particular implies that H1 (D) is not isomorphic to a complemented subspace of an L1 (μ) space, because such spaces do not contain complemented Hilbert spaces (cf. [21]). Clearly any multiplier from H1 (D) into H2 (D) is also a multiplier into H1 (D), but obviously it is not all. The important class of multipliers from H1 (D) into H1 (D) is given by the following T HEOREM 18 (Stein, cf. [12]). Let Λ = (λn )∞ n=0 be a bounded sequence such that supn0 (n + 1)|λn+1 − λn | < ∞. Then Λ is a multiplier from H1 (D) into H1 (D). Note that this theorem provides an H1 version of the Littlewood–Paley decomposition. Let us deﬁne multipliers Λn = (λnk )∞ k=0 by the formula ⎧ ⎨1 if 2n k 2k+1 , n λk = 0 if k 2n−1 or k 2n+2 , ⎩ linear otherwise, 2n then for each sequence εn = ±1 the multiplier ∞ the assumptions of n=0 εn Λ ∞ satisﬁes the above Theorem 18, so for each f ∈ H1 (D) the series n=0 Λ2n (f ) converges unconditionally to f . Bergman spaces. Multipliers between Bergman spaces were investigated quite intensively recently. The formula (8) can be very useful to see how they look like (at least for p > 1). Note that this formula shows that the real difﬁculty in describing multipliers from Bp (D) into sequence spaces or into Bq (D) is the description of Fourier multipliers of polynomials in Lp (T). This is a demanding problem of harmonic analysis. There is a full description of multipliers from Bq (D) into Bp (D) for 0 < p 2 q < ∞ given in T HEOREM 19 ([52]). The sequence (λn )∞ n=0 is a multiplier from Bq (D) into Bp (D), 0 < p 2 q < ∞, if and only if sup 2k n<2k+1

n−1/r |λn |

∞ k=1

∈ r

with r −1 = p−1 − q −1 .

Various other multipliers are described (in various terms) in [6,20,51] and in references given there. It seems that from the point of view of geometry of Banach spaces multipliers on Bp are less useful then on Hp , largely because isomorphic structure of Bp (D) is clear. Let us conclude those remarks by stating that the sequence ((n + 1)−1/p )∞ n=0 is, for 1 p 2, a multiplier from Bp (D) into Hp (D). For p = 2 it is obvious while for p = 1 can be easily derived from (8). The rest follows by interpolation. This example was ﬁrst observed in [31].

Spaces of analytic functions with integral norm

1683

4.2. Composition operators Let D ⊂ Cn be an open subset and let Φ : D → D be an analytic map. Then the composition operator CΦ (f )(z) =: f (Φ(z)) acts on H(D). It is a natural area of investigation what are the properties of such operators on various Banach spaces of holomorphic functions on D. Naturally a lot of effort went into the Hilbert space problems (cf. [56,45]). In this survey we will discuss only the most basic results in the context of Hp (D). The fundamental fact is T HEOREM 20 (Littlewood subordination principle). Let Φ : D → D be an analytic function. For all p > 0 the operator CΦ is bounded on every Hp (D) and on each Hp (D) we have CΦ

1 + |Φ(0)| . 1 − |Φ(0)|

(20)

P ROOF. We write Φ = M ◦ ϕ where M is a Möbius transformation with M(0) = Φ(0) and ϕ(0) = 0. A direct computation gives CM 1+|M(0)| 1−|M(0)| . Now it follows from the Schwarz lemma that ϕ maps each disk |z| < r into itself. For a given f ∈ Hp (D) and r < 1 let h(z) be the harmonic function in |z| r which for |z| = r equals |f (z)|p . Since |f (z)|p is subharmonic we see that |f (ϕ(z))|p h(ϕ(z)). Thus 1 2π

2π 0

2π it

p it

f ϕ r e dt 1 h ϕ r e dt = h ϕ(0) = h(0) 2π 0 2π 2π it p

1 1 f r e dt. = h r eit dt = 2π 0 2π 0

Since CΦ = Cϕ ◦ CM the claim follows.

If we are interested to what operator ideals a composition operator belongs then the results are very nice for ideal of compact operators. Namely we have T HEOREM 21. Let Φ : D → D. The following conditions are equivalent: (1) CΦ is compact on H2 (D), (2) CΦ is compact on Hp (D) for some p, 1 p < ∞, (3) CΦ is compact on Hp (D) for all p ∈ [1, ∞). P ROOF. Fix 0 < p, q < ∞ and suppose that CΦ is compact on Hp (D). Since bounded sequences in Hq (D) are normal families we need to show that CΦ (fn )q → 0 on a subsequence for fn convergent to 0 uniformly on compact subsets of D and fn q 1. We factorise fn = In · Fn into inner and outer parts. Clearly there is a subsequence so that both In and Fn converge almost uniformly on this subsequence to I∞ and F∞ respectively, and q/p at least one of those functions equals 0. Put Gn = Fn . Since CΦ is compact on Hp (D) q/p we see that CΦ (In ) converges to CΦ (I∞ ) and CΦ (Gn ) converges to CΦ (F∞ ) in the q/p norm of Hp (D). If F∞ = 0 then CΦ (fn )q CΦ (Fn )q = CΦ (Gn )p so converges

1684

P. Wojtaszczyk

to 0. If I∞ = 0 then In (Φ(eit )) → 0 a.e. because one checks that compactness of CΦ imq plies |Φ(eit )| < 1 a.e. We have CΦ (fn )q = T |In (Φ(eit ))|q |Gn (Φ(eit ))|p dλ(t), where the ﬁrst factor in the integrand converges to 0 a.e. and the second is uniformly integrable because CΦ (Gn ) converges in Hp . So we get CΦ (fn )q → 0. To get the intuition of how the size of CΦ reﬂects properties of Φ note that if Φ is inner then CΦ is an isometric embedding, while if Φ∞ < 1 then CΦ is nuclear. Also note, what we used in the previous proof, that CΦ compact implies |Φ(eit )| < 1 a.e. Thus the intuition suggests that for small operators the function Φ(eit ) touches T rather rearly. The natural measure here turns out to be the Nevanlinna counting function of Φ deﬁned for z ∈ D \ {Φ(0)} by NΦ (z) =: {− log |w|: f (w) = z}. Using this function we can express when CΦ is compact or in p-Schatten class. T HEOREM 22. (a) ([44]) CΦ is compact on H2 (D) (and so on all Hp (D), p ∈ [1, ∞)) if and only if lim|w|→1− |log |w||−1 NΦ (w) = 0. (b) ([30]) CΦ is in Schatten p-class 0 < p < ∞ (in particular nuclear for p = 1) on H2 (D) if and only if

log |z|−1 NΦ (z) p/2 1 − |z|2 −2 dν(z) < ∞. D

For other, more Banach space operator ideals the situation is less clear. There is a lot of open problems and a good place to start looking at this subject is [19]. What seems to be of special importance here is the notion of order boundedness, i.e., when CΦ ({f ∈ Hp (D): f p 1}) is an order bounded subset of some Lr (T). We say that CΦ is β-order bounded if the above holds with r = βp, 0 < β < ∞. It is known (cf. [19, Theorem 9] that this notion does not depend on p and is equivalent to (1 − |Φ|2 )−1 ∈ Lβ (T). With this notion we can get some fragmented information about absolutely summing operators. Namely (cf. [46,19]) P ROPOSITION 23. The following are equivalent: (1) operator CΦ is 1-order bounded, (2) operator CΦ : Hp (D) → Hp (D) is p-nuclear for some (and then for all) 2 p < ∞, (3) operator CΦ : Hp (D) → Hp (D) is p-absolutely summing for some (and then for all) 2 p < ∞. 5. Isomorphisms of H1 Let us consider f (z) = !f (z) + i'f (z) ∈ H1 (C+ ). Clearly the function f (z) is determined by !f (in general 'f is determined by !f up to a constant but for f ∈ H1 (C+ ) both functions have integral 0 so the constant is also determined). So about the boundary value f (t) we can think as !f (t) + i'f (t) where 'f is a boundary value of a harmonic conjugate of the harmonic extension of !f (t). We can reverse this process. For a realvalued function f ∈ L1 (R) let F (z) denotes the harmonic extension of f to C+ and let G(z) denotes the harmonic conjugate of F (z) in C+ . Let us put f˜(t) = limy→0+ G(t + iy).

Spaces of analytic functions with integral norm

1685

We see from the above that we can naturally identify H1 (C+ ) with the space !H1 (R) of all functions f ∈ L1 (R) such that f˜(t) exists a.e. and is in L1 (R). We equipp the space !H1 (R) with the norm f 1 + f˜1 . Now this identiﬁcation establishes an isomorphism between !H1 (R) and H1 (C+ ) but considered as Banach spaces over reals. If one insists (as is quite natural) on complex scalars one has to extend the deﬁnition of !H1 (R) the space !C H1 (R) of complex-valued functions F = f + ig with f, g ∈ !H1 (R). Then H1 (C+ ) is (via boundary values) a closed subspace of !C H1 (R). An operator P (f + ig) =: 1 ˜ ˜ is a complex-linear bounded projection from !C H1 (R) onto H1 (C+ ) 2 [f + if + i(g + ig)] (remember that f˜˜ = − f ). One checks that ker P = {f ∈ !C H1 (R): f¯ ∈ H1 (C+ )}, so clearly !C H1 (R) = H1 (C+ ) ⊕ H1 (C+ ). In the sequel we will usually deal with the space !C H1 (R) but for simplicity of notation we will drop the subscript C. D EFINITION 24. The function f on R belongs to the space BMO(R) if and only if 1 sup |I I ⊂R |

|f − fI | = f ∗ < ∞,

I

where I ⊂ R is an interval in R and fI =

(21)

1 |I | I

f (t) dt.

The space BMO(R) is a Banach space with the above norm if we identify functions differing by a constant a.e. The classical John–Nierenberg inequality implies that for each p, 1 p < ∞, the expression

1 sup I ⊂R |I |

1/p

|f − fI |

p

(22)

I

deﬁnes an equivalent norm on BMO(R). Clearly L∞ (R) ⊂ BMO(R). It is quite easy to check that log |x| ∈ BMO(R). A deep theorem of Fefferman and Stein asserts that (!H1 (R))∗ = BMO(R). This assertion requires certain care. For f ∈ !H1 (R) and g ∈ BMO(R) the product f · g need not be integrable. Thus the way to understand the Fefferman–Stein theorem is to observe that the Schwartz class S is dense in !H1 (R) and ∗ say that for each x ∗ ∈ (!H 1 (R)) there exists a unique element g ∈ BMO(R) such that for ∗ f ∈ S we have x (f ) = R f (t)g(t) dt and conversely every element g ∈ BMO(R) gives by this formula a continuous linear functional on !H1 (R). A very useful reformulation of the Fefferman–Stein theorem is the atomic decomposition. A function a(t) is called a p-atom supported on an interval I ⊂ R if supp a ⊂ I and a(t) dt = 0 and moreover ap |I |(1/p)−1 . Our aim now is to provide an outline of the following theorem which provides a set of characterisations of !H1 (R). T HEOREM 25. The following conditions are equivalent for a real-valued function on R: (1) f ∈ L1 (R) and f˜ ∈ L1 (R), i.e., f ∈ !H1 (R), (2) there is an analytic function F (z) ∈ H1 (C+ ) such that f (x) = !F (x), (3) f ∗ (x) =: sup|x−y|
1686

P. Wojtaszczyk

(4) there exists a sequence of ∞-atoms aj such that f = j λj aj with j |λj | < ∞, (5) there exists a sequence of p-atoms, 1 < p ∞, aj such that f = j λj aj with j |λj | < ∞. Before proceeding with the proof let us note that conditions (4) and (5) allow us to describe the dual of !H1 (R). Disregarding some minor technical problems it is obvious from (5) that given p, 1 < p ∞, !H1 (R)∗ consists of all functions g on R such that g(x)a(x) dx: a(x) is a p-atom < ∞ sup R

(23)

and the theorem implies that this space does not depend on p with 1 < p ∞. This easily gives that !H1 (R)∗ = BMO(R) and a proof of (22). P ROOF. From our earlier discussion it is clear that (1) ⇔ (2). (2) ⇒ (3) Clearly it sufﬁces to show that F ∗ (x) = sup|x−y|

|x − y| + 2 Mf (y), t

(24)

where Mf is the classical Hardy–Littlewood maximal function. Inequality (24) follows immediately if we efﬁciently write Pt ∞ j =0 bj χIj where Ij are increasing intervals and 0, x − y ∈ I0 . Applying (24) to V (x) (i.e., the boundary values of V ) we get V ∗ (x) 3MV (x) for x ∈ R. Since the Hardy–Littlewood maximal function is bounded on L2 (R) we get R |V ∗ (x)|2 dx C R |V (x)|2 dx. But (use analyticity) F ∗ (x) F1∗ (x) = [V ∗ (x)]2 so 2

∗ 2 ∗ V (x) dx C V (x) dx F (x) dx R R R = C F1 (x) dx = C F (x) dx. R

R

(3) ⇒ (4) Let us treat f ∗ Pt (x) as a harmonic function F on C+ and let us put Ek = + {x ∈ R: f ∗ (x) > 2k } for k ∈ Z. Since F is continuous kon C we see that Ek is an open set, so it is a union of disjoint open intervals Ek = j Ij . Let us deﬁne hk =

|Ijk |−1 f

Ijk

f (x) dx

on Ijk , outside Ek .

(25)

Note that if above each interval Ijk we build an isosceles triangle with base Ijk and angles of 45◦ at the base, the function F (z) is 2k on the arms of this triangle. Using contour

Spaces of analytic functions with integral norm

1687

integration along this triangle and harmonicity of F (z) we get 1 C2k . f (x) dx |I k | k I j j Since f (x) =

∞

−∞ (hk+1 − hk )

we can write f (x) =

∞

k=−∞

(26) j

bk,j where bk,j = χI k · j

(hk+1 − hk ). Note that bj,k = 0, supp bj,k ⊂ Ijk and |bj,k | C2k , so each aj,k = C −1 2−k |Ijk |−1 bj,k is an ∞-atom. Thus ∞

f (x) =

C2k Ijk aj,k

k=−∞ j

and ∞

∞ 2k Ijk = 2k |Ek | CF ∗ 1 ,

k=−∞ j

k=−∞

so we have the desired atomic decomposition. (4) ⇒ (5) is obvious. (t ) (5) ⇒ (1) is a direct calculation using the formula f˜(x) = p.v. R fx−t dt and the fact ˜ that f p Cp f p for 1 < p < ∞. More precisely, if a(t) is a p-atom supported on [−ε, ε] (by translation such are enough) then

2ε −2ε

a(t) ˜ dt

2ε

p a(t) ˜ dt

−2ε (1/p)−1

C(2ε)

1/p (4ε)1/q Cap (4ε)1/q

(4ε)1/q = C.

For |x| > 2ε we use the cancelation property of the atom to get 1/q ε 1 1 1 q 1 − a(x) ˜ = a(t) dt ap − x − t dt x x −t −ε −ε x 2ε ε Cap 2 · ε1/q C 2 x x ∞ so |x|>2ε |a(t)| ˜ dx C ε xε2 dx C. Thus for every p-atom a, both a and a˜ are in L1 (R) with uniformly bounded norms, so the same is true for f which is an absolutely convergent sum of atoms.

ε

An entirely analogous considerations can be applied to the circle. The notion of a harmonic conjugate is valid in the disc D and it allows us to deﬁne a complex space !H1 (T) of all those functions on T such that f 1 + f˜1 < ∞. We have !H1 (T) = H1 (T) ⊕ H10 (T) where H10 (T) = {f ∈ H1 (T): f (0) = 0}. The notion of a p-atom has a natural analogue on T but we have to consider also a constant function as an atom. With those conventions we have the following analog of Theorem 25.

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P. Wojtaszczyk

T HEOREM 26. The following conditions are equivalent for a real-valued function on T: (1) f ∈ L1 (T) and f˜ ∈ L1 (T), i.e., f ∈ !H1 (T), (2) there is an analytic function F (z) ∈ H1 (D) such that f (x) = !F (x), (3) f ∗ (θ ) =: supz∈S(θ) |F (z)| ∈ L1 (T) where F (z) is a harmonic extension of f to D and S(θ ) = conv({|z| 1/2} ∪ {eiθ }), (4) there exists a sequence of ∞-atoms aj such that f = j λj aj with j |λj | < ∞, (5) there exists a sequence of p-atoms, 1 < p ∞, aj such that f = j λj , aj with j |λj | < ∞, (6) !H1 (T)∗ = BMO(T) with natural duality. A more detailed presentation of the ideas involved in the above theorems can be found in [26] or in most of modern books on harmonic analysis, e.g., [48]. The main use of atoms is that to check continuity of a linear operator deﬁned on H1 it sufﬁces to check that it is uniformly bounded on atoms. As an interesting example of this method we present the proof of Fefferman’s Theorem 14. In our presentation we follow [49]. P ROOF OF T HEOREM 14. In this proof we will use the standard harmonic analysis notation that the n-th Taylor (or Fourier) coefﬁcient of a function g is denoted by g(n). ˆ First let us check that condition (15) is necessary. We will consider shifted Fejer kernels

Gm (t) =

[3m/2]+m

1−

[3m/2]−m

|n − [3m/2]| int e . m

Clearly Gm ∈ H1 (T) and it is well known that Gm = 1. For f from the disc algebra we put f1 (t) = f (2mt)Gm (t). One checks that f1 ∈ H1 (T) and f1 1 f ∞ Gm 1 = f ∞ . We note that there are no cancelations of Taylor coefﬁcients in the product deﬁning f1 so |fˆ1 (n)| 12 |fˆ(k − 1)| for (2k − 1)m < n 2km and k = 1, 2, . . . . This implies ∞ 1 ˆ f (k − 1) 2 k=1

2km n=(2k−1)m+1

|λn |

∞

2km

λn fˆ1 (n)

k=1 n=(2k−1)m+1 ∞ λn fˆ1 (n) Cf1 1 Cf ∞ . n=0

∞ This means that the sequence ( 2km n=(2k−1)m+1 |λn |)k=1 is a continuous multiplier from the disc algebra into 1 and by the classical theorem of Paley (cf. [39]) is square summable. 2km ∞ 2(k+1)m 2 2 So we obtain ∞ k=1 ( n=(2k−1)m+1 |λn |) < ∞. The sum k=1 ( n=2km+1 |λn |) is estimated analogously, so we get the necessity of condition (15). To prove the sufﬁciency of (15) we will embed H1 (T) into !H1 (T). Because the multiplier is rotation invariant it sufﬁces to check that it is uniformly bounded on all ∞-atoms

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1689

supported on intervals I centered at 1. For such an atom a(t) we have following estimates a(n) ˆ n|I |

for n > 0,

(27)

∞ 2 a(n) ˆ |I |−1 , n=1 ∞

(28)

2 a(n ˆ + 1) − a(n) ˆ |I |/8.

(29)

n=0

π Since a(n) ˆ = −π a(eit ) e−int dλ(t) the estimate (27) follows by integration by parts. Es 2 a2 . The inequality (29) we prove as foltimate (28) is clear since ∞ ˆ n=1 |a(n)| 2 lows: ∞ ∞ 2 2 a(n a(n ˆ + 1) − a(n) ˆ ˆ + 1) − a(n) ˆ n=−∞

n=0

=

π

−π

a(t) eit −a(t)2 dλ(t)

|I |/2 it 1 e −12 dλ(t) = 2 |I | −|I |/2 |I |/2 t 2 dλ(t) |I |/8. −|I |/2

We will also need the observation that if (λn )∞ n=0 satisﬁes (15) then sup m−1 m2

m

n|λn | < ∞.

(30)

n=2

This is true because for 2j −1 < m 2j m n=2

n|λn |

j

k

2

n|λn |

k=1 n=2k−1 +1

Now we are ready to estimate write

∞

j

k

2

k=1

ˆ n=2 |λn a(n)|.

2

k

|λn | C

n=2k−1 −1

n=2

2k Cm.

k=1

Fix m such that |I |−1 m < 2|I |−1 and

∞ m ∞ (k+1)m λn a(n) λn a(n) λn a(n) ˆ = ˆ + ˆ . n=2

j

k=1 n=km+1

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From (27) and (30) we get m m λn a(n) ˆ |I | na(n) ˆ |I |Cm C. n=2

n=2

ˆ k )| = maxkm
k=1

(k+1)m

|λn |

n=km+1

∞ 2 a(n ˆ k )

1/2 >

k=1

(k+1)m ∞ k=1

∞ 2 a(n ˆ k ) C

2 ?1/2 |λn |

n=km+1

1/2 .

(31)

k=1

Since for each l = 1, 2, . . . , m we have (k+1)m a(n a(n ˆ + 1) − a(n) ˆ ˆ + l) + ˆ k ) a(km n=km+1

we get (k+1)m 2 2 2 a(n a(n ˆ + 1) + 2m ˆ + 1) − a(n) ˆ ˆ k ) 2a(km n=km+1

and averaging this over l we obtain 2 2 a(n ˆ k ) m

(k+1)m

(k+1)m 2 2 a(n) a(n ˆ + 2m ˆ + 1) − a(n) ˆ .

n=km+1

n=km+1

Thus we get ∞ ∞ ∞ 2 2 2 2 a(n a(n ˆ k ) a(n) ˆ + 2m ˆ + 1) − a(n) ˆ m k=1

n=m+1

n=m+1

∞ so (28), (29) and the choice of m gives ˆ k )|2 C. From (31) we get the n=1 |a(n claim. Let us note that Corollary 15 follows, apart from the constant, either from the formulation of Theorem 14 or can be easily obtained directly using the estimates we have. For an

Spaces of analytic functions with integral norm

1691

atom a(t) supported on an interval I centered at 1 and for m ∼ |I |−1 we have from (27) and (28) ∞ m ∞ |an | |an | |an | + n+1 n+1 n+1 n=0

n=0

n=m+1

m n|I | + n+1 n=0

∞

1/2 |an |

n=m+1

2

∞

1/2 (n + 1)

−2

n=N+1

(m + 1)|I | + Ca2 (m + 1)−1/2 C. If we want to use atoms to estimate operators from H1 into H1 we need a way to decompose functions into atoms. In general this is embodied into the theory of molecules (cf. [12]) but as an example we will give the following simple fact: ∞ L EMMA 27. If f ∈ L1 (R) and −∞ f (t) dt = 0 and f 2 K1 and |f (x)| K2 |x|−2 for |x| 16 then F H1 C = C(K1 , K2 ). P ROOF. We take In = [−2n , 2n ] and write g n =: (f − fIn )1In . The desired atomic decom position into 2-atoms is given by f = g4 + ∞ n=4 (gn+1 − gn ). Using those ideas it is relatively easy to show that !H1 (R) has an unconditional basis. T HEOREM 28. Let Ψ (x) be a wavelet such that Ψ (x) exists in each point and max |Ψ (x)|, |Ψ (x)| C(1 + |x|2)−1 . Then the wavelet basis Ψj k = 2j/2 Ψ (2j x − k) for k, j ∈ Z is an unconditional basis in !H1 (R). For the deﬁnition, most basic properties of wavelets and an argument that wavelets as described in this theorem exist, the reader may consult [15] and for more detailed treatment, e.g., [54]. P ROOF ( SKETCH ). Using the appropriate change of variables it sufﬁces to take an ∞-atom a(t) with supp a ⊂ [−1, 1] and show that H1 -norms of all functions j,k ±a, Ψj k Ψj k are uniformly bounded. First note that from Lemma 27 we get Ψj k H1 C2−j/2 .

(32)

For the proof we split our sum into three sums j 0,k

+

j >0,|k|<5·2j

+

j >0,|k|5·2j

±a, Ψj k Ψj k = S1 + S2 + S3 .

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Using the assumption about the derivative of the wavelet we integrate by parts and get |a, Ψj k | 23j/2 (1 + |2j − k|)−2 . From this and (32) we infer that the sum S1 is absolutely convergent in H1 . The sum S3 is also absolutely convergent. We see directly that

a, Ψj k 2j/2 1 + |2j − k| −2 .

(33)

This and (32) yields the absolute convergence. For S2 we can show (after some calculations using (33)) that for |x| > 16 we have the pointwise estimate j 0 |k|>5·2j |a, Ψj k | |Ψj k (x)| C|x|−2 . This allows us to apply Lemma 27. It is interesting to note that H1 (T) not only has an unconditional basis but also has certain universality property in this respect (cf. Proposition 37). We can develop a parallel theory for the Haar wavelet h (cf. [15] or [54, 1.1]). This theory has deep connections with martingale theory. What is important from our point of view can be summarised in the following T HEOREM 29. For a function f ∈ L1 (R) the following conditions are equivalent: (1) R ( j k∈Z |f, hj k hj k |2 )1/2 < ∞, (2) there exists a p, with 1 < p ∞ such that f = j λj aj with j |λj | < ∞ and aj ’s dyadic p-atoms, i.e., p-atoms supported on some dyadic interval, (3) for every p, with 1 < p ∞ we have f = j λj aj with j |λj | < ∞ and aj ’s dyadic p-atoms, i.e., p-atoms supported on some dyadic interval, (4) the series j k f, hj k hj k converges unconditionally in L1 (R) to f . The space of functions satisfying any of the above conditions is denoted as H1 (δ). Clearly the Haar wavelet basis is an unconditional basis in H1 (δ). One can deﬁne an analogous space of functions on [0, 1]. It is denoted by H1 [0, 1] and actually equals span{hj k : supp hj k ⊂ [0, 1]}. As spaces of functions spaces !H1 (R) and H1 (δ) are different. The important fact is that they are isomorphic. T HEOREM 30. Let Ψ be a wavelet as in Theorem 28. The map Ψj k → hj k extends by linearity to an isomorphism from !H1 (R) onto H1 (δ). The argument for this theorem is very similar to the proof of Theorem 28. Actually this proof also shows that the map f → j k f, Ψj k hj k is bounded from !H1 (R) into H1 (δ). Conversely for a dyadic ∞-atom a(t) supported on [0, 1] we see that j k a, hj k Ψj k can be estimated exactly like S2 in the above proof. The details of the proof of this theorem can be found in [54]. From this easily follows C OROLLARY 31. The space H1 (D) is isomorphic to the space H1 (δ). The success of the atomic decomposition in !H1 (R) lead to the vast generalisation of the atomic approach. A natural framework are the so called spaces of homogeneous type, cf. [12].

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D EFINITION 32. A space of homogeneous type (X, d, μ) is a set X equipped with the quasi-metric d and a positive Borel measure μ such that there exists a constant C such that for all x ∈ X and r > 0 we have

μ B(x, 2r) Cμ B(x, r) < ∞. Once we have a space of homogeneous type (X, d, μ) we can deﬁne atoms as Borel measurable functions a(t) on X, such that supp a(t) ⊂ B(x, r), |a(t)| μ(B(x, r))−1 and X a(t) dμ(t) = 0. We deﬁne H1 (X, d, μ) as the space of all functions f such that f = λ a where a ’s are atoms and j j j j |λj | < ∞. The norm f H1 is deﬁned as j inf j |λj | where the inf is taken over all atomic representations of the function f . The following examples are important in our context: (1) X = R, d(x, y) = |x − y| and μ the Lebesgue measure. In this case H1 (R, d, λ) equals !H1 (R). (2) X = T, d is the arc-length distance and μ is the normalised Lebesgue measure on T. In this case H1 (T, d, μ) equals !H1 (T). (3) X is either R or the interval [0, 1), d is the dyadic distance, i.e., if x = ∞ k=−∞ xk · 2−(k+1) where xk = 0 or 1 then we put d(x, y) =

∞

|xk − yk |2−(k+1).

k=−∞

The measure μ is the Lebesgue measure. In this case we get the dyadic Hardy space H1 (δ). (4) X = Sn the unit sphere in Cn , d(z, w) = |1 − z, w| and μ = σn . In this case we get the space of functions H1 (Sn , d, σn ) on Sn . It contains (boundary values of) the space H1 (Bn ) as a closed, complemented subspace. This example is investigated and the facts mentioned here are proven in [12] and [17]. A deep theorem of Paul Müller describes the isomorphic type of spaces H1 (X, d, μ). Before we can formulate it we need to introduce ﬁnite-dimensional analogues of Hardy n −1 spaces. By H1n , n = 1, 2, . . . , we denote the space span{hj k }n−1,2 j =0,k=0 ⊂ H1 (δ). From Theorem 30 we see that isomorphically we obtain the same sequence of spaces when we replace the Haar wavelet by any basis discussed in Theorem 28. It is interesting that the deep result of Boˇckariov [8] asserts that the Banach–Mazur distance between H1n and n −1 ⊂ H1 (D) is uniformly bounded. span{zj }2j =1 T HEOREM 33 (Müller [35]). Let (X, d, μ) be a space of homogeneous type and let H1 (X, d, μ) be inﬁnite-dimensional. Then H1 (X, d, μ) is isomorphic to H1 (δ) if and only if μ({x ∈ X: μ({x}) = 0}) > 0. When μ({x ∈ X: μ({x}) = 0}) = 0 then H1 (X, d, μ) is n isomorphic either to 1 or to ( ∞ n=1 H1 )1 . The proof of this theorem is very ingenious and complicated. To give even the sketch is well beyond the scope of this survey. When we apply this theorem to the case described in (4) above we see that H1 (Bn ) is isomorphic to a complemented subspace of H1 (δ). Next we show that H1 (Bn ) contains a

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P. Wojtaszczyk

complemented copy of H1 (δ). The natural way to prove it is to use the inner function in the ball: there exists a bounded function ϕ(z) holomorphic on Bn such that ϕ(0) = 0 and for ζ ∈ Sn we have |ϕ(ζ )| = 1 σn -a.e. (cf. [42]). One checks that span{ϕ n (z)}∞ n=0 ⊂ H1 (Bn ) is isometric to H1 (D) (the isometry is given by ϕ n ↔ zn ). Let Σ1 be the sub σ -algebra of the σ -algebra Σ of all measurable subsets of Sn generated by the function ϕ. One checks that the conditional expectation operator from L1 (Sn , Σ, σn ) onto L1 (Sn , Σ1 , σn ) gives a projection from H1 (Bn ) onto span{ϕ n }∞ n=0 . So the decomposition argument (use also Theorem 39) gives T HEOREM 34. The spaces H1 (Bn ) for n = 1, 2, . . . are all isomorphic to the space H1 (δ) and also to !H1 (R) and H1 (D). R EMARK 35. The ﬁrst correct proof of this theorem was given by Wolniewicz in [55]. This result can be greatly generalised. The deﬁnition of H1 space can be naturally extended to H1 (Ω) where Ω is a bounded strictly pseudoconvex domain in Cn . Then the boundary ∂Ω can be made into a space of homogeneous type in such a way that H1 (Ω) is a complemented subspace of the atomic H1 (∂Ω). Also such spaces H1 (Ω) are isomorphic to H1 (δ) as was shown by entirely different methods in [2] and [35]. The situation for polydiscs is dramatically different. T HEOREM 36 ([9,10]). The spaces H1 (Dn ) and H1 (Dm ) are non-isomorphic for n = m. Let us give the idea of the proof of this theorem when n = 1 and m = 2. Suppose that 2j there is an isomorphism T : H1 (D2 ) → H1 [0, 1]. Let (ϕj k )∞ j =0 k=1 be an unconditional basis in H1 (D) equivalent to the normalised Haar basis in H1 [0, 1]. Then the subspace of H1 (D2 ) l spanned by the functions ϕj k (z)w2 where j = 0, 1, 2, . . . , k = 1, 2, . . . , 2j , l = 1, 2, . . . , is complemented in H1 (D2 ) – this follows from the Paley’s theorem. Now ﬁx N and look l at functions T (ϕj k (z)w2 ) with j = 0, 1, . . . , N and k, l as previously. Since for a ﬁxed l l j, k the w-liml→∞ ϕj k (z)w2 = 0, the same holds for T (ϕj k (z)w2 ). Thus, starting from j = N and k = 2N and going backward, we see that there are integers l = l(j, k) such that p(j,k) l(j,k) (up to a small perturbation) T (ϕj k (z)w2 ) = s=s(j,k),r as,r hs,r where

s N, 2N < p N, 2N < s N, 2N − 1 < p N, 2N − 1 < · · · < s(N, 1)

< p(N, 1) < s N − 1, 2N−1 < p N − 1, 2N−1 < · · · < s(0, 1) < p(0, 1). The point is that (when we look at j, k only) we invert the order; the last function N ϕN,2N (z)wl(N,2 ) is mapped by T to a function whose expansion in H1 [0, 1] is at the beginning, while the ﬁrst function ϕ0,1(z)wl(0,1) has the expansion starting very far away. Nevertheless the spaces spanned by (ϕj,k (z)w2

l(j,k)

)N j =0

2j k=1

are uniformly in N comple-

mented in H1 (D2 ) and this basis is equivalent to the basis (ϕj,k (z))N j =0

2j k=1

in H1 (D).

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1695

l(j,k)

Thus T (ϕj k (z)w2 ) span uniformly complemented subspace of H1 [0, 1] and this ba2j sis is equivalent to (hj,k )N j =0 k=1 . But this contradicts the fact that T inverts the order, i.e., T (ϕj k (z)w2

l(j,k)

)=

p(j,k)

as,r hs,r .

s=s(j,k),r

This is the main technical argument in the proof. To give the reader an idea why it is so, let us consider the model case (which reﬂects the general situation although it is by no l(j,k) means easy to see). Let us consider the case when T (ϕj,k (z)w2 ) = hj,k · rm(j,k) where r denotes the Rademacher functions and m(j, k) = 10N − 2j − k. (Clearly the exact value of m(j, k) is not essential – the important thing is that it reverse the order.) We have N 2j 1/2 1 aj k hj,k rm(j,k) ∼ |aj k |2 |hj k |2 |rm(j k) |2 0 j =0 k=1

j,k

H1

=

1 0

1/2 |aj k | |hj k | 2

2

N 2j = aj k hj,k . j =0 k=1

j,k

H1

j

2 This shows (and we already know it) that (hj,k · rm(j,k) )N j =0 k=1 in H1 ([0, 1), δ) are unij

2 formly in N equivalent to (hj,k )N j =0 k=1 . But they are not complemented (uniformly in N ) because in the dual space BMO the equivalence breaks down. Let h∗j,k denote the Haar basis normalised in L∞ , i.e., biorthogonal functionals to hj,k . We have

N aj h∗j,0 j =0

∼ sup |aj |.

(34)

BMO

∗ On the other hand for f = N j =0 aj hj,0 rm(j,0) we use the deﬁnition of the dyadic BMO to −N get (for the interval [0, 2 ]) f BMO 2N

2−N 0

1/2 |f − f[0,2−N ] |

2

.

But for j = 0, 1, . . . , N we have m(j, 0) > N so we infer that f[0,2−N ] = 0, and also that the Rademacher functions we use are orthogonal on the interval [0, 2−N ]. Thus we get f BMO 2N

2−N 0

1/2 |f |

2

=

N

1/2 |aj |

2

.

j =0

This together with (34) clearly shows that there is no equivalence in BMO.

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The technical side of the above proof was greatly simpliﬁed by Müller [36]. It is easy to see that the space !H1 (R) is not a Banach lattice in the natural order ∞ because lattice operations do not preserve the condition −∞ f (t) dt = 0. Actually !H1 (R) is not isomorphic to any non-atomic Banach lattice [23]. There is however a closely related Banach lattice of functions on C+ introduced and studied in greater generality in [11]. It is the tent space T 1 (C+ ) deﬁned as the space of all functions on C+ such that

∞ −∞

|x−y|<αt

f (y + it)2 t −2 dy dt

1/2 dx < ∞.

(35)

One proves (cf. [11]) that this condition is independent of α > 0, so that (35) gives equivalent norms for different α’s. Clearly the space T 1 (C+ ) is a Banach lattice of functions. We can embed !H1 (R) as a complemented subspace of T 1 (C+ ) as follows: for f ∈ !H1 (R) ) let u(y, t) be its Poisson integral and put F (y, t) = t ∂u(y,t ∂t . The map f → F is the desired isomorphic embedding (cf. [11]). A bit more transparent complemented embedding of H1 (δ) into T 1 (C+ ) can be realised as follows: let for j, k ∈ Z squares Aj k ⊂ C+ be deﬁned as [k2j , (k + 1)2j ] ⊗ i[2j , 2j +1 ] and let Fj k = 1Ajk 2−j/2 . One checks (cf. [23]) that hj k → Fj k is an isomorphic embedding onto a complemented subspace of T 1 (C+ ). Actually (and it is easy to believe if one understands the above embedding of H1 (δ) into T 1 (C+ )) the space T 1 (C+ ) is isomorphic to the Hilbert space valued H1 -space H1 (δ, L2 ) k 2 or (what is the same) span{zj w2 }∞ j k=0 in H1 (T ). It follows from the Bourgain’s arguments indicated above in the proof of Theorem 36 that the space H1 (δ, L2 ) is not isomorphic neither to H1 (T) nor to H1 (T2 ).

6. Isomorphic structure of H1 We have seen in the previous sections that many H1 -type spaces are isomorphic. In this section we want to elaborate more fully on the isomorphic structure of this space, which we will generically denote by H1 . H1 is a dual space. Let D be any of our standard domains in Cn . Then H1 (D) is a dual space because it is a ω∗ -closed subspace of C(∂D)∗ . In the special case of H1 (D) we obtain from the F. and M. Riesz theorem that H1 (D) = (C(T)/A0 )∗ . Here by A0 we mean the subspace of the disc algebra of all functions vanishing at 0. The predual C(T)/A0 can be isometrically described as a space of compact Hankel operators on the Hilbert space. For a function f ∈ C(T) we deﬁne the Hankel operator with index f , Hf (g) = (I − R)(fg) where R is the Riesz projection. Clearly Hf : H2 (T) → H2 (T)⊥ . If f1 − f2 ∈ A0 then Hf1 = Hf2 and we infer that Hf f C/A0 . Also since f ∈ C(T) we can approximate it by trigonometric polynomials, so we can approximate Hf by ﬁnite-dimensional operators thus Hf is compact. On the other hand if f ∈ C(T) with f C/A0 = 1 then there exists h ∈ H1 (T) such that h1 = 1 and f h = 1. Using the canonical factorisation we write h = h1 · h2 where h1 , h2 ∈ H2 (T) and h1 2 = h2 2 = 1. Then we have ! " Hf Hf (h1 ) 2 Hf (h1 ), h2 = f h1 h2 = 1.

Spaces of analytic functions with integral norm

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This gives that the map f → Hf is an isometry from C/A0 into the space of compact operators on a Hilbert space. When we use the space BMO as a dual of H1 then we can describe the predual of H1 (but only isomorphically) as a space VMO of functions of vanishing mean oscillation where 1 VMO = f ∈ BMO: lim |f − fI | = 0 . |I |→0 |I | I Subspaces and complemented subspaces. Let us ﬁrst discuss subspaces of H1 . The next proposition shows that many subspaces of L1 are also subspaces of H1 . P ROPOSITION 37. Let X ⊂ L1 (T) be a closed subspace. Suppose that X either has an unconditional basis or X is reﬂexive. Then X is isomorphic to a subspace of H1 (T). To see the ﬁrst part ﬁx an unconditional basis (xn ) in X and assume (perturbing it slightly) that each xn (t) is a trigonometric polynomial and ﬁx a sequence kn such that for some strictly increasing sequence of integers ln we will have Λ2ln (eikn t xn ) = eikn t xn , where Λs are multipliers discussed after Theorem 18. Then using the unconditionality of (xn ) and Theorem 18 we get 2 1/2 2 1/2 ikn t e a a a x ∼ x (t) dt ∼ x (t) dt n n n n n n n

1

T

n

ikn t an e xn ∼

T

n

n

which shows that eikn t xn is a basic sequence in H1 (T) equivalent to (xn ). For the second part recall that by Rosenthal’s theorem a reﬂexive subspace of L1 is a subspace of Lp for some p > 1, that such Lp is a subspace of L1 [40], and that Lp for p > 1 has an unconditional basis (cf., e.g., [15]). As a corollary of the above we get that a reﬂexive complemented subspace of H1 is isomorphic to 2 (cf. [27, Corollary 2.1]). It was also shown in [27] that every Hilbertian subspace of H1 contains an inﬁnite-dimensional complemented subspace. To see it take (fn )∞ n=1 , a sequence of functions in H1 (D) equivalent to the unit vector basis in 2 and let V : 2 → H1 (D) be deﬁned as V (en ) = fn . Think that H1 (D) = X∗ where X ⊂ K(2 ) the space of all compact operators on 2 . We have V = U ∗ where U : X → 2 is onto. Take xn ∈ X such that xn C and U (xn ) = en . Then for some subsequence zk = xn2k − xn2k+1 is weakly null. Now we work in K(2 ) and see that zk either contains a subsequence zks equivalent to the unit vector basis in c0 (what in our case leads to a contradiction) or to 2 This gives that P (f ) = s f, zks fn2ks is the desired projection. Now let us consider subspaces of H1 (D) which are invariant under rotation. Such a subspace is described by a subset Λ ⊂ N and equals span{zn : n ∈ Λ}. Also, by invariance, if such a subspace is complemented, it is complemented by a multiplier 1Λ . We have seen examples of such multipliers in Paley’s theorem. Other easily checked examples are arithmetic progressions intersected with N. All the other examples are build from the above ones.

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T HEOREM 38 (Klemes [25]). A subspace span{zn : n ∈ Λ} ⊂ H1 (D) is complemented in H1 (D) iff Λ is a ﬁnite Boolean combination of lacunary sets, ﬁnite sets and arithmetical progressions intersected with N. The version (much more difﬁcult) of Klemes’ theorem for H1 (R), i.e., a characterisation of translation invariant complemented subspaces of H1 (R) was given by Alspach [1]. No extension of those results to several variables are known. Clearly the existence of unconditional basis gives many projections. In particular the following result easily follows from the form of Haar basis in H1 (δ). T HEOREM 39. The space H1 is isomorphic to ( ∞ n=1 H1 )1 , its inﬁnite 1 sum. Actually the Haar basis in H1 (δ) and the Haar basis in H1 [0, 1] are permutatively equivalent and each of them is permutatively equivalent to its inﬁnite 1 sum. P ROOF. Let us identify the normalised in H1 (δ) Haar function with its support I and denote it by hI . Let O = {hI : I ⊂ [0, 1]} and let Dn = {hI : I ⊂ [1 − 2−n+1 , 1 − 2−n ]} with n = 1, 2, . . . Clearly the basis Dn is isometrically .∞ ∞ equivalent to the Haar basis in 1] and D is isometrically equivalent to ( H1 [0, n=1 n n=1 Dn )1 . One easily observes that O\ ∞ D is a basis equivalent to the unit vector basis in 1 . Since each Dn contains a n=1 n we infer that O is permutatively equivsubsequence equivalent to the unit vector basis in 1 alent to ( O)1 . To treat the case of H1 (δ) we deﬁne O = {hI : I ⊂ R} and deﬁne Dn = {hI : I ⊂ [2n−1 , 2n ] ∪ [−2n , −2n−1 ]} where n = 1, 2, . . . and D0 = {hI : I ⊂ [−1, 1]}. The argument now is analogous. The main unsolved problem about inﬁnite-dimensional complemented subspaces of H1 (D) is whether there are inﬁnitely many isomorphic types of them. The easy ones are obtained from the above theorem, Paley’s theorem which implies that 2 is complemented in H1 (D) and ﬁnite-dimensional spaces H1n spanned by increasing subsets of unconditional basis. A routine n argument n yields 10non-isomorphic n types ofn them as , , ⊕ , ( ) , ( ) ⊕ , ( ) , ( H ) , ( H 1 )1 ⊕ 2 , follows 1 2 1 2 1 1 2 2 1 1 n 2 1 ( H1n )1 ⊕ ( 2 )1 , H1 . Two essentially new examples were obtained by a more reﬁned martingale techniques by Müller and Schechtman [37]. One of them, called Y1 in [37], can 2 1/2 < ∞}. This space is not isomorphically be described as {(αn ): ∞ n=1 min{|αn |, |αn | n} isomorphic to H1 but contains subspaces isomorphic to all p with 1 p 2. The other one is a sum of independent copies of H1n and can be described as span{Gn }∞ n=1 where each Gn is an isometric copy of H1n but different Gn ’s consist of statistically independent functions. To be more explicite let (rn )∞ n=1 be the sequence of Rademacher functions n and let Gn be the span in H1 [0, 1] of all Walsh functions of the form rkn11 · · · · · rkjj where ni = 0, 1 and n2 ki < (n + 1)2 . Clearly one can use those two new spaces to form direct sums with old ones to get a more extensive (but still ﬁnite) list of all known complemented subspaces of H1 . One should also note that the Haar basis in H1 (δ) gives only three isomorphic types T HEOREM 40 ([34]). Let Λ ⊂ Z × Z bean inﬁnite subset. Then the subspace span{hλ : λ ∈ n Λ} ⊂ H1 (δ) is isomorphic to 1 or to ( ∞ n=1 H1 )1 or to H1 (δ).

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The proof of this theorem is quite complicated and technical so we will only indicate how different possibilities arise. In order to make things more transparent let us consider H1 [0, 1]. If we take any subset B of Haar functions whose supports are in [0, 1] we deﬁne the set σ (B) as {t: t ∈ supp hI for inﬁnitely many hI ∈ B}. If |σ (B)| > 0 then span{hI ∈ B} ∼ H1 . The proof builds a block basic sequence of {hI ∈ B} which is very close in the H1 norm and distribution to the original Haar system. This gives that our space contains complemented H1 so by decomposition we get the claim. −1 If |σ (B)| = 0 but sup{|I | I ∈ B} ∼ hJ ∈B,J ⊂I |J |: hI ∈ B} = ∞ then we have span{h ( H1n )1 . We show that in this case our space contains complemented ( H1n )1 and is contained as a complemented subspace in one. If |σ (B)| = 0 but sup{|I |−1 hJ ∈B,J ⊂I |J |: hI ∈ B} < ∞ then a direct calculation shows that the basis B is equivalent to the unit vector basis in 1 . The condition distinguishing cases in the situation when |σ (B)| = 0 is a martingale version of the Carleson condition (cf. [26]). Note also that there is a close similarity between conditions and conclusion of this theorem and Theorem 33. This is not accidental and actually the methods of proof of Theorem 33 are an outgrowth and elaboration of the methods used to prove Theorem 40. There are also some general results about complemented subspaces of H1 . Let us formulate some of them as one theorem. T HEOREM 41. (a) ([34]) Let X ⊂ H1 be isomorphic to H1 . Then there exists Y ⊂ X complemented in H1 and isomorphic to H1 . (b) ([34]) The space H1 is primary, i.e., whenever H1 = X ⊕ Y then either X or Y is isomorphic to H1 . (c) ([37]) A complemented subspace H1 either contains 2 or is isomorphic to a X of n) . complemented subspace of ( ∞ H n=1 1 1 Approximation property. It is easy to see that H1 (D) has the bounded approximation property. The easiest argument is using Fejer’s kernels but it also trivially follows from the existence of unconditional basis. The analogous problem for BMO(R) is much harder. It was solved by Jones [22] who proved T HEOREM 42. The space BMO(R) and thus also H1 (D) has the uniform bounded approximation property. 7. Isometric questions The most natural isometric question about a Banach space is to describe all isometries of the space. In the case of H1 spaces (and actually for all Hp spaces) this question has a very satisfactory answer. For D it is given in the following theorem, but for other spaces the description is analogous.

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T HEOREM 43. The operator T : H1 (D) → H1 (D) is an isometry into iff T (f )(z) = F (z) · f (ϕ(z)) where ϕ(z) is an analytic map from D onto D, i.e., an inner function, and F (z) ∈ H1 (D) such that for every bounded Borel function h(t) on T we have h(t) dλ(t) = (h ◦ ϕ)F (t) dλ(t). T

T

Such an operator T is onto iff ϕ is a Möbius transformation and F (z) = αϕ (z) with α ∈ C and |α| = 1. Another natural problem is the description of the extreme structure of the unit ball. The study of extreme points of unit the ball in H1 (D) was done in [28]. In particular, the extreme points in the unit ball of H1 (D) are described as outer functions of norm 1. To see this take f = I · F with I a non-constant inner function and adjust the constants so that 2π it it 0 |f (e )|I (e ) dt is purely imaginary. Then the decomposition 1 1 1 2 2 f= F (1 + I ) − F (1 − I ) 2 2 2 shows that f is not an extreme point. Conversely if outer f0 = 12 (f1 + f2 ) with fi = 1 and f1 = f2 , we infer that both functions fj (eit ) have the same argument. But then fj /f0 are bounded analytic functions in D with real boundary values, thus constants. This gives a contradiction so f0 is an extreme point. No analogous characterisation is known for other H1 (D) spaces. Some effort went to describe exposed and strongly exposed points in various H1 spaces, but only various sufﬁcient conditions are known (cf. [24,43,50]). Another related problem of describing extreme points in the unit ball of the space VMO and BMO with the natural p-mean oscillation norm was studied in [3]. Very little is known about norm one projections in H1 spaces. All known such projections are restrictions of norm one projections on L1 which leave H1 invariant. It is not known if there are any others. In particular it is unknown if there is a norm one ﬁnitedimensional projection on H1 (D) whose whose range has dimension > 1. Also it is unknown if H1 (D) has a monotone basis. Another classical isometric result about H1 (D), but valid also for other spaces H1 (D) is the fact (cf. [38]) that if fn converges weakly in H1 (D) to f and fn = f for all n then actually fn converges to f in norm. This means that the natural norm on H1 (D) has the Kadec–Klee property. Actually a bit stronger result holds: for each ε > 0 there is a δ > 0 such that for every weak∗ convergent (with respect to the natural (H1 , C/A0 ) duality) sequence fn → f such that fn 1 and f 1 − δ we have lim infn=m fn − fm ε. From this result follows that H1 (D) has so called ω∗ -normal structure. This implies that every non-expansive map deﬁned on a ω∗ -closed bounded convex subset of H1 (D) has a ﬁxed point (cf. [5]). References [1] D.E. Alspach, A characterization of the complemented translation-invariant subspaces of H 1 (R), Trans. Amer. Math. Soc. 323 (1) (1991), 197–207. [2] H. Araki, Degenerate elliptic operators, Hardy spaces and diffusions on strongly pseudoconvex domains, Tohoku Math. J. 46 (4) (1994), 469–498.

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[3] S. Axler and A. Shields, Extreme Points in VMO and BMO, Indiana Univ. Math. J. 31 (1982), 1–6. [4] D. Bekollé and A. Bonami, Estimates for the Bergman and Szegö projections in two symmetric domains in Cn , Colloq. Math. 68 (1) (1995), 81–100. [5] M. Besbes, S.J. Dilworth, P.N. Dowling and C.J. Lennard, New convexity and ﬁxed point properties of Hardy and Lebesgue–Bochner spaces, J. Funct. Anal. 119 (2) (1994), 340–357. [6] O. Blasco, Multipliers on spaces of analytic functions, Canad. J. Math. 47 (1995), 44–64. [7] R.P. Boas, Isomorphism between Hp and Lp , Amer. J. Math. 77 (1955), 655–656. [8] S.V. Boˇckariov, Construction of polynomial bases in ﬁnite-dimensional spaces of functions analytic in the disk, Proc. Steklov Inst. of Math. (1985), 55–81. [9] J. Bourgain, Non-isomorphism of H 1 -spaces in one and several variables, J. Funct. Anal. 46 (1982), 45–57. [10] J. Bourgain, The non-isomorphism of H 1 -spaces in different number of variables, Bull. Soc. Math. Belg. Sér. B 35 (1983), 127–136. [11] R.R. Coifman, Y. Meyer and E. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335. [12] R.R. Coifman and G. Weiss, Extensions of Hardy spaces and theory use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645. [13] F. Delbaen, H. Jarchow and A.Pełczy´nski, Subspaces of Lp isometric to subspaces of p , Positivity 2 (1998), 339–367. [14] P. Duren, Theory of H p Spaces, Academic Press, New York (1970). [15] T. Figiel and P. Wojtaszczyk, Bases in function spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 561–597. [16] T. Gamelin and S.V. Kislyakov, Uniform algebras and spaces of analytic functions in the supremum norm, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 671–706. [17] J.B. Garnett and R.H. Latter, The atomic decomposition for Hardy spaces in several complex variables, Duke Math. J. 45 (1978), 845–915. [18] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs (1962). [19] H. Jarchow, Some functional analytical properties of composition operators, Quaestiones Math. 18 (1995), 229–256. [20] M. Jevti´c and M. Pavlovi´c, Coefﬁcient multipliers on spaces of analytic functions, Preprint. [21] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1–84. [22] P.W. Jones, BMO and the Banach space approximation problem, Amer. J. Math. 107 (4) (1985), 853–893. [23] N.J. Kalton and P.Wojtaszczyk, On nonatomic Banach lattices and Hardy spaces, Proc. Amer. Math. Soc. 120 (1994), 731–741. [24] A. Kheifets, Nehari’s interpolation problem and exposed points of the unit ball in the Hardy space H 1 , Israel Math. Conf. Proc. 11 (1997), 145–151. [25] I. Klemes, Idempotent multipliers of H 1 (T), Canad. J. Math. 39 (1987), 1223–1234. [26] P. Koosis, Introduction to Hp Spaces, London Math. Soc. Lecture Notes Ser. 40, Cambridge Univ. Press, Cambridge (1980). [27] S. Kwapie´n and A. Pełczy´nski, Some linear topological properties of the Hardy spaces H p , Compositio Math. 33 (1976), 261–288. [28] K. deLeeuw and W. Rudin, Extreme points and extremum problems in H 1 , Paciﬁc J. Math. 8 (1958), 467– 485. [29] J. Lindenstrauss and A. Pełczy´nski, Contributions to the theory of the classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249. [30] D.H. Luecking, Composition operators belonging to the Schatten ideals, Amer. J. Math. 114 (5) (1992), 1127–1145. [31] T. MacGregor and K. Zhu, Coefﬁcient multipliers between Bergman and Hardy spaces, Mathematika 42 (2) (1995), 413–426. [32] M. Mateljevi´c and M. Pavlovi´c, Lp -behaviour of the integral means of analytic functions, Studia Math. 77 (1984), 219–237.

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[33] P.F.X. Müller, On subsequences of the Haar basis in H1 (δ) and isomorphisms between H 1 -spaces, Studia Math. 85 (1987), 73–90. [34] P.F.X. Müller, On subspaces of H 1 isomorphic to H 1 , Studia Math. 88 (1988), 121–127. [35] P.F.X. Müller, The Banach space H 1 (X, d, μ). II, Math. Ann. 303 (1995), 523–544. [36] P.F.X. Müller, A simpliﬁcation in the proof of the non-isomorphism between H 1 (δ) and H 1 (δ 2 ), Studia Math. 150 (2002), 13–16. [37] P.F.X. Müller and G. Schechtman, On complemented subspaces of H 1 and VMO, Lecture Notes in Math. 1376, Springer (1989), 113–126. [38] D.J. Newman, Pseudo-uniform convexity in H1 , Proc. Amer. Math. Soc. 14 (1963), 676–679. [39] R.E.A.C. Paley, A note on power series, J. London Math. Soc. 7 (1932), 122–130. [40] H.P. Rosenthal, On subspaces of Lp , Ann. of Math. 97 (1973), 344–373. [41] W. Rudin, Function Theory in the Unit Ball of Cn , Springer, Berlin (1980). [42] W. Rudin, New Constructions of Functions Holomorphic in the Unit Ball of Cn , CBMS Regional Conf. Ser. in Math. 63, Providence (1986). [43] D. Sarason, Exposed points in H 1 , Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl. 48, Birkhäuser (1990), 333–347. [44] J.H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), 375–404. [45] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer (1993). [46] J.H. Shapiro and P.D. Taylor, Compact, nuclear and Hilbert–Schmidt composition operators on H 2 , Indiana Univ. Math. J. 23 (1973/74), 471–496. [47] W.T. Sledd and D.A. Stegenga, An H 1 multiplier theorem, Ark. Mat. 19 (1981), 265–270. [48] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ (1993). [49] S.J. Szarek and T. Wolniewicz, A proof of Fefferman’s theorem on multipliers, Preprint N.209, Institute of Math. Polish Academy of Sciences (1980). [50] D. Temme and J. Wiegerinck, Extremal properties of the unit ball in H 1 , Indag. Math. (N.S.) 3 (1) (1992), 119–127. [51] D. Vukoti´c, On the coefﬁcient multipliers of Bergman spaces, J. London Math. Soc. 50 (1994), 341–348. [52] P. Wojtaszczyk, On multipliers into Bergman spaces and Nevanlinna class, Canad. Math. Bull. 33 (1990), 151–161. [53] P. Wojtaszczyk, On unconditional polynomial bases in Lp and Bergman spaces, Constr. Approx. 13 (1997), 1–15. [54] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge Univ. Press, Cambridge (1997). [55] T. Wolniewicz, On isomorphisms between Hardy spaces on complex ball, Ark. Mat. 27 (1) (1989), 155–168. [56] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, Inc., New York (1990).

CHAPTER 40

Extension of Bounded Linear Operators M. Zippin Mathematics Department, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) The injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) Separably injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (c) The class of compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (d) Lifting of operators and extension of isomorphisms to automorphisms . . . . . . . . (e) Extension into C(K) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (f) Extension of operators from subspaces of a space of type 2 into a space of cotype 2 . 2. The injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Separably injective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Extension of compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Lifting of operators and extension of isomorphisms to automorphisms . . . . . . . . . . Remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Extension of operators into C(K) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Extension of operators from subspaces of a space of type 2 into a space of cotype 2 . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction In this chapter we discuss various extension problems concerning bounded linear operators in Banach spaces. Let X and Y be Banach spaces and let E be a subspace of X. An operator : X → Y is said to be an extension of an operator T : E → Y if T e = T e for all e ∈ E. T The general problem discussed below is the following: when does every member of a class : X → Y (of the same class)? of operators T : E → Y admit an extension operator T We will introduce here the basic deﬁnitions and describe the main extension problems. Each of these problems is discussed, in detail, in one of the next sections. In this chapter an “operator” means a bounded linear operator. The starting point of all extension theories is the following, well-known, perfect extension theorem for linear functionals, T HE H AHN –BANACH THEOREM ([22,5,8]). Let X be a Banach space over the real or complex ﬁeld F and let E be a subspace of X. Then every bounded linear functional e∗ : E → F can be extended to a linear functional x ∗ : X → F with x ∗ = e∗ . Unfortunately, such perfection is rare; few extension theories which deal with more general operator extension problems can avoid compromises. For example, as shown in Section 6, the Hahn–Banach theorem, valid for operators of rank 1, is false for operators of rank 2 once we replace F by some two-dimensional space F . Any attempt to generalize the Hahn–Banach theorem necessarily requires some restrictions: restrictions on the spaces X and E, restrictions on the range space Y , relaxation of the norm preservation condition T = T or restrictions on the class of operators to be extended. We start with a discussion about conditions on the domain space E which ensure, in full generality, the existence of bounded extensions of operators.

(a) The injective spaces Given a Banach space X, a subspace E of X and λ 1, we say that the pair (E, X) has the λ-extension property (λ-EP, in short) if, for every Banach space Y , every operator T : E → Y admits an extension T : X → Y with T λT . The pair (E, X) is said to have the Extension Property (EP in short) if, for every space Y , every operator T : E → Y : X → Y . admits an extension T These two properties are closely connected to the following: a Banach space E is called λ-injective or a Pλ space, if for every space X containing E there is a projection P of X onto E with P λ. E is called injective (or a P space) if, for every X containing E, there is a projection of X onto E. The relations between the above properties are formulated in the following P ROPOSITION 1.1. Let E be a Banach space and let λ 1. The following three assertions are equivalent. (1.1) E is λ-injective. (1.2) For every space X containing E, (E, X) has the λ-EP.

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(1.3) For every pair of spaces Y ⊃ X, every operator T : X → E admits an extension : Y → E with T λT . T P ROOF. (1.1) follows from either (1.2) or (1.3) by extending the identity T = IE . To prove that (1.1) ⇒ (1.2), let X ⊃ E and T : E → Y . Let P be a projection of X onto E with = T P is the desired extension. It remains to establish (1.1) ⇒ (1.3). Let P λ then T ∗ Γ = Ball(E ) and let j : E → ∞ (Γ ) be the isometric embedding of E into ∞ (Γ ) deﬁned by j (e)(e∗ ) = e∗ (e) for all e ∈ E and e∗ ∈ Ball(E ∗ ). Let Y ⊃ X and let T : X → E be an operator. Let S = j T : X → j (E) ⊂ ∞ (Γ ) and extend S to an operator S : Y → l∞ (Γ ) as follows: For each e∗ ∈ Γ , the functional T ∗ e∗ on X may be extended to a functional ye∗∗ ∈ Y ∗ with ye∗∗ = T ∗ e∗ by the Hahn–Banach theorem. Deﬁne S : Y → ∞ (Γ ) by Sy(e∗ ) = ∗ S is linear, S = T and S extends S. Now deﬁne the desired extension ye∗ (y) then = j −1 P S, where P is a projection of l∞ (Γ ) onto j (E) with T : Y → E of T by T P λ. Proposition 1.1 shows that E is injective if and only if (E, X) has the EP for every X containing E and this property is equivalent to (1.3) with the condition T λT omitted. By using the third property (1.3) it is easily proved that every injective space is λ-injective for some λ 1. A direct consequence of the proof of Proposition 1.1 is that, for every set Γ , the space ∞ (Γ ) is a P1 space. The spaces L∞ (Ω, μ) = L∗1 (Ω, μ) are 1-injective, too. To show this one uses the following, well-known compactness argument: let ω = (Ω1 , Ω2 , . . . , Ωm ) denote a ﬁnite partition of Ω into mutually disjoint measurable sets of positive measure. Let A denote the collection of all such partitions, ordered as follows: (Ω1 , Ω2 , . . . , Ωm ) = ω < γ = (Γ1 , Γ2 , . . . , Γn ) if n > m and each Ωi is a union of members of γ . Clearly, A is directed by < and each subspace Eω = [χΩi ]m i=1 of L∞ (Ω, μ) is isometric to ∞ (1, 2, . . . , m). Hence Eω is a P1 space and, whenever X ⊃ L∞ (Ω, μ) there is a projection Pω of X onto Eω with Pω = 1. It is clear that ω Eω is dense in L∞ (Ω, μ). Regarding L∞ (Ω, μ) as a linear topological space under the ω∗ topology (induced by L1 (Ω, μ)), the unit ball U = Ball(B(X, L∞ (Ω, μ))) of the space of bounded operators from X into L∞ (Ω, μ), under the pointwise ω∗ topology, is compact. Since {Pω } is a net in U (directed by <) it contains a converging subnet with limit P ∈ U . It is easy to check that P is a projection of X onto L∞ (Ω, μ) with P = 1 and hence L∞ (Ω, μ) is a P1 space. The family of P1 spaces will be characterized in Section 2. The problem of characterizing the P spaces is a long standing open problem in Banach space theory. Interesting partial results are discussed in detail in Section 2. The reader may wonder about the phenomenon demonstrated by the above two families of examples of P1 spaces, ∞ (Γ ) and L∞ (Ω, μ), concerning the linear dimension of a Pλ space: these examples are either ﬁnite-dimensional or non-separable. This turns out to be true in general: for every λ 1, a Pλ space is either ﬁnite-dimensional or non-separable. This leads us to the question, what is the relevant formulation of injectivity if one is interested in separable spaces E?

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(b) Separably injective spaces An inﬁnite-dimensional separable Banach space is called separably injective if, for every separable space X containing E, there is a projection P of X onto E. Given λ 1, E is called λ-separably injective if the above condition is satisﬁed with the additional restriction P λ. The following is analogous to Proposition 1.1. P ROPOSITION 1.2. Let E be a separable Banach space and λ 1. Then the following assertions are equivalent: (1.4) E is λ-separably injective. (1.5) For every separable X containing E, the pair (E, X) has the λ-EP. (1.6) Let Y ⊃ X be any separable spaces. Then every operator T : X → E admits an : Y → E with T λT . extension T R EMARK 1.3. The proof of Proposition 1.3 is similar to that of Proposition 1.1 with the following difference: in (1.4) ⇒ (1.6) the projection from l∞ (Γ ) onto j (E) should be replaced by a projection from the separable subspace span {j (E), S(Y )} onto j (E). The proof shows also that a separable space E is separably injective if and only if, for every separable space X containing E, (E, X) has the EP and this property is equivalent to (1.6) with the restriction T λT omitted. Again, using this third property it is easy to show that if X is separably injective then it is λ-separably injective for some λ 1. The characterization problem of separably injective spaces has been completely solved. T HEOREM 1.4 ([66,74]). An inﬁnite-dimensional separable space E is separably injective if, and only if, it is isomorphic to c0 . We will present the solution in Section 3. In the same section we will discuss Rosenthal’s approach to separable injectivity which extends this notion to non-separable spaces. In between linear functionals, for which norm preserving extensions always exist, and bounded operators T : E → Y , which admit extensions T : X → Y , for all Y and every X ⊃ E, only when E is a Pλ space, one may consider an intermediate class of operators.

(c) The class of compact operators An extension theory for compact operators was developed by Lindenstrauss [42]. This theory establishes relations between extension properties of compact operators with intersection properties of balls and the special structure of the domain or range space X. The extension theory of compact operators will be discussed in Section 4. Some extension theorems are closely related to the following properties of the special spaces c0 , 1 and ∞ .

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(d) Lifting of operators and extension of isomorphisms to automorphisms A Banach space Z is said to have the lifting property if, for every pair of spaces X and Y and every operator S from Y onto X, the following condition holds: for every operator : Z → Y such that T = S T. It is well-known and easy to T : Z → X there is an operator T prove (see, e.g., [49], Proposition 2, f.7) that the space 1 (Γ ) has the lifting property and, moreover, whenever S : Y → X is a quotient map then, for every ε > 0, T : 1 (Γ ) → Y may be chosen so that T (1 + ε)T . We will present in Section 5 generalizations of this phenomenon. Equally easy is the following observation which demonstrates the role of 1 as a “universal” separable space with respect to extension of operators into a space Y . P ROPOSITION 1.5. Let W be a subspace of a Banach space X and let Q be an operator from a space Z onto X so that Q = 1 and Q(Ball(Z)) ⊃ δ Ball(X). Let Y be any Banach space and suppose that every operator T : Q−1 (W ) → Y admits an extension T : Z → Y with T λT . Then any S : W → Y admits an extension S : X → Y with S −1 λδ S. P ROOF. Given an operator S : W → Y , consider the operator SQ : Q−1 (W ) → Y . If S : Z → Y extends SQ and S λS, then, since S vanishes on ker Q, S induces an S λδ −1 S. operator S from X ∼ Z/ ker(Q) into Y so that SQ = S and S δ −1 The signiﬁcance of the above fact is the following: since every separable space X is a quotient space of Z = 1 , for any subspace W of X, the understanding of the extension properties of the pair (Q−1 (W ), 1 ) with respect to operators into a space Y sheds light on the same extension properties of the pair (W, X) regarding operators into Y . This will be useful in Section 6 below. The main problems discussed in Section 5 are the following two: P ROBLEM 1.6. Let X = c0 (X = ∞ , resp.). Let E be a subspace of X and let T be an on X? isomorphism from E into X. When can T be extended to an automorphism T It turns out ([47]) that, in all “reasonable” situations an extending automorphism does exist, thus demonstrating the surprising richness of the class of automorphisms on X. The second problem is, in a sense, dual to Problem 1.6. It concerns the possibility of lifting an isomorphism between quotient spaces of 1 to an automorphism on 1 . P ROBLEM 1.7. Let E and F be inﬁnite-dimensional subspaces of 1 and let ϕ : 1 → 1 /E and ψ : 1 → l/F be quotient maps. Suppose that T is an isomorphism from 1 /E onto 1 /F . Does there exist an automorphism T on 1 so that ψ T = T ϕ? Again, surprisingly, the answer is positive and provides a useful tool for extension of operators from subspaces of 1 .

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(e) Extension into C(K) spaces We will see in Section 4 below that, restricting the range space Y to the class of L1 (Ω, μ) preduals, we can nicely extend any compact operator T : E → Y to a compact operator : X → Y with T = T , whenever E ⊂ X. Does the same restriction on Y make T any operator T : E → Y extendable? The answer is negative in general. However, positive partial results can be obtained in the special case of Y = C(K), the space of continuous functions on a compact Hausdorff space K. For example, Lindenstrauss and Pełczy´nski proved the following: T HEOREM 1.8 ([45], Theorem 3.1). Let E be a subspace of c0 and let Y = C(K), for some compact Hausdorff space K. Then any operator T : E → Y admits, for every ε > 0, an extension T : c0 → Y with T (1 + ε)T . This result opens the door onto a new area. We start with the following: D EFINITION 1.9. Let X be a Banach space, let E be a subspace of X and let λ 1. We say that the pair (E, X) has the λ-C(K) Extension Property (λ-C(K) EP, for short) if for every compact Hausdorff space K, any operator T : E → C(K) admits an extension : X → C(K) with T λT . The pair (E, X) has the C(K) EP if it has the λ-C(K) T EP for some λ 1. The restriction of the range space to the family of C(K) spaces provides us with a simple but effective tool in the form of the following. E XTENSION C RITERION 1.10. Let X be a Banach space, let E be a subspace of X and let λ 1. The pair (E, X) has the λ-C(K) EP if and only if there is a ω∗ -ω∗ continuous function ϕ : Ball(E ∗ ) → λ Ball(X∗ ) which extends functionals (i.e., ϕ(e∗ )(e) = e∗ (e) for all e ∈ E and e∗ ∈ Ball(E ∗ )). The proof is elementary. To establish sufﬁciency, let T : E → C(K) be an operator with T = 1. Then the function ψT : K → Ball(E ∗ ) deﬁned by ψT (k)(e) = (T e)(k), k ∈ K, is clearly ω∗ continuous. Hence ψ = ϕ ◦ ψT : K → λ Ball(X∗ ) is ω∗ continuous. De : X → C(K) by the equality (Tx)(k) = ψ(k)(x), then T is linear because, for ﬁne T each k ∈ K, ψ(k) is a linear functional; T λ because ψ(k) ∈ λ Ball(X∗ ) and so extends T because ϕ extends functionals: if e ∈ E then ψ(k) λ and, ﬁnally, T (T e)(k) = ψ(k)(e) = (ϕ ◦ ψT (k))(e) = ψT (k)(e) = (T e)(k) for all k ∈ K. Conversely, assume that (E, X) has the λ-C(K) EP and put K = Ball(E ∗ ) under the ω∗ topology. Let T : E → C(K) denote the natural isometric embedding deﬁned by (T e)(e∗ ) = e∗ (e) for all e∗ ∈ Ball(E ∗ ). Let T : X → C(K) denote an extension of T with T λT . Denoting by δe∗ the point evaluation functional on C(K) at e∗ , we deﬁne ϕ : Ball(E ∗ ) → λ Ball(X∗ ) by ϕ(e∗ ) = T∗ (δe∗ ). It is easy to check that ϕ is ω∗ -ω∗ continuous and extends functionals. An immediate consequence of the Extension Criterion 1.10 is

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C OROLLARY 1.11. Let 1 < p < ∞ and let E be a subspace of p . Then (E, p ) has the 1-C(K) EP. Indeed, the uniform convexity of the unit ball of q (where 1/p + 1/q = 1) yields the existence of a unique functional ϕ(e∗ ) on p which extends any non-zero functional e∗ ∈ Ball(E ∗ ) with ϕ(e∗ ) = e∗ . It is not hard to check that ϕ : Ball(E ∗ ) → Ball(q ) (with ϕ(0) = 0) is ω∗ -ω∗ continuous and hence, by the Extension Criterion, (E, p ) has the 1-C(K) EP. Very little is known about the C(K) EP. In view of the Extension Criterion, the C(K) EP depends on the behavior of the ω∗ topologies of Ball(E ∗ ) and Ball(X∗ ). On the other hand, as is demonstrated in Theorem 6.6 below, the “local” structure of E and X plays a role in connection with the C(K) EP, although this role is much less decisive than the role it plays in the case of the extension of compact operators.

(f) Extension of operators from subspaces of a space of type 2 into a space of cotype 2 One of the most elegant extension theorems is based on the notions of Gaussian type 2 and cotype 2. Let {ψi (t)}∞ i=1 denote a sequence of independent normalized Gaussian random variables on a probability space (Ω, μ). A Banach space X is said to be of Gaussian type 2 (respectively, cotype 2) if there is a constant M > 0 so that, for every ﬁnite sequence {xi } ⊂ X, the following inequality holds: 2 (1.7) xi 2 xi ψi (t) dμ(t) M 2 Ω

(respectively,

xi 2 M 2

2 xi ψi (t) dμ(t)).

(1.8)

Ω

γ

The Gaussian type 2 constant T2 (X) of X is the smallest M for which (1.7) holds. The γ Gaussian cotype 2 constant C2 (X) is the smallest M for which (1.8) holds. The notions of “Gaussian type” and “Gaussian cotype” are equivalent to those of “type” and “cotype”, respectively. These notions are considered in the article “Basic concepts in the geometry of Banach spaces” ([27], Section 8). For more details about these properties the reader is referred to [16], Chapter 11 and pages 249–251. A beautiful by-product of the concepts of Gaussian type 2 and cotype 2 is M AUREY ’ S EXTENSION THEOREM ([53]). Let X be a Banach space of Gaussian type 2 and let Y be a space of Gaussian cotype 2. Then there is a constant C = C(X, Y ) such that, for every subspace E of X, every operator S : E → Y admits an extension S :X → Y with S CS. An immediate corollary of Maurey’s Extension Theorem is the following: let X be a Banach space of type 2 and let E be a subspace of X which is isomorphic to a Hilbert

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space H . Then there is a bounded projection P of X onto E. This corollary is drawn from Maurey’s Extension Theorem by putting Y = E and extending the identity IE . A proof of Maurey’s Extension Theorem is given in Section 7. In most of the results described above, given the Banach spaces E ⊂ X and Y we are interested in extending members T : E → Y of a class of operators to operators T : X → Y . However, in many cases we may be satisﬁed with an extension of an individual operator T into a larger space. The following result suggests a “canonical” way of doing that. be Banach spaces, assume that X is a subspace of X, and L EMMA 1.12. Let X, Y , and X let T : X → Y be an operator. Then there exists a Banach space Y containing Y such that /Y and there is a norm preserving extension T : X →Y of T . Let q X/X is isometric to Y (resp. q) ˜ be the quotient map of X onto X/X (Y onto Y /Y , resp.), let j (resp. j˜) denote (Y to Y , resp.) then there is an isometry I so that the the natural embedding of X into X following diagram commutes: j q X −−−−→ X˜ −−−−→ X/X ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ TA IA TA ˜

j −−−q˜−→ Y /Y Y −−−−→ Y

Moreover, if T is an isomorphism and T x γ x for all x ∈ X then T is an isomor phism with Tx ˜ γ x ˜ for all x˜ ∈ X. The detailed proof of the ﬁrst part can be found in [16], p. 316. The “moreover” part is proved in [35]. Actually, the argument is very simple: we assume that T = 1, deﬁne = (X ⊕ Y )1 /W . We identify (X ⊕ {0})1 ⊕ Y )1 : x ∈ X} and put Y W = {(x, −T x) ∈ (X and Y with its isometric image ({0} ⊕ Y )1 /W . Let f : (X ⊕ Y ) 1 → Y be the quowith X = f |(X⊕{0}) . Since T(x, 0) ∈ (X ⊕ {0}1 /W ⊂ ({0} ⊕ Y )1 /W ∼ Y tient map, then put T 1 /Y so that I q = q˜ T . The norm infor every x ∈ X, there is a unique map I : X/X →Y equalities and the fact that I is an isometry are easy to check. Signiﬁcant parts of the theory described below have been well presented in various books. When discussing these parts we state the results and refer the reader to the books in which detailed proofs are given. N OTATION . We use standard Banach space theory notation as can be found in [48] and [49]. In particular, the ω∗ topology on a bounded subset of a dual space X∗ is the σ (X∗ , X) topology. A net {xα∗ } ⊂ X∗ ω∗ -converges to x ∗ if and only if xα∗ (x) → x ∗ (x) for all x ∈ X. 2. The injective spaces The problem of characterizing the injective spaces seems to be very hard. However, two special cases are completely understood. The ﬁrst one is the case of P1 spaces. Before stating the result, let us mention two properties used in the characterization of P1 spaces.

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A compact Hausdorff space K is called extremally disconnected if the closure of every open subset of K is open. As is well-known (see, e.g., Theorem 17 of [18]) K is extremally disconnected if, and only if, every non-void subset of C(K) which has an upper bound (with respect to the natural order of C(K)) has a least upper bound. The second relevant notion is the binary intersection property: a Banach space X has the binary intersection property if every family of mutually intersecting closed balls has a common point. T HEOREM 2.1. Let X be a Banach space over the real numbers then the following statements are equivalent: (a) X = C(K) where K is an extremally disconnected compact Hausdorff space. (b) X has the binary intersection property. (c) X is a P1 space. P ROOF. Let us start with the history of this result. Goodner [19] and Nachbin [56] independently proved the implication (a) ⇒ (c). Both proved also (c) ⇒ (a) under the assumption that Ball(X) has an extreme point. Nachbin established the equivalence (b) ⇔ (c). Finally, Kelley [37] settled (c) ⇒ (a) in full generality. We outline his argument here. (a) ⇒ (b) Let u denote the constant function 1 on K. A closed ball B(x, r) with center x and radius r in C(K) is exactly the order segment [x − ru, x + ru] = {y: x − ru y x + ru}. If A = [xα , yα ]α∈A is a collection of mutually intersecting segments then, for every α, β ∈ A, there is a zα,β such that xα , xβ zα,β yα , yβ . Hence {xα }α∈A is orderbounded from above and, by the preceding remark, x = sup{xα : α ∈ A} exists and is a common point for all segments in A. (b) ⇒ (c) By Zorn’s lemma, it sufﬁces to show that if Z ⊃ Y, dim(Z/Y ) = 1 and : Z → X T : Y → X is an operator with norm T = 1 then T admits an extension T with T = 1. Let z ∈ Z \ Y and consider the family {B(T y, z − y): y ∈ Y } of balls in X. Any two of these balls intersect because, for y1 , y2 ∈ Y, T y1 − T y2 y1 − y2 z − y1 + z − y2 . Therefore there is a point e common to all balls of this family. Deﬁne T : Z → X by T(az + y) = ae + T y for all az + y ∈ Z. It is easily checked that T extends T and T = T = 1. (c) ⇒ (a) Let Ω0 denote the set of extreme points of Ball(X∗ ) equipped with the ω∗ ω∗ topology and put Ω = Ω 0 = the ω∗ closure of Ω0 . By Zorn’s lemma, there exists a ω∗ open subset K0 of Ω, maximal with respect to the property that −K0 ∩ K0 = ∅. Let ω∗ K = K 0 ; we claim that K is extremally disconnected and X = C(K). The claim easily follows from the following L EMMA 2.2. Suppose that X is a P1 space and let U and V be two ω∗ open subsets of Ω satisfying the two conditions U ∩ V = ∅ = −(U ∪ V ) ∩ (U ∪ V )

(2.1)

and −(U ∪ V ) ∪ (U ∪ V )

is ω∗ dense in Ω.

(2.2)

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ω∗

Let G = ({0} × U ) ∪ ({1} × V ) and let J : X → C(G) denote the natural isometric embedding (J (x)(g) = u(x) if g = (0, u) and J (x)(g) = v(x) if g = (1, v)). Finally, let E ⊂ Ball(C(G)∗ ) denote the set of point-evaluation functionals on C(G). Then J is a ω∗ ω∗ surjective isometry and J ∗ maps −E ∪ E ω∗ homeomorphically onto −(U ∪ V ) ∪ ω∗ ω∗ ω∗ ω∗ ω∗ ω∗ ω∗ ω∗ (U ∪ V ). Moreover, U ∩ V = ∅ = −(U ∪ V ) ∩ (U ∪ V ). Let us ﬁrst show that Lemma 2.2 implies the claim preceding the statement of the lemma. The set −K0 ∪ K0 is weak∗ dense in Ω. Applying Lemma 2.2 to U = K0 and V = ∅ we get that −K ∩ K = ∅ and that K is weak∗ open in Ω. Moreover, the embedding J : X → C(K) is surjective. To see that K is extremally disconnected, pick a ω∗ open subset U of K and ω∗ ω∗ ω∗ ω∗ put V = K \ U . Lemma 2.2 implies that U ∩ V = ∅ hence U is open. P ROOF OF L EMMA 2.2. Clearly, G is a compact Hausdorff space under the topology ω∗ ω∗ induced by U and V . Let e(0, u) ∈ C(G)∗ and e(1, v) ∈ C(G)∗ denote the evaluation ω∗ ω∗ functionals at (0, u) and (1, v), respectively, for every u ∈ U and v ∈ V . Kelley’s argument consists of seven easily checked steps: ω∗ ω∗ (i) J ∗ e(0, u) = u and J ∗ e(1, v) = v for all u ∈ U and v ∈ V . (ii) By a standard extreme point argument, if u0 ∈ U ∩Ω0 and y ∗ is an extreme point of ω∗ (J ∗ )−1 (u0 ) ∩ Ball C(G)∗ then y ∗ = ±e(0, u) for some u ∈ U or y ∗ = ±e(1, v) ω∗ for some v ∈ V . Hence, by (i), u0 = J ∗ y ∗ = ±J ∗ (e(0, u)) = ±u and therefore y ∗ = e(0, u0 ). Similarly, if v0 ∈ V ∩ Ω0 and y ∗ is an extreme point of (J ∗ )−1 (v0 ) ∩ Ball C(G)∗ then y ∗ = e(1, v0 ). (iii) Let B denote Ball(C(G)∗ ). The Krein–Milman theorem and (ii) imply that −1 ∗ −1 (u0 ) ∩ B = e(0, u0 ) and J ∗ (v0 ) ∩ B = e(1, v0 ) J

(2.3)

for all u0 ∈ U ∩ Ω0 and v0 ∈ V ∩ Ω0 . Note that J ∗ |B is a ω∗ continuous function which maps B onto Ball X∗ and, in particular, maps the extreme points e(0, u0 ): u0 ∈ U ∩ Ω0 ∪ e(1, v0 ): v0 ∈ V ∩ Ω0 in a one-to-one fashion onto (U ∪ V ) ∩ Ω0 . The fact that X is a P1 space provides us with a ω∗ continuous inverse as follows: (iv) Since X is a P1 space there exists a projection P of C(G) onto J (X) with P = 1. Let S = J −1 P then S : C(G) → X is a surjective operator with S ∗ (Ball(X∗ )) ⊂ Ball(C(G)∗ ). Moreover, S ∗ is a norm isometry, J ∗ S ∗ = IX∗ and, in view of (2.3), we have that S ∗ u = e(0, u) and S ∗ v = e(1, v)

(2.4)

for all u ∈ U ∩ Ω0 and v ∈ V ∩ Ω0 . (v) S ∗ is a ω∗ homeomorphism of Ball(X∗ ) into Ball(C(G)∗ ) and it maps the ω∗ dense subset [−(U ∪ V ) ∪ (U ∪ V )] ∩ Ω0 of Ω onto a ω∗ dense subset of (−E) ∪ E. Since both Ω and (−E) ∪ E are weak∗ compact, S ∗ (Ω) = (−E) ∪ E.

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(vi) Let u ∈ U ∩ Ω0 and v ∈ V ∩ Ω0 then, by (i) and (2.4), S ∗ J ∗ (e(0, u)) = S ∗ u = e(0, u) and S ∗ J ∗ (e(1, v)) = S ∗ v = e(1, v). It follows that S ∗ J ∗ is the identity on a dense subset of (−E) ∪ E. Consequently, S ∗ is a ω∗ homeomorphism on Ω and J ∗ is, on (−E) ∪ E, the inverse homeomorphism, which maps E onto ω∗ ω∗ ω∗ ω∗ U ∪ V . Since {0} × U and {1} × V and, respectively, E and −E are ∗ ∗ ω ω ω∗ ω∗ ω∗ ω∗ ∗ disjoint ω compact subsets, U ∩ V = ∅ = −(U ∪ V ) ∩ (U ∪ V ). (vii) To see that J is surjective, note that S ∗ (Ball X∗ ) is convex and weak∗ compact and, by (vi), each extreme point of Ball(C(G∗ )) is in S ∗ (Ball X∗ ). Hence S ∗ (Ball X∗ ) = Ball(C(G∗ )) by the Krein–Milman theorem. A standard separation argument shows that J (X) = C(K). This proves Lemma 2.2. The equivalence (a) ⇔ (c) in the complex scalars case was established by Hasumi in [23]. Another case of a Pλ space which has a complete description is that of a ﬁnitedimensional Pλ space X with λ close to 1. It turns out that such a space is “close” to X , independently of the dimension. More precisely, dim ∞ T HEOREM 2.3 ([75,76]). There exists a positive function ϕ(λ), deﬁned for 1 < λ < 1.001, such that, for every n 1 and 1 k n and for every projection P on n∞ with rank(P ) = k and P = λ < 1.001, the Banach–Mazur distance d(P (n∞ ), k∞ ) < ϕ(λ). Moreover, limλ→1 ϕ(λ) = 1. Most of the research on Pλ spaces revolves around the following C ONJECTURE 2.4. Every Pλ space is isomorphic to a C(K) space for some extremally disconnected compact Hausdorff space K. For general results on Pλ spaces the reader is referred to an excellent discussion in [48], pp. 190–194 and [49], I, pp. 105–106. The main topics considered there are summarized in T HEOREM 2.5. (a) ([61]) Let X be an inﬁnite-dimensional Pλ space. Then X contains a subspace isomorphic to ∞ . Moreover, if X contains a subspace isomorphic to c0 (Γ ) for some inﬁnite Γ then it contains also a subspace isomorphic to ∞ (Γ ). (b) ([62]) While every conjugate C(K) space X is a Pλ space (because then X is complemented in X∗∗ = L∞ (μ) which is a P1 space) there exist P1 spaces which are not isomorphic to dual spaces. (c) ([3]) Let X be a C(K) space which is also a Pλ space. Then K contains an extremally disconnected dense open subset G. (d) ([2,25]) If a C(K) space X is a Pλ space for λ < 2 then K is extremally disconnected. For more information about compact Hausdorff spaces K for which C(K) is a Pλ space the reader is referred to [2–4] and [70–72]. The research on the injectivity property restricted to the class of Banach lattices has been more successful because of the extra structure at hand.

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D EFINITION 2.6. A Banach lattice X is called lattice injective (L-injective, in short) if it is complemented by a positive projection in every Banach lattice containing it as a sublattice. The lattice X is called (λ-L)-injective if, for every Banach lattice Y containing X as a sublattice, there is a positive projection P of Y onto X with P λ. Lotz started the research on L-injective lattices and showed ([51]) that every L-injective Banach lattice is (λ-L)-injective for some λ 1. He proved also that, in addition to P1 spaces, the class of (1-L)-injective lattices contains the spaces L1 (μ) and is closed under ∞ -direct sums. Cartright [13] characterized the (1-L)-injective lattices by using the following notion: D EFINITION 2.7. A Banach lattice X is said to have the splitting property if for every positive elements x1 , x1 and y and positive numbers r1 and r2 which satisfy the inequalities xi ri and x1 + x2 + y r1 + r2 there exist positive y1 and y2 in X with y1 + y2 = y such that xi + yi ri for i = 1, 2. Cartwright’s characterization is the following T HEOREM 2.8 ([13]). A Banach lattice X is (1-L)-injective if, and only if, the following two conditions hold: (a) there is a positive projection P of X∗∗ onto X with P = 1. (b) X has the splitting property. He used these ideas to prove that every ﬁnite-dimensional (1-L)-injective lattice is one m(j ) of the spaces ( nj=1 ⊕1 )∞ . Haydon proved in [24] a general representation theorem for (1-L)-injective lattices which uses vector bundles. Unfortunately these isometric tools are not suitable for the isomorphic classiﬁcation problem of (λ-L)-injective lattices. That problem was solved in [50] in the ﬁnite-dimensional case as follows: T HEOREM 2.9. There is a function ϕ(λ) 216 λ27 such that every ﬁnite-dimensional (λ-L)-injective Banach lattice X is order isomorphic to a (1-L)-injective Banach lattice Y with d(X, Y ) ϕ(λ). A Banach lattice is called discrete if it coincides with the band generated by its atoms. Lindenstrauss and Tzafriri extend the ﬁnite-dimensional methods and prove in [50] also that every discrete L-injective Banach lattice is isomorphic to a (1-L)-injective one. In particular, T HEOREM 2.10. Let X be a discrete injective Banach lattice with countably many atoms. Then X is order isomorphic to one of the following six lattices ∞ , 1 , ∞ ⊕ 1 ,

∞ n=1

⊕n1

, 1 ⊕ ∞

∞ n=1

⊕n1

,

∞

∞ n=1

⊕1

. ∞

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The case of injective order continuous lattices was considered by Mangheni [52]. He proved T HEOREM 2.11. Let X be an injective order continuous lattice. Then X is order isomorphic to an L1 (μ) space.

Remarks and open problems R EMARK 2.12. A different approach to the equivalence (a) ⇔ (c) of Theorem 2.1 can be found in [36] and [39]. The nature of a ﬁnite-dimensional Pλ space is a fascinating mystery. The precise statement of the problem is the following: P ROBLEM 2.13. Does there exist a function ϕ(λ) such that, for every n 1, n > k 1 and every projection P on n∞ with rank(P ) = k and P = λ, d(P (n∞ ), k∞ ) ϕ(λ)? A small indication of a positive solution is the result, independently proved by Bourgain [10] and Johnson and Schechtman [29], which states that there is a C(λ) 1 such that every ﬁnite-dimensional Pλ space E contains a subspace F of dim(F ) C(λ)−1 dim(E) ) such that d(F, dim(F ) C(λ). ∞ There are some indications that the problem may be easier to handle if the matrix representing P is an orthogonal projection on n2 . This leads to P ROBLEM 2.14. Does there exist a function ϕ(λ) such that for every n 1 and every projection P of n∞ there is an orthogonal projection Q in some N ∞ space such that Q ϕ(λ) and d(P (n∞ ), Q(N )) ϕ(λ)? ∞ The isomorphic identity of the ﬁnite-dimensional Pλ spaces is unknown even in some natural special cases. P ROBLEM 2.15. Does there exist a function ϕ(λ) such that, for every ﬁnite Abelian group G and any translation invariant projection P on ∞ (G) with P = λ and rank(P ) = k, d(P (∞ (G)), k∞ ) ϕ(λ)? 3. Separably injective spaces The purpose of this section is to present a solution of the characterization problem of separable separably injective spaces and discuss a generalization of this property to nonseparable spaces. It was proved by Sobczyk [66] in 1941 that c0 is separably injective. An elegant proof of this fact, due to Veech [69], is presented above in the introductory article [27]. We will prove the converse, namely,

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T HEOREM 3.1. Let X be a separable separably injective space. Then X is isomorphic to c0 . The proof of Theorem 3.1 is based on the knowledge accumulated in Banach space theory in the sixties and early seventies. It heavily relies on the following facts which are listed in the order in which they are used in the following argument. FACT (a). Every inﬁnite-dimensional quotient space of c0 is isomorphic to a subspace of c0 ([31]). FACT (b). Every inﬁnite-dimensional complemented subspace of c0 is isomorphic to c0 ([57]). FACT (c). The spaces C[0, 1] and C(ωω ) are not separably injective ([2]). FACT (d). A complemented subspace of C[0, 1] which has a non-separable dual is isomorphic to C[0, 1] ([63]). FACT (e). If a complemented subspace E of C[0, 1] has a separable dual E ∗ then E ∗ is isomorphic to 1 ([40]). FACT (f). Let X be a Banach space with a separable dual. If X∗ has a basis then X has a basis such that its biorthogonal functionals form a basis of X∗ ([28]). FACT (g). Let X be a separable space and let W be a subspace of X which is isomorphic to C(ωω ). Then W contains a subspace W0 which is complemented in X and is isomorphic to C(ωω ) ([59]). We prove the theorem by embedding the given separably injective space X into a C(F ) space in a very special position, which makes X (non-linearly) “close” to a certain subspace of C(F ) which is isomorphic to c0 . Our ﬁrst step will be to show that the existence of this embedding, described in the next lemma, implies Theorem 3.1. L EMMA 3.2. Let X be a γ -separably injective space and let 0 < ε < (8γ )−1 . Then there exist a compact metric space F , a subspace Y of C(F ) which is isomorphic to c0 and an embedding i of X into C(F ) such that (1 − ε)x i(x) x for all x ∈ X. Moreover, for each x ∈ X there is a y ∈ Y with i(x) − y εi(x). Step 1. Lemma 3.2 implies Theorem 3.1 P ROOF. Since X is γ -separably injective, then, with μ = (1 − ε)−1 , i(X) is μγ -separably injective and hence there is a projection P of C(F ) onto i(X) with P μγ . Let Q = P |Y ; let 0 = x ∈ X and pick y ∈ Y such that i(x) − y εi(x). Then Qy − i(x) = P y − P i(x) P εi(x) εμγ i(x). But μγ ε < 1/2, therefore, clearly, Q maps Y

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onto i(X). Hence X is isomorphic to a quotient space of c0 , and, by Fact (a), X is isomorphic to a subspace of c0 . Since X is separably injective, it is isomorphic to a complemented subspace of c0 and, by Fact (b), X is isomorphic to c0 , as claimed in Theorem 3.1. It thus remains to prove Lemma 3.2, a long argument which is divided into the next six steps. Step 2. The construction of F The space X, being separably injective, is a complemented subspace of C[0, 1] which, by Fact (c), cannot be isomorphic to C[0, 1]. It then follows from Fact (d) that X∗ is separable, and according to Fact (e), this dual space is isomorphic to 1 . Hence there is a ∗ constant ν > 0 (which depends on X) and a basis {ϕn }∞ n=1 of X such that ν −1

∞ n=1

∞ ∞ |an | a n ϕn |an | n=1

(3.1)

n=1

∞ for all real {an }∞ n=1 . We now use Fact (f) which states that X has a normalized basis {xn }n=1 ∞ with basis constant M, say, and with biorthogonal functionals {θn }n=1 which form a basis of X∗ . Following ([6], Proposition 1), given 0 < δ < εM −1 we deﬁne, for every n 1, Mn = 2(n+2)δ −1 M, Cn = {j/Mn : j is an integer, −Mn < j Mn }, Hn = {θ ∈ Hn is ω∗ Ball(X∗ ): θ (xj ) ∈ Cj for all 1 j n} and F = ∞ n=1 Hn . Since each set ∞ ∗ compact so is F . Moreover, F is a δ net in Ball(X ). Indeed, let ψ = i=1 ai θi ∈ (1 − δ/2)Ball(X∗ ) then |ai | = |ψ(xi )| 1 − δ/2 for all i 1. Fix n 1 and choose the integer −Mn < j (n) Mn for which (j (n) − 1)Mn−1 < an = ψ(xn ) j (n)Mn−1 . Then ∞ ∞ ∞

−1 −1 j (n)M θ j (n)M θ − a − a 2M Mn−1 δ/2. n n n n n n n=1

n=1

n=1

∞

Hence θ = n=1 j (n)Mn−1 θn ∈ Ball(X∗ ), θ − ψ δ/2 and, because θ (xn ) = j (n)Mn−1 for all n 1, θ ∈ F . Equipped with the ω∗ topology, F is a compact metric space and, for each x ∈ X (3.2) (1 − δ)x Sup ψ(x): ψ ∈ F x. It follows that the embedding i : X → C(F ), deﬁned by (ix)(ψ) = ψ(x), satisﬁes the desired inequality (1 − ε)x i(x) x for x ∈ X. (3.3) Step 3. The topological properties of F shown above, {Ci }∞ i=1 is a family of ﬁnite sets of numbers and, for each ψ = As ∞ a θ ∈ F, a = ψ(x ) ∈ Ci for all i 1. Suppose that n 1 and ai ∈ Ci for 1 i n. i i i i i=1 Put A(a1 , a2 , . . . , an ) = θ ∈ F : θ (xi ) = ai for all 1 i n .

Extension of bounded linear operators

1719

Then A(a1 , a2 , . . . , an ) is a clopen subset of F . The collection A = {A(a1, a2 , . . . , an ): n 1, ai ∈ Ci for 1 i n} is clearly a base for the ω∗ topology of F and satisﬁes the following conditions: F=

A(a1 )

and A(a1 , a2 , . . . , an ) =

a1 ∈C1

A(a1 , a2 , . . . , an , an+1 )

an+1 ∈Cn+1

(3.4) and

if an+1 = an+1 are in Cn+1

then

A(a1, a2 , . . . , an , an+1 ) ∩ A a1 , a2 , . . . , an , an+1 = ∅.

(3.5)

This establishes the fact that F is a Cantor set. Step 4. The Szlenk index of F Let 0 < δ < 1. Following [67] we assign to each ordinal α, by transﬁnite induction, a set Fα (δ) as follows: F0 (δ) = F and Fα+1 (δ) = {γ ∈ X∗ : there exists a sequence {yn }∞ n=1 in Ball(X) and γn ⊂ Fα (δ) with ω∗ -lim γn = γ , ω-lim yn = 0 and limn→∞ γn (yn ) δ}. If α is a limit ordinal then Fα (δ) = β<α Fβ (δ). Put η = η(δ, F ) = sup{α: Fα (δ) = ∅}. The following properties are proved in ([67], Lemma 1.5 and Prop. 1.4): L EMMA 3.3. The sets Fα (δ) satisfy the following: Each Fα (δ) is ω∗ compact,

(3.6)

Fα (δ) ⊃ Fα+1 (δ)

(3.7)

for all α

and η < ω1

and Fη (δ) = ∅.

(3.8)

The main purpose of this part of the proof is to show that there is an integer N such that FN+1 (δ) = 0. To prove this claim we need the following fundamental result of Alspach [1]. L EMMA 3.4. Regard X as a subspace of C[0, 1] and let P : C[0, 1] → X be a projection onto X. If Fn (δ) = ∅ for all integers n 1 then there is a subspace Z of C[0, 1] such that Z is isomorphic to C(ωω ) and the restriction P |Z is an isomorphism. The existence of an integer N for which FN+1 (δ) = 0 is a consequence of the following argument: If this is not the case then, by Lemma 3.4, X contains a subspace W (W = P (Z) in the above notation) which is isomorphic to C(ωω ). But Fact (g) establishes the existence of a subspace W0 of W which is still isomorphic to C(ωω ) and is complemented in X. This means that C(ωω ) is separably injective, contradicting Amir’s argument (Fact (c)).

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Step 5. The construction of Y Let N denote the smallest integer for which FN+1 (δ) = ∅ and, forthe sake of simplicity, put Fi = Fi (δ) for 1 i N . Suppose that 0 k N and let θ = ∞ 1 ai θi ∈ Fk − Fk+ . C LAIM . There is an integer m = m(θ ) such that diam A(a1 , . . . , am ) ∩ Fk 4(M + 1)δ, where M is the basis constant of {xn }∞ n=1 . Indeed, if no such m exists, then, for every n 1, diam A(a1 , . . . , an ) ∩ Fk > (M + 1)δ n and there is a γn = ∞ i=1 bi θi ∈ A(a1 , . . . , an ) ∩ Fk for which γn − θ 2(M + 1)δ. Because both γn and θ are in A(a1, . . . , an ), bin = ai for all 1 i n. Hence ∞

2(M + 1)δ γn − θ bin − ai θi . ∞

i=n+1

Pick zn = ∈ Ball(X) such that (γn − θ )(zn ) > (3/2)(M + 1)δ, then ∞ n n i=n+1 ci xi M + 1 and, hence, if yn = (M + 1)−1 ∞ i=n+1 ci xi then yn 1 and −1 |(γn −θ )(yn )| = (M + 1) |(γn − θ )(zn )| (3/2)δ. Moreover, because {θi )∞ i=1 is a basis, lim ∞ a θ = 0 and, therefore, if n is large enough then i i i=n+1 ∞ γn (yn ) (γn − θ )(yn ) − θ (yn ) (3/2)δ − ai θi (yn ) δ. n i=1 ci xi

i=n+1

Finally, because θi (yn ) = 0 for n > i, ω-lim yn = 0. We have thus constructed a sequence ∞ ∗ (γn )∞ n=1 ∈ Fk with θ = ω -lim γn and a sequence {yn }n=1 ⊂ Ball(X) with ω-lim yn = 0 such that γn (yy ) δ. This means, by deﬁnition, that θ ∈ Fk+1 − a contradiction which proves our Claim. The Claim implies that, for each 0 k N , the set Fk − Fk+1 can be covered by a k union of a sequence {Aki }∞ i=1 of members of A for which diam(Ai ∩ Fk ) 4(M + 1)δ. Moreover, because of the set theoretical relations among the members of A, for every k , Ak ⊂ A k ∈ A with Aki ∈ A with diam(Aki ∩ Fk ) 4(M + 1)δ, there is a maximal set A i i i k and A k , say, are either k ∩ Fk ) 4(M + 1)δ. Any two such maximal sets, A diam(A i i j disjoint or identical and the union of these maximal sets covers Fk − Fk+1 . Hence we may assume, without loss of generality, that for each 0 k N there is a sequence {Aki }∞ i=1 of members of A such that the following conditions hold: ∞

Aki ⊃ Fk − Fk+1 ,

(3.9)

i=1

diam Aki ∩ Fk 4(M + 1)δ

for all i 1,

(3.10)

Aki ∩ Akj = ∅ if i = j

(3.11)

for each i 1, Aki is a maximal member of A which satisﬁes (3.10).

(3.12)

and

Extension of bounded linear operators

1721

Since FN = FN − FN+1 is ω∗ compact, we may assume that {AN i } is a ﬁnite sequence. We now claim that if h < k and Ahi ∩ Akj = ∅ then Ahi ⊂ Akj .

(3.13)

Indeed, since Fh ⊃ Fk , diam(Ahi ∩ Fk ) diam(Ahi ∩ Fh ) 4(M + 1)δ. Hence Ahi ⊂ Akj because, by (3.12), Akj is maximal. Let us denote Ahi ∩ Fh by Bih for all 1 h N and i ∞ h 1 and deﬁne G0 = F, GN+1 = ∅ and, for 1 k N, Gk = Fk − k−1 i=1 Bi . Clearly, h=1 N Gk ⊃ Gk+1 for all 0 k N and F = k=0 (Gk − Gk+1 ). Moreover, since Fh ⊃ Fk whenever h < k we have that Gk = Fk −

k−1 ∞

k−1 ∞

h

Ahi ∩ Fh = Fk − Ai ∩ Fk .

h=1 i=1

h=1 i=1

Because Ahi ∩ Fk are relatively open in Fk , Gk is a closed set. We claim that ∞

Aki ∩ Gk ⊃ Gk − Gk+1

for all 0 k N.

(3.14)

i=1

∞ h k / k−1 Bi . We must show that θ ∈ ∞ Indeed, let θ ⊂ Gk − Gk+1 then θ ∈ i=1 i=1 Ai . h=1 ∞ k k ∞ ∞ k / i=1 Bi and therefore, θ ∈ / Bh. Assuming that θ ∈ / i=1 Ai we know that θ ∈ k ∞h=1 h i=1 i But, by (3.9), θ ∈ Gk − (Fk − Fk+1 ) ⊂ Fk+1 . Hence, θ ∈ Fk+1 − h=1 i=1 Bi = Gk+1 – a contradiction which proves (3.14). Finally, if h < k we have that Ahj ∩ Gk = Ahi ∩ Fh ∩ Gk = Bih ∩ Gk = ∅.

(3.15)

Let B = {Aki ∩ Gk : 0 k N, i 1}. We have just proved that B is a decomposition of F into pairwise disjoint closed sets satisfying the condition diam(Aki ∩ Gk ) 4(M + 1)δ for all i 1 and 0 k N . We deﬁne Y = {y ∈ C(F ): y is a constant function on each set Aki ∩ Gk , 0 k N, i 1}. Step 6. i(X) is close to Y In view of Lemma 3.2, we must prove that for every x ∈ X, if i(x) = 1 then there is a y ∈ Y with y − i(x) < ε. In view of (3.10), assuming that 4(M + 1)δ < ε, it sufﬁces to prove that, if f (θ ) ∈ C(F ) satisﬁes sup oscillationAk ∩Gk f (θ ) < ε, i,k

i

(3.16)

then there is a y ∈ Y with y − f < ε. This statement is a consequence of the Approximation Lemma of [74].

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Step 7. Y is isomorphic to c0 Pick a point ωik from each set Aki ∩ Gk and deﬁne ωˆ ik in Y ∗ by ωˆ ik (y) = y(ωik ) for = ωˆ k : ωk ∈ Ω. Because each all y ∈ Y . Now let Ω = {ωik ; 0 k N, i 1} and Ω i i k k y ∈ Y is constant on Ai ∩ Gk , the deﬁnition of ωˆ i does not depend on ωik . Clearly y = for all y ∈ Y . We claim that Ω is a closed set in the ω∗ topology sup{|ω(y)|: ˆ ωˆ ∈ Ω} k(n) ∗ ∗ ∗ ∗ of Y . Indeed, if ω lim ωˆ i(n) = y ∈ Y , by passing to a subsequence, we may assume that lim ωi(n) = ω in F , hence, y(ω) = y ∗ (y) for all y ∈ Y . Moreover, in view of (3.14), The space Ω is a countable ω ∈ Aki ∩ Gk for some k and i and therefore y ∗ = ωˆ ik ∈ Ω. ∗ ∗ ω compact subset of Y and the map j : Y → C(Ω) deﬁned by (j (y))(ωˆ ik ) = ωˆ ik (y) is an which separates points and isometric embedding. Moreover, j (Y ) is a subalgebra of C(Ω) In order to prove that C(Ω) ∼ c0 it sufﬁces to show, contains the unit hence j (Y ) = C(Ω). is by the classical Mazurkiewicz–Sierpinski theorem [54], that the (N +1)-derived set of Ω (k) empty. To establish this fact we will prove the following: if ωˆ ∈ Ω (= the k-th derivative then there is a θ ∈ Gk such that ωˆ = θˆ . Since the number of sets AN is ﬁnite, it would of Ω) i (N+1) = ∅. The assertion is clearly true for k = 0 because G0 = F . then follow that Ω (k+1) be a limit of a sequence (ωˆ k(n) ) Assume for k and proceed by induction. Let ωˆ ∈ Ω i(n) k(n) (k) of distinct points of Ω . We may assume, by the induction hypothesis, that (ω ) ⊂ Gk k(n)

i(n)

k(n) and, by passing to a subsequence, we may assume that limn ωi(n) = θ ∈ F . Therefore, for every y ∈ Y, y(θ ) = ω(y). ˆ Since Gk is closed, θ ∈ Gk . But θ ∈ / Gk − Gk+1 ; indeed, if θ ∈ Gk − Gk+1 then, by (3.14), θ ∈ Aki ∩ Gk for some i. The set Aki is clopen, hence k(n) k(n) ωi(n) ∈ Aki ∩ Gk for large enough n and so, ωˆ ik = ωˆ i(n) eventually, contradicting the fact k(n)

that the points ωˆ i(n) are distinct. It follows that θ ∈ Gk+1 and the assertion is proved. This completes the proof of Lemma 3.2 and thus Theorem 3.1 is proved. Rosenthal [64] has recently suggested the following generalization of separable injectivity which is very natural, in view of Sobczyk’s approach and Proposition 1.2, and opens the door to new problems. Let λ 1. The space E is said to have the λ-Separable Extension Property (λ-SEP, in short) if it satisﬁes (1.6). Note that this deﬁnition makes sense for non-separable spaces E. An example of a 1-SEP space which is not injective is the following: let Γ be an uncountable set and let E = C ∞ (Γ ) = the subspace of ∞ (Γ ) containing all countably supported functions on Γ . E has the 1-SEP because, if X is separable and T : X → C ∞ (Γ ) is any operator, then, clearly, there is a countable Γ0 ⊂ Γ such that T (X) is supported on Γ0 . Since ∞ (Γ0 ) is 1-injective, it follows that, for every Y ⊃ X, T admits an extension C C T : Y → ∞ (Γ0 ) ⊂ C ∞ (Γ ). To see that ∞ (Γ ) is not injective, note that c0 (Γ ) ⊂ ∞ (Γ ). If P is a projection of ∞ (Γ ) onto E then P |c0 (Γ ) is the identity and hence, by [61] there is a subset Γ ⊂ Γ with the same cardinality so that P |∞ (Γ ) is an isomorphism into E. Obvious density character considerations lead to a contradiction. The following are recent results of Rosenthal [64] which generalize Sobczyk’s theorem on the separable injectivity of c0 .

Extension of bounded linear operators

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T HEOREM 3.5. (a) Let {Ej }∞ and let X and Y be j =1 be a sequence of 1-injective spaces ∞ Banach spaces so that X ⊂ Y and Y/X is separable. Put E = ( 1 ⊕Ej )c0 and let ε > 0. Then every operator T : X → E extends to an operator T : Y → E with T (2 + ε)T . (b) Let Z ⊃ E and Z/E be separable. Then there is a projection P of Z onto E with P 2 + ε. ∞ (c) Let {Fj }∞ j =1 be a sequence of spaces with the λ-SEP. Then F = ( j −1 ⊕Fj )c0 has the (λ2 + λ + ε)-SEP for every ε > 0. R EMARK 3.6. Rosenthal originally proved (c) with λ = 1. The extension to λ > 1 is an observation of Oikhberg (see the remark following the proof of Theorem 1.5 of [64]). 4. Extension of compact operators While extension theories for general bounded operators leave basic questions unanswered, once we restrict our attention to the extension of compact operators we arrive at a remarkably complete theory [42]. This study, inspired by earlier work of Grothendieck ([20,21]), establishes beautiful relations among three ﬁelds: (a) Extension of compact operators. (b) The local structure of the domain space (or, respectively, the range space). (c) Intersection properties of balls in the domain space (or, respectively, the range space). Let us start by explaining what we mean by “local structure” and what special kind of local structure is relevant here. The work of Grothendieck in the ﬁfties directed attention to the role that is played in Banach space theory by ﬁnite rank operators and ﬁnitedimensional subspaces. These ideas, developed in the sixties by Lindenstrauss, Pełczy´nski and Rosenthal ([44,46]) culminated in the theory of Lp spaces. Lindenstrauss was the pioneer of this approach in Banach space theory and completed Grothendieck’s work on the connection between extension of compact operators and the L∞ spaces. A Banach space X is a L∞,λ space if it is paved by a family of ﬁnite-dimensional spaces {Xα }α∈I α) with d(Xα , dim(X ) < λ. It is called a L∞ space if it is a L∞,λ space for some λ > 1. We ∞ have seen the role of the binary intersection property of balls in the theory of P1 spaces. Lindenstrauss showed that weaker intersection properties of balls of a space X are equivalent to the existence of compact norm preserving extensions of compact operators from X (or, respectively, almost norm preserving compact extensions of compact operators into the space X). Recall that a Banach space X is said to have the 4-2 intersection property if every collection of four mutually intersecting balls has a non-empty intersection. The main result of [42] is the following fundamental characterization of compact operators extension properties (Y and Z denote Banach spaces). T HEOREM 4.1. The following statements on a Banach space X are equivalent: (4.1) X∗∗ is a P1 space. (4.2) X∗ is an L1 (μ) space. (4.3) Let Z ⊃ Y and let ε > 0. Then every compact operator T : Y → X admits a compact extension T : Z → X with T < (1 + ε)T .

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(4.4) Let Z ⊃ Y and let dim Y 3, dim(Z/Y ) = 1 and ε > 0. Then every operator T : Y → X admits an extension T : Z → X with T < (1 + ε)T . : Z → X∗∗ (4.5) Let Z ⊃ Y . Then every operator T : Y → X admits an extension T with T = T . (4.6) Let Z ⊃ Y and assume that Ball(Z) = conv{Ball(Y ), F } where F ⊂ Z is a ﬁnite set. Then every operator T : Y → X admits an extension T : Z → X with T = T . (4.7) Let Z ⊃ X and let Y be a dual space. Then every operator T : X → Y admits an : Z → Y with T = T . extension T (4.8) Let Z ⊃ X then every compact operator T : X → Y admits a compact extension T : Z → Y with T = T . (4.9) Let Z ⊃ X then every weakly compact operator T : X → Y admits a weakly : Z → Y with T = T . compact extension T (4.10) Let Z ⊃ X and assume that dim(Z/X) = 1 and dim Y 3. Then every operator T : X → Y admits an extension T : Z → Y with T = T . (4.11) The space X has the 4-2 intersection property. (4.12) Every family of mutually intersecting balls {B(xα , rα }α∈A where {xα }α∈A is conditionally norm compact has a non-empty intersection. (4.13) X is a L∞,1+ε space for every ε > 0. The equivalences (4.1) ⇔ (4.2) ⇔ (4.3) ⇔ (4.5) were proved by Grothendieck ([20, 21]). The rest of the equivalences are proved in [42], where the reader may ﬁnd a detailed study of the solution. A summary can be found in [48], pp. 157–160. The implication (4.1) ⇒ (4.13) is a consequence of the principle of local reﬂexivity ([16], p. 178) and Theorem 2.1. The converse implication follows by a ω∗ compactness argument (see [42], pp. 12–13) from the fact that, by (4.13), X is paved by a family {Xα }α∈I of subspaces, with n(α) n(α) = dim Xα < ∞ and limα d(Xα , ∞ ) = 1. Hence, whenever X∗∗ ⊂ Y there is a net {Pα } of projections of Y into X∗∗ with limα Pα = 1 which converges to a projection of Y onto X∗∗ . In the isomorphic setting there is a similar characterization of spaces whose second duals are Pλ spaces in terms of extension properties. T HEOREM 4.2. Let X be a Banach space. Then the following seven assertions are equivalent. Moreover, the validity of (4.14) for some λ 1 is equivalent to the statement “X∗ is a L1 space”. (4.14) X∗∗ is a Pλ space. : Z → X∗∗ with (4.15) Let Z ⊃ Y then every operator T : Y → X admits an extension T T λT . (4.16) Let Z ⊃ X and assume that Y is a dual space. Then every operator T : X → Y : Z → Y with T λT . admits an extension T (4.17) Let Z ⊃ Y and let ε > 0. Then every compact operator T : Y → X admits a : Z → X with T (λ + ε)T . compact extension T (4.18) Let Z ⊃ Y, let dim(Z/Y ) < ∞ and let ε > 0. Then every compact operator T : Y → X admits an extension T : Z → X with T (λ + ε)T .

Extension of bounded linear operators

1725

(4.19) Let Z ⊃ X then every compact operator T : X → Y admits a compact extension : Z → Y with T λT . T (4.20) Let Z ⊃ X then every weakly compact T : X → Y admits a weakly compact extension T : Z → Y with T λT . (4.21) Let Z ⊃ X and let Y be of ﬁnite dimension. Then every operator T : X → Y admits an extension T : Z → Y with T λT . The implications (4.14) ⇔ (4.15) ⇔ (4.16), (4.17) ⇒ (4.18) ⇒ (4.14), (4.16) ⇒ (4.19) and (4.16) ⇒ (4.20) are proved in [42], pp. 12–15. The rest of the implications were established by Johnson (private communication). Johnson’s arguments are presented below. The “Moreover” part is a straightforward consequence of Theorems I and III of [46] and the principle of local reﬂexivity. P ROOF OF (4.14) ⇒ (4.17). Embed X∗∗ in a large enough ∞ (Γ ) space W . Then there is a projection P of W onto X∗∗ with P λ and W has the Bounded Approximation Property (BAP). By local reﬂexivity, X has the BAP and hence, given a compact T : Y → X and ε > 0, there exists a sequence {Tn }∞ n=1 of operators of ﬁnite rank, Tn : Y → X, such that ∞ n : Z → W T = n=1 Tn and Tn < T + 12 ελ−1 . Extend each Tn to an operator T −(n+2) −1 ∗∗ of ﬁnite rank with Tn < Tn + ε2 λ . Then P Tn : Z → X extends Tn and has norm P Tn λTn λTn + ε2−(n+2) . The principle of local reﬂexivity yields the existence of an operator Tn : Z → X of ﬁnite rank which extends Tn and satisﬁes Tn < = ∞ λTn + ε2−(n+1) . The operator T is clearly the desired extension. T n n=1 Clearly, each of the two assertions (4.19) and (4.20) formally implies (4.21). It remains to discuss T HE PROOF OF (4.21) ⇒ (4.14). Let Z denote any C(K) space which contains X isometrically. For any ﬁnite-dimensional subspace E of X∗ , let JE : E → X∗ denote the natural embedding and let TE : X → E ∗ be the operator deﬁned by TE (x)(e) = e(x) for all e ∈ E and x ∈ X. Then, clearly, TE∗ = JE . Use (4.21) to obtain an extension TE : Z → E ∗ with TE λTE = λ. Deﬁne JE = (TE )∗ then JE is a lifting of JE to the L1 (μ)space Z ∗ . If Q denotes the TE

X Z

TE

∗ E

JE

E

X∗

Q ∗

JE Z

natural quotient map of Z ∗ onto X∗ then JE = QJE . We now extend both JE and JE to (non-linear) maps on the whole space X∗ by putting both JE (x ∗ ) = 0 and JE (x ∗ ) = 0 for all x ∗ ∈ X∗ \ E. Consider the set E = {E ⊂ X∗ : E a ﬁnite-dimensional subspace} directed by inclusion. The net {JE }E∈E is pointwise convergent to the identity on X∗ . The standard, well known ω∗ -compactness argument (described in detail in Theorem 5.1 below) yields the existence of a subnet {JE }E∈Δ which converges ω∗ -pointwise to a linear operator T : X∗ → Z ∗ with T λ. Since Q is the dual of the natural embedding of X

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into Z, it is ω∗ continuous. Therefore the equality JE = QJE for all E ∈ E implies that IX∗ = QT . Hence X∗ is λ-complemented in the L1 (μ)-space Z ∗ and therefore X∗∗ is λ-complemented in L∞ (μ) and is thus a Pλ space. R EMARK 4.3. It is shown in [42] that each of (4.14)–(4.20) implies (4.22) Let Z ⊃ X, let dim(Z/X) < ∞ and let ε > 0. Then every compact operator T : X → X admits a compact extension T : Z → X with T (λ + ε)T . The statement (4.22) formally implies (4.23) with X, Z and ε > 0 as above, every uniform limit T : X → X of ﬁnite rank operators admits a compact extension T : Z → X with T (λ + ε)T . While it is unknown whether (4.22) implies (4.14), Pisier constructed in Theorem 3.7 of [60] an example of a separable space X which fails the approximation property (and hence does not satisfy (4.14)) but every uniform limit T : X →X of ﬁnite rank operators on X ∞ ∗ ∗ admits a nuclear representation T = ∞ x i=1 i ⊗ xi with i=1 xi xi λT . Clearly, ∞ if zi∗ ∈ Z ∗ is a Hahn–Banach extension of xi∗ then T = i=1 zi∗ ⊗ xi is an extension of T , T : Z → X and T λT . Consequently, X satisﬁes (4.23). R EMARK 4.4. A separable space X whose second dual is a P1 space is a precise quotient space of C(Δ) where Δ = Cantor’s set ([30]) and contains c0 ([73]). Bourgain and Delbaen ([11]) constructed, for every λ > 1, a separable space Y with Y ∗∗ a Pλ space which is not a quotient space of a C(K) space and does not contain c0 . Bourgain and Pisier ([12]) showed that for every λ > 1 and every separable space E there is a separable L∞,λ space X such that X/E has the RNP and the Schur property. Thus a space whose second dual is a Pλ space need not be isomorphic to a space whose second dual is a P1 space.

5. Lifting of operators and extension of isomorphisms to automorphisms We mentioned in the Introduction the “lifting property” of 1 (Γ ): for every pair of Banach spaces X and Y such that there is a surjective operator q : Y → X, every operator T : 1 (Γ ) → X can be “lifted” through Y , i.e., there is an operator T : 1 (Γ ) → Y such = T . We start with a generalization of the lifting property of 1 (Γ ) due to Lindenthat q T strauss [41], who introduced a beautiful idea of using ω∗ compactness and the abundance of “good”, ﬁnite rank linear operators to prove the linearity of a certain map, which is originally deﬁned by a composition with a non-linear function. T HEOREM 5.1. Let Y and X be Banach spaces such that there is a surjective operator q : Y → X. Assume that kernel(q) is complemented in its second dual. Let E be any : E → Y such that L1 space. Then every operator T : E → X admits a lifting operator T qT = T . P ROOF. Let {Eα }α∈A denote a net, directed by inclusion, of ﬁnite-dimensional subspaces Eα ) = γ0 < ∞. Let Tα = T |Eα then, of E such that α∈A Eα = E and supα d(Eα , dim 1 dim Eα α : Eα → Y such that implies the existence of an operator T the lifting property of 1

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α = Tα and Tα γ T where γ is any number greater than γ0 . Because q is surqT jective, there is a homogeneous mapping ϕ : X → Y (which is not necessarily linear or continuous) such that qϕ(x) = x for all x ∈ X and ϕ(x) μx, where μ depends on q only. Let K = Kernel(q) and, for every r > 0 let B(r) = r Ball(K ∗∗ ). Equipped with the ω∗ topology induced by K ∗ , B(r) is compact and hence, by the Tychonoff theo* rem, the product Π = e∈E B((γ + μ)T e) is compact. This ω∗ compactness implies that the net πα (e) deﬁned, for every α ∈ A and e ∈ E, by πα (e) = Tα (e) − ϕ(T (e)) if e ∈ Eα and πα (e) = 0 otherwise, has a limit point π(e). Let P be the assumed projection is the deof K ∗∗ onto K and deﬁne T by Te = P π(e) + ϕ(T (e)). Let us show that T sired operator. For every α ∈ A and e ∈ E, πα (e) = Tα (e) − ϕ(T (e)) ∈ K and therefore πα (e) ∈ K ∩ B((γ + μ)T e). Therefore, the limit point π(e) ∈ B(γ + μ)T e and P π(e) ∈ K ∩ B((γ + μ)P T e). It follows that T (γ + 2μ)P T e for all e ∈ E. Clearly, T is homogeneous and it remains to show that T is additive. Because α∈A Eα is dense in E and the family {Eα } directed by inclusion, it sufﬁces to show that (f ). For every α such that for every e, f in some subspace Eα0 , T(e + f ) = T(e) + T α (e + f ) − Tα (e) − Tα (f ) = 0 and ϕ(T (e + f )) − ϕ((T (e)) − ϕ(T (f )) ∈ K Eα ⊃ Eα0 , T hence, πα (e + f ) − πα (e) − πα (f ) = −[ϕ(T (e + f )) − ϕ(T (e)) − ϕ(T (f ))]. Passing to the limit, we get that

π(e + f ) − π(e) − π(f ) = − ϕ T (e + f ) − ϕ T (e) − ϕ T (f ) ∈ K. Therefore, since P is a projection onto K,

P π(e + f ) − P π(e) − P π(f ) = − ϕ T (e + f ) − ϕ T (e) − ϕ T (f ) which proves the additivity of T.

The following variation of the lifting property for abstract L-spaces which is an easy consequence of [40] is presented in [35]. P ROPOSITION 5.2. Let Y and X be Banach spaces such that there is a surjective operator q : Y → X. Assume that X has the Radon–Nickodym property. Then, for every abstract =T. L-space E, every operator T : E → X admits an operator T : E → Y such that q T The rest of this section is presented in detail in [49], I, pp. 108–111. We will only state the results, which were originally proved in [47]. It turns out, in response to Problem 1.7 of the Introduction, that every isomorphism between two quotient spaces of 1 which are not isomorphic to 1 can be “lifted” to an automorphism of 1 . T HEOREM 5.3. Let: q1 : 1 → X1 and q2 : 1 → X2 be quotient maps and assume that neither X1 nor X2 is isomorphic to 1 . Let T be an isomorphism of X1 onto X2 . Then there on 1 such that q2 T = T q1 . In particular, Kernel(q1 ) is isomorphic is an automorphism T to kernel(q2 ). Problem 1.6 of the Introduction has the following complete solutions, both for c0 and ∞ .

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T HEOREM 5.4. Let Y and Z be subspaces of c0 , both of inﬁnite codimension. Assume that T : Y → Z is a surjective isomorphism. Then T extends to an automorphism of c0 . The analogous statement for the space ∞ uses the well-known terminology of Fredholm operators. An operator T on a Banach space X is called a Fredholm operator if both spaces kernel T and X/T (X) are ﬁnite-dimensional. The integer i(T ) = dim(kernel(T )) − dim(X/T (X)) is called the index of T . T HEOREM 5.5. Let Y and Z be subspaces of ∞ of inﬁnite codimension and let T be an isomorphism of Y onto Z. Then (a) T can be extended to an automorphism of ∞ if both ∞ /Y and ∞ /Z are nonreﬂexive. (b) T cannot be extended to an automorphism of ∞ if exactly one of the spaces ∞ /Y and ∞ /Z is reﬂexive. (c) If both ∞ /Y and ∞ /Z are reﬂexive then every extension T of T to ∞ is a Fredholm operator. Its index i(T) is an integer valued invariant i(T ) of T and does not depend on the particular extension. The operator T extends to an automorphism of ∞ if and only if i(T ) = 0. Remarks and open problems R EMARK 5.6. The lifting property characterizes the spaces 1 (Γ ) [38] up to isomorphism. For example, in the case of a separable space E with the lifting property, let X = E, q : 1 → E be a quotient map and let I : E → X be the identity. If I: E → 1 lifts I so that q I = I then, clearly, I is an isomorphism of E into 1 and I q is a projection of 1 onto a subspace isomorphic to E. Hence E is isomorphic to 1 , by [57]. R EMARK 5.7. Lindenstrauss’s argument presented in the proof of Theorem 5.1 was originally used in [41] to construct the ﬁrst example of a subspace U of 1 which is not complemented in any dual space and which does not have an unconditional basis. Let q : 1 → L1 be the natural quotient mapping which maps u2n +k−1 , the (2n + k − 1)k th natural basis element, onto the indicator function of the interval [ k−1 2n , 2n ] where n n = 0, 1, 2, . . . , k = 1, 2, . . . , 2 . If U = kernel(q) were complemented in a dual space then, by Theorem 5.1, the identity I : L1 → L1 could be lifted through 1 thus leading to the contradiction that L1 is isomorphic to a complemented subspace of 1 . The subspace U of 1 is a L1,2 space. It is not known whether the pair (U, 1 ) has the C(K) EP (see Section 6 below for the deﬁnition). R EMARK 5.8. Lemma 6.5 and Theorem 6.4 below provide tools which may replace Lemma 1 of [47] for the purpose of extending Theorem 5.4 to the case of c0 (Γ ) with Γ uncountable. We believe that such an extension is valid but have not checked it. R EMARK 5.9. Theorem 5.4 is false if we replace c0 by p (1 p 2), Lp (1 p ∞) and C(K) (if K is a compact Hausdorff space for which C(K) is not isomorphic to c0 ),

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see [47]. The validity of this statement for 1 is easy to see. Let q1 : 1 → c0 be a quotient map and consider the quotient map q2 : 1 ⊕ 1 → c0 ⊕ 1 deﬁned by q2 = q1 ⊕ I where I denotes the identity on 1 . It follows that there is an isomorphism T of kernel q1 onto a subspace Y of 1 = 1 ⊕ 1 such that 1 /Y ∼ c0 ⊕ 1 . Since c0 is not isomorphic to c0 ⊕ 1 , T cannot be extended to an automorphism of 1 . However, in certain important cases, isomorphisms admit extensions to automorphisms. A stronger version of Theorem 5.3 is true if we know more about X1 and X2 , for example: T HEOREM 5.10 ([43]). For i = 1, 2 let qi : 1 → Xi be a quotient map onto a L1 space Xi . Let Ei = kernel(qi ) and assume that Ei is inﬁnite-dimensional. Then E1 is isomorphic to E2 if and only if X1 is isomorphic to X2 . Moreover every isomorphism of E1 onto E2 extends to an automorphism on 1 . It is unknown if Theorems 5.3 and 5.4 characterize 1 and c0 respectively. P ROBLEM 5.11. Let X be an inﬁnite-dimensional separable Banach space. Suppose that for every separable space Y which is not isomorphic to X and for every pair of surjective operators q1 : X → Y and q2 : X → Y there is an automorphism T on X with q1 = q2 T . Is X isomorphic to either 1 or 2 ? P ROBLEM 5.12. Let X be a separable inﬁnite-dimensional Banach space. Assume that, for every pair of isomorphic subspaces Y and Z of X with inﬁnite codimension there is an automorphism T of X such that T (Y ) = Z. Is X isomorphic to either 2 or c0 ? R EMARK 5.13. Recently Ferenczi ([17]) has constructed an example of a space X and its subspace E such that any isomorphic embedding T of E into X is of the form T = J + S, where J is the natural isometric embedding of E into X and S is strictly singular. It is therefore conceivable that the answers to Problems 5.11 and 5.12 may be negative.

6. Extension of operators into C(K) spaces We have seen the effectiveness of the simple Extension Criterion 1.10 in establishing the fact (Corollary 1.11) that, for every 1 < p < ∞ and every subspace E ⊂ p , the pair (E, p ) has the 1-C(K) EP. Another family of classical pairs of spaces which share this property is {(E, L1 ): E ⊂ L1 and dim(E) < ∞}. E XAMPLE 6.1. Let E be any ﬁnite-dimensional subspace of L1 = L1 [0, 1]. Then (E, L1 ) has the 1-C(K) EP. P ROOF. Pick a basis {fj }nj=1 of E and let {fj∗ }∞ j =1 denote the corresponding biorthogonal functionals. Let J : E → L1 be the natural isometric embedding then J ∗ is a ω∗ continuous mapping of Ball(L∞ ) onto Ball(E ∗ ). Let Σ denote the Lebesgue σ -ﬁeld in

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[0, 1] and put i(A) = 1A for A ∈ Σ then i(Σ) ⊂ Ball L ψ : i(Σ) → Ball(E ∗ ) by ∞n. Deﬁne ∗ ψ(1A ) = J (1A ) then, as is easily checked, ψ(1A ) = j =1 ( A fj dμ)fj∗ for all A ∈ Σ. It is proved in [65] that the set function Ψ : ψ(i(Σ)) → (i(Σ) deﬁned by Ψ (e∗ ) = ψ −1 (e∗ ) admits a ω∗ continuous selection ϕ such that ψ(ϕ(e∗ )) = e∗ for all e∗ ∈ ψ(i(Σ)). Now consider the subset E = {2 · 1A − 1: A ∈ Σ} which is nothing but the subset of extreme points of Ball(L∞ ). Let β : E → Ball(E ∗ ) be deﬁned by β = J ∗ |E . Then clearly, 1 β = 2ψ − nj=1 ( 0 fj dμ)fj∗ . It is easily checked that β(E) = Ball(E ∗ ) and hence, the set function β −1 = J ∗(−1) ∩ E admits the obvious ω∗ continuous selection 2ϕ − 1 from Ball E ∗ into E which extends functionals. It follows from the Extension Criterion 1.10 that (E, L1 ) has the 1-C(K) EP. The Extension Criterion 1.10 may create the false impression that “almost” every pair (E, X) has the C(K) EP. Let us discuss some counter examples. It is obvious that if H and K are compact Hausdorff spaces and E is a subspace of C(H ) which is isomorphic to C(K) then (E, C(H )) has C(K) EP if and only if E is complemented in C(H ). An example of subspaces E of C[0, 1] which are isomorphic to C[0, 1] but uncomplemented was constructed in [2]. One can use Proposition 1.5 to construct a subspace F of 1 such that (F, 1 ) does not have the C(K) EP. Indeed, with the notations of Proposition 1.5, let X = C[0, 1], let E be the above mentioned example of Amir and denote by Q a quotient mapping of 1 onto X. Let F = Q−1 (E), then, by Proposition 1.5, the pair (F, 1 ) does not have the C(K) EP. E XAMPLE 6.2. There is a two-dimensional subspace E of C[−1, 1] such that (E, C[−1, 1]) does not have the 1-C(K) EP. P ROOF. Let E be the subspace of C[−1, 1] spanned by the functions f (t) = t 2 if −1 t 0 and f (t) = t if 0 t 1 and g(t) = −t if −1 t 0 and g(t) = t 2 if 0 t 1. If (E, C[−1, 1]) has the 1-C(K) EP then, by the Extension Criterion 1.10, there is a ω∗ continuous function ϕ : Ball(E ∗ ) → Ball(C[−1, 1]∗ ) which extends functionals. Let δγ denote the point evaluation functional on C[−1, 1] at γ ∈ [−1, 1] and let δγ0 denote its restriction to E. It is easy to check that ϕ(δγ0 ) must be δγ for all γ close to 1 and 0 0 all γ close to −1. But δ−1 = δ10 , δγ0 → δ10 , and δγ0 −→ δ−1 hence δ1 = limγ →1 δγ = γ →1

γ →−1

limγ →1 ϕ(δγ0 ) = limγ →−1 ϕ(δγ0 ) = limγ →−1 δγ = δ−1 a contradiction. Note that this argument shows that for every subspace F ⊃ E of C[−1, 1], if δ1 |F = δ−1 |F then (E, F ) does not have the 1-C(K) EP. Moreover, it follows that the identity I : E → E ⊂ C(Ball E ∗ ) does not admit a norm preserving extension T : C[−1, 1] → E because, if such extension T existed, then ϕ = T ∗ |Ball(E ∗ ) would be a functional extending ω∗ continuous function into Ball(C[−1, 1]∗ ). Let us now take a look at Theorem 1.8 of the Introduction from the point of view of the Extension Criterion. We know that for every ε > 0 and every subspace E of c0 , the pair (E, c0 ) has the (1 + ε)-C(K) EP. This means that there exists a ω∗ continuous, functional extending function ϕε : Ball(E ∗ ) → (1 + ε)Ball(1 ). It turns out that the following stronger statement is true.

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T HEOREM 6.3 ([77]). Let ε > 0 and let E be a subspace of c0 . Then there exists a ω∗ continuous functional extending function ϕε : Ball(E ∗ ) → (1 + ε)Ball(1 ) such that ϕ(e∗ ) (1 + ε)e∗ for all e∗ ∈ Ball(E ∗ ). The proof of Theorem 6.3 is based on Michael’s Continuous Selection Theorem [55]. The main part of the proof is the following: it is shown that for every subspace E ⊂ c0 , the carrier Φ of Ball(E ∗ ) into the convex subsets of (1 + ε)Ball(1 ) deﬁned by Φ(0) = {0} and, for e∗ = 0, Φ(e∗ ) = {x ∗ ∈ 1 : x ∗ extends e∗ and x ∗ (1 + ε)e∗ } is ω∗ l.s.c. Michael’s theorem ensures the existence of a ω∗ continuous selection ϕ of Φ, which is the desired function. This completes the proof of Theorem 6.3. Theorem 6.3 is an essential tool in the proof of the following generalization of Theorem 1.8. T HEOREM 6.4 ([32]). Let Γ be an uncountable set, let ε > 0 and let E be a subspace of c0 (Γ ). Then the pair (E, c0 (Γ )) has the (1 + ε)-C(K) EP. P ROOF. The ﬁrst step is a decomposition lemma (which remains true in any space with an extended shrinking basis). L EMMA 6.5 ([32, Lemma 2]). Let Γ be an uncountable set and let E be a subspace of c0 (Γ ). Then Γ can be decomposed into a family {Γα }α∈A of pairwise disjoint countable sets such that if α ∈ A and Eα = {x ∈ E: support(x) ⊂ Γα } then, for every x ∈ E, the restriction x|Γα of x to Γα is in Eα . To prove Theorem 6.4, let {Γα }α∈A and {Eα }α∈A be the decomposition guaranteed by Lemma 6.5. The dual space E ∗ is identical with the 1 (A)-direct sum of {Eα∗ }α∈A . Since each Eα is a subspace of c0 (Γα ) = c0 , by Theorem 6.3, there exists a ω∗ continuous function ϕα : Ball(Eα∗ ) → (1 + ε)Ball(1 (Γα )) which extends functionals and satisﬁes the inequality ϕα (eα∗ ) (1 + ε)eα∗ for every eα∗ ∈ Ball(Eα∗ ). Deﬁne ϕ : Ball(E ∗ ) → ∗ (1 + ε)Ball(1 (Γ )) by ϕ(e ) = α∈A ϕα (e∗ |Eα ) where e∗ ∈ Ball(E ∗ ). It is easily checked that ϕ is ω∗ continuous and extends functionals. The Extension Criterion 1.10 now gives the desired conclusion. The special role of 1 in extension of operators into C(K) spaces has been explained in Proposition 1.5. This result demonstrates the importance of examining those subspaces E of 1 for which (E, 1 ) has the C(K) EP. At the moment, the most general class of subspaces of 1 which are known to share this property is the family of ω∗ closed subspaces of 1 . This statement is a special case of the following ∞ T HEOREM 6.6 ([33]). Let {X n }n=1 be ﬁnite-dimensional spaces let ε > 0 and let E be a ∗ ω closed subspace of X = ( Xn )1 , regarded as the dual of ( Xn∗ )c0 . Then (E, X) has the (3 + ε)-C(K) EP. Moreover, if E has the approximation property, then (E, X) has the (1 + ε)-C(K) EP.

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Remarks and open problems Because so little is known about the C(K) Extension Property, there are many open problems. P ROBLEM 6.7. Let W be a subspace of a reﬂexive space F . Does (W, F ) have the C(K) EP? What if F is superreﬂexive? What if F is Lp , 1 < p = 2 < ∞? P ROBLEM 6.8. Let E be a reﬂexive subspace of a separable space X. Does (E, X) have the C(K) EP? What if E is only isomorphic to a conjugate space? What if E = 1 ? (The separability assumption is needed here because the Dunford–Petis property of ∞ and [61] imply that if E is a separable reﬂexive subspace of ∞ and J : E → C([0, 1]) is an isometric embedding then J cannot be extended to an operator from ∞ into C([0, 1])). If E is a subspace of c0 , then (E, c0 ) has the (1 + ε)-C(K) EP for every ε > 0 ([45]) but need not have the 1-C(K) EP ([32]). We do not know if this phenomenon can occur in the setting of “nice” spaces: P ROBLEM 6.9. If X is a reﬂexive smooth space and (E, X) has the (1 + ε)-C(K) EP. For every ε > 0, does (E, X) have the 1-C(K) EP? The following observation gives an afﬁrmative answer to Problem 6.9 in a special case. P ROPOSITION 6.10 ([33]). If X is uniformly smooth and (E, X) has the (1 + ε)-C(K) EP for every ε > 0, then (E, X) has the 1-C(K) EP. P ROOF. Since X is uniformly smooth, given ε > 0 there exists δ > 0 so that if x ∗ , y ∗ in X∗ and x in X satisfy x ∗ = x = 1 = x ∗ , x = y ∗ , x with y ∗ < 1 + δ, then x ∗ − y ∗ < ε. Letting φn : Ball(E ∗ ) → (1 + n−1 )Ball(X∗ ) be a weakly continuous extension mapping and letting f : Sphere E ∗ → Sphere X∗ be the (uniquely deﬁned, by smoothness) Hahn–Banach extension mapping, we conclude that

lim sup φn x ∗ − f x ∗ : x ∗ ∈ Sphere E ∗ = 0.

n→∞

∗ That is, {φn |Sphere E ∗ }∞ n=1 is uniformly convergent to f |Sphere E . Since each φn is weakly continuous, so is f |Sphere E ∗ . If E is ﬁnite-dimensional, then clearly the positively homogeneous extension of f to a mapping from Ball E ∗ into Ball X∗ is a weakly continuous extension mapping. So assume that E has inﬁnite dimension. But then Sphere E ∗ is weakly dense in Ball E ∗ , so by the weak continuity of the φn ’s and the weak lower semicontinuity of the norm, we have

sup φn x ∗ − φm x ∗ : x ∗ ∈ Ball E ∗

= sup φn x ∗ − φm x ∗ : x ∗ ∈ Sphere E ∗ ,

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which we saw tends to zero as n, m tend to inﬁnity. That is {φn }∞ n=1 is a uniformly Cauchy sequence of weakly continuous functions and hence its limit is also weakly continuous. It is apparent from the proof of Proposition 6.10 that the 1-C(K) EP is fairly easy to study in a smooth reﬂexive space X because every extension mapping from Ball E ∗ to Ball X∗ is, on the unit sphere of E ∗ , the unique Hahn–Banach extension mapping. Let us examine this situation a bit more in the general case. Suppose E is a subspace of X and let A(E) be the collection of all norm one functionals in E ∗ which attain their norm at a point of Ball E. The Bishop–Phelps theorem [7,15] says that A(E) is norm dense in Sphere E ∗ , hence, if E has inﬁnite dimension, A(E) is weak∗ -dense in Ball E ∗ . Therefore (E, X) has the 1-EP if and only if there is a weak∗ -continuous Hahn–Banach selection mapping φ : A(E) → Ball X∗ which has a weak∗ -continuous extension to a mapping φ ω∗ from A(E) = Ball E ∗ to Ball X∗ , since clearly φ will then be an extension mapping. The existence of φ is equivalent to saying that whenever {xα∗ } is a net in A(E) which weak∗ converges in E ∗ , then {φxα∗ } weak∗ converges in X∗ (see, for example, [9], I.8.5). Now, when X is smooth, there is only one mapping φ to consider, and in this case the above discussion yields the next proposition when dim E = ∞ (when dim E < ∞ one extends ω∗ from Sphere E ∗ = A(E) to Ball E ∗ by homogeneity). P ROPOSITION 6.11 ([33]). Let E be a subspace of the smooth space X. The pair (E, X) fails the 1-C(K) EP if and only if there are nets {xα∗ }, {yα∗ } of functionals in Sphere X∗ which attain their norm at points of Sphere E and which weak∗ converge to distinct points x ∗ and y ∗ , respectively, which satisfy x ∗ |E = y ∗ |E . An immediate, but surprising to us, corollary to Proposition 6.11 is: C OROLLARY 6.12 ([33]). Let E be a subspace of the smooth space X. If the pair (E, X) fails the 1-C(K) EP, then there is a subspace F of X of codimension one which contains E so that (F, X) fails the 1-C(K) EP. P ROOF. Get x ∗ , y ∗ from Proposition 6.11 and set F = span E ∪ (ker x ∗ ∩ ker y ∗ ).

P ROBLEM 6.13. Is Corollary 6.12 true for a general space X? In contrast to Corollary 1.11 we have the following P ROPOSITION 6.14 ([33]). For 1 < p = 2 < ∞, Lp has a subspace E for which (E, Lp ) fails the 1-C(K) EP. It is stated in [45] that, for every subspace E of c0 , not only does (E, c0 ) have the (1 + ε)-C(K) EP but, in addition, if Y is an L1 (μ)-predual, then every operator T : E → Y : c0 → Y with T (1 + ε)T . The proof of Theorem 6.6 points in an extends to a T analogous direction: if E is a ω∗ closed subspace of 1 then, in addition to (E, 1 ) having the (3 + ε)-C(K) EP, for every L∞,λ space Y , every operator T : E → Y extends to an operator T : 1 → Y with T| λ(3 + ε)T .

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P ROBLEM 6.15. Let X be a Banach space, E ⊂ X and let (E, X) have the C(K) EP. Let Y be a L∞ space. Does every operator T : E → Y extend to T : X → Y ?

9

A different point of view on certain operator extension problems is that of splitting twisted sums. Recall that a Banach space Z is called a twisted sum of spaces Y and X (denoted by Z = Y X) if j

q

0→Y → Z → X→0

9

is a short exact sequence, i.e., j is an isometric embedding, q is a quotient map onto X and j (Y ) = kernel(q). We will identify j (Y ) with Y below. The twisted sum Y X is said to split if there is an operator T : X → Y X such that qT = IX . Note that in this case P = I − T q is a projection of Y X onto j (Y ). Conversely, if P is a projection of Y X onto j (Y ) deﬁne the operator T : X → Y X for all x ∈ X by T x = z − P z where z ∈ Y X is any element for which q(z) = x (T is well deﬁned because, if q(w) = 0 then P w = w). 9

9

9

9

9

9

P ROPOSITION 6.16 ([35]). Let F be a Banach space and let ϕ : 1 (Γ ) → F be a quotient map. Let E = kernel(ϕ). Then, for every Banach space Y , the following two assertions are equivalent (a) Every bounded operator S : E → Y extends to a bounded operator S : 1 (Γ ) → Y . (b) Every twisted sum Y F splits.

9

9

P ROOF. (b) ⇒ (a) By Lemma 1.12, given an operator S : E → Y , there is a twisted sum Y F such that S extends to an operator S1 : 1 (Γ ) → Y F . Let q be the quotient map of Y F onto F then, since (b) implies that Y F splits, there exists an operator T : Y → Y F such that qT = IF . Let P = I − T q, be the above mentioned projection of Y F onto Y . Then S = P S1 is the desired extension of S. (a) ⇒ (b) Let Y F be any twisted sum and let q : Y ∈ F → F be the quotient map with kernel(q) = Y . Given μ > 1, the lifting property of 1 (Γ ) implies the existence of an operator ψ : 1 (Γ ) → Y F so that qψ = ϕ where ψ < μ. Because E ∈ kernel(ϕ), ψ(E) ∈ kernel(q) = Y . Let ψ0 = ψ|E : E → Y and use (a) to extend ψ0 to an operator ψˆ 0 : 1 (Γ ) → Y . The operator ψ − ψˆ 0 maps 1 (Γ ) into Y F . Since kernel (ψ − ψˆ 0 ) ⊃ E = kernel(ϕ), we may deﬁne the operator u : F → Y F by u(e) = (ψ − ψˆ 0 )(x) if e = ϕ(x) for some x ∈ 1 (Γ ). It follows that uϕ = ψ − ψˆ 0 and qu = 0. Hence, quϕ = qψ = ϕ. But ϕ(1 (Γ )) = F , therefore qu = IF and Y F splits. 9

9

9

9

9

9

9 9

9

In the terminology of twisted sums, the ﬁrst part of Problem 6.8 has a positive solution if so does the following

9

P ROBLEM 6.17. Let F be a reﬂexive space and Y = C(K) for some compact Hausdorff space. Does every twisted sum Y F split?

Extension of bounded linear operators

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9

Indeed let F be a reﬂexive space, let W ⊂ F and let Γ be so large that there is a quotient map Q : 1 (Γ ) → F . Put E = Q−1 (W ). By Proposition 1.5, (W, F ) has the C(K) EP if (E, 1 (Γ )) does. But, by Proposition 6.16, if every twisted sum Y F splits then (E, 1 (Γ )) has the C(K) EP. The above mentioned algebraic point of view was a useful tool in [35] to show that if q : L1 → Y is a quotient mapping and c0 ⊂ Y then kernel(q) is not a L1 space. Johnson [26] went deeper into the algebra involved in [35] and showed that kernel(q) does not have the GL-lust. R EMARK 6.18. Recently Kalton [34] proved the following partial inverse of Theorem 6.6: let E be a subspace of 1 such that (E, 1 ) has the C(K) EP and 1 /E has an unconditional ﬁnite-dimensional decomposition. Then there is an automorphism T on 1 such that T (E) is ω∗ closed.

7. Extension of operators from subspaces of a space of type 2 into a space of cotype 2 The purpose of this section is to prove Maurey’s Extension Theorem stated in Section 1 (f). The proof presented here is based on Maurey’s argument [53] and the approach of [68]. There is a conceptual difference between this extension theorem and the extension theorems of Sections 4 and 6. In the above sections the desired extension is into a space Y which is paved by a family of ﬁnite-dimensional subspaces {Yα } directed by inclusion, d(α) where α Yα = Y , and each Yα is ∞ with d(α) = dim Yα . Since each Yα is a P1 space, the extension of an operator into Yα is trivial. The difﬁculty is in the passage from the d(α) ﬁnite-dimensional ∞ to the inﬁnite-dimensional Y . On the other hand, in the present case, the ﬁnite-dimensional construction is where most of the action is while the passage to the inﬁnite-dimensional case is achieved by an ultraproduct argument. This is demonstrated in the ﬁnal part of the proof of the following L EMMA 7.1. Let X and Y be a Banach spaces and let E ⊂ X. Let c > 0 and let T : E → Y be an operator satisfying the following condition: n ∗ 2 ∗ 2 ⊂ E and {xi }ni=1 ⊂ X, if m (7.1) for any ﬁnite sets {ei }m i=1 |x (ei )| i=1 |x (xi )| i=1 m n ∗ ∗ 2 1/2 2 1/2 for every x ∈ X then ( i=1 T ei ) cT ( j =1 xj ) . Then there is a Hilbert space H and an operator S : X → H such that S c and T e Se for all e ∈ E. P ROOF. Pick a ﬁnite-dimensional subspace F of X and let M = Ball(F ∗ ). Let m n ∗ ∗ 2 ∗ f (xi )2 , K = ϕ:M → R | ϕ f = f (ei ) − i=1

where m i=1

{ei }m i=1

⊂ E ∩ F,

T ei 2 > c2

n i=1

i=1

{xi }ni=1

xi 2 .

⊂ F and

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M. Zippin

It is easy to verify that K is a convex subset of C(M) and, in view of (7.1), each ϕ ∈ K attains a positive maximum in M. Hence if G = {ψ ∈ C(M): ψ(f ∗ ) < 0 for all f ∗ ∈ C(M)} then G ∩ K = ∅ and, by the standard separation theorem, there is a non-trivial separating measure μF ∈ C(M)∗ and α ∈ R such that μF (ψ) < α μF (ϕ) for all ψ ∈ G and ϕ ∈ K. Since G is the negative cone, α 0 and hence μF is a positive measure on M. It follows that 0 < β = sup{ M |f ∗ (f )|2 dμF (f ∗ ): f ∈ Ball(F )} μF (M). Put νF = β −1 c2 μF then ∗ 2

f (f ) dνF f ∗ : f ∈ Ball(F ) . (7.2) c2 = sup M

Now deﬁne an operator SF : F → L2 (M, νF ) by (SF f )(f ∗ ) = f ∗ (f ). It follows from (7.2) that SF c. Moreover, suppose that f ∈ E ∩ F and Tf > c and let x ∈ Ball F . Then the function ϕ(f ∗) = |f ∗ (f )|2 − |f ∗ (x)|2 belongs to K and, by the separating property of νF , SF f 2 − M |f ∗ (x)|2 dνF (f ∗ ) = M ϕ(f ∗ ) dνf (f ∗ ) > 0. Passing to the sup on Ball F we get that SF f 2 sup{ |f ∗ (x)|2 dνF (f ∗ ): x ∈ Ball F } = c2 . It follows that SF (f ) c whenever Tf > c and hence Tf SF f cf

for all f ∈ F ∩ E.

(7.3)

We turn now to the ultra product argument. We use here only basic facts which can be found in Section 9 of [27]. Let I = {F ⊂ X: F a ﬁnite-dimensional subspace of X} and let U be an ultraﬁlter on*I which contains, for each ﬁnite-dimensional F ⊂ X the set {Z ∈ I : Z ⊃ F }. Let H = ( i L2 (νF ))U be the ultra product of the L2 (νF ) spaces constructed above. As is well-known, H = L2 (μ) for some measure μ, by the stability of Lp spaces under ultraproducts (see, e.g., [16], Chapter 8). Let SF : X → L2 (νF ) be deﬁned by SF x = SF x if x ∈ F and SF x = 0 otherwise, and deﬁne S : X → H by Sx = { SF x}F ∈I . Then, clearly, S is a linear operator and Sx cx for all x ∈ X. Moreover, because T x SF x whenever x ∈ F ∩ E, we get that ce Se for all e ∈ E. C OROLLARY 7.2. Let X and Y be Banach spaces and let E be a subspace of X. Let c > 0 and suppose that T : E → Y is an operator satisfying (7.1). Then there is a Hilbert space H and operators S : X → H and U : H → Y such that U S : X → Y extends T , U 1 and S c. P ROOF. By Lemma 7.1 there is a Hilbert space H and an operator S : X → H such that Se T e for all e ∈ E. Let P be the orthogonal projection of H onto S(E), deﬁne U0 : S(E) → Y by U0 Se = T e for all e ∈ E and put U = U0 P . Then clearly U 1 and U Se = T e whenever e ∈ E. Before we proceed to the last step of the proof of Maurey’s theorem let us remind the reader of some special operator ideal norms and relations between them. For more details the reader is referred to Section 10 of [27], [16] or [68]. An operator T : X → Y is called 2-summing if π2 (T ) =def sup{( ni=1 T xi 2 )1/2 : n = 1, 2, . . .} < ∞ where the sup is taken over all ﬁnite sets {xi }ni=1 ⊂ X for which sup{( ni=1 |x ∗ , xi |2 )1/2 : x ∗ ∈ Ball X∗ } 1.

Extension of bounded linear operators

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It is easy to see that π2 (T ) is an ideal norm, i.e., a norm which satisﬁes the inequality π2 (U T V ) U π2 (T )V for every pair of spaces X0 and Y0 and operators U : Y → Y0 and V : X0 → X. The second idealnorm we need is the -norm (see [68], p. 80) deﬁned as follows: for T : n2 → Y, (T ) = ( R n T x2 dγ (x))1/2 where γ is the standard Gaussian measure on R n , the density of which is given by (2π)−n/2 exp(− ni=1 21 |ti |2 ). It is easy n to see that, for every U : m 2 → 2 , (T U ) U (T ). For a general operator S : X → n Y, (S) = sup{(SU ) : U ∈ L(2 , X), U = 1, n = 1, 2, . . .}. Again, it is easily checked that (S) is an ideal norm. The relations between the -norm and the notions of Gaussian type and cotype are the consequences of the identity n 2 1/2 ψi (t)T ui dγ (t) Rn

(T ) =

i=1

for every T : n2 → Y , where {ui }ni=1 is the unit vector basis of n2 . This identity follows from the fact that ψi (t) = t, ui , i = 1, 2, . . . , n, are independent standard Gaussian variables on R n . We treat here Banach spaces over the real numbers but, with the corresponding deﬁnitions for complex numbers, all of the following results hold in the complex case. We need the following basic facts about the π2 and norms. L EMMA 7.3. (a) Let X and Y be Banach spaces and assume that either X or Y is ﬁnitedimensional. Then, for every operator S : X → Y and ε > 0, there is an integer m, opm m m erators U : X → m ∞ and V : 2 → Y , and a diagonal operator Δ : ∞ → 2 such that S = V ΔU and V ΔU < (1 + ε)π2 (S). (b) Let V : n2 → X be the operator deﬁned by V ui = xi , 1 i n, where {ui }ni=1 de notes the unit vector basis of n2 . Then π2 (V ∗ ) ( n1 xi 2 )1/2 . The proof of (a) is based on the Pietch factorization theorem ([27], Section 10) and the fact that X isometrically embeds into a C(K) space, where K is a compact Cantor space. This C(K) space is paved by a family of m ∞ spaces, each spanned by m disjointly m supported functions {fi }m i=1 . The formal identity I : C(K) → L2 (K, μ) maps {fi }i=1 onto numerical multiples of an orthonormal basis where μ is the measure provided by Pietch’s theorem. Part (b) is obtained by a straightforward computation of π2 (V ∗ ). L EMMA 7.4. Let X be a Banach space of Gaussian type 2 and let S : n2 → X be an γ operator. Then (S) T2 (X)π2 (S ∗ ). P ROOF. By Lemma 7.3(a), given ε > 0 there is an integer m, operators U : X∗ → m n m m ∗ m ∞ , V : 2 → 2 and a diagonal operator Δ : ∞ → 2 such that S = V ΔV and ∗ V ΔU (1 + ε)π2 (S ). Denoting by JX the natural embedding of X into X∗∗ m ∗ we get that JX S = U ∗ Δ∗ V ∗ . Because Δ∗ : m 2 → 1 is a diagonal operator, Δ = m γ γ ( i=1 Δ∗ ui 2 )1/2 . The principle of local reﬂexivity implies that T2 (X∗∗ ) = T2 (X). These two facts, together with the fact that is an ideal norm, imply the following in-

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M. Zippin

equality:

(S) = (JX S) V ∗ U ∗ Δ∗ 2 1/2 m ∗ ψi (t)U ∗ Δui dγ (t) = V R n i=1 1/2 n ∗ ∗ 2 ∗ γ ∗∗ U Δ ui V T X 2

i=1

n 1/2 ∗ ∗ γ 2 V U T2 (X) Δui i=1

γ γ = V ∗ U ∗ ΔT2 (X) (1 + ε)T2 (X)π2 S ∗ . Since ε > 0 is arbitrary, (S) T2 (X)π2 (S ∗ ) as claimed. γ

P ROOF OF M AUREY ’ S EXTENSION THEOREM . Let X and Y be Banach spaces of Gaussian type 2 and cotype 2, respectively, and let E be a subspace of X. In view of Corollary 7.2 it sufﬁces to show that every operator T : E → Y satisﬁes condition (7.1) with γ γ n c = T2 (X)C2 (Y ). To do that choose any two ﬁnite sets {ei }m i=1 ⊂ E and {xj }j =1 ⊂ X. Such that m !

n " ! ∗ " x ∗ , ei x , xj for all x ∗ ∈ X∗ .

(7.4)

j =1

i=1

We may assume, without loss of generality, that m = n because adding a few zero’s to one of the sets will affect neither (7.1) nor (7.3). Deﬁne the operators U : n2 → E and V : n2 → X by U ui = ei and V ui = xi for 1 i n. Denoting by J the natural embedding of E into X we see that, by (7.3), U ∗ J ∗ x ∗ 2 = sup{x ∗ (J U (u)): u ∈ Ball(n2 )}2 = n n ∗ 2 ∗ 2 ∗ ∗ 2 ∗ ∗ i=1 |x , ei | i=1 |x , xi | = V x for every x ∈ X . Therefore there is an n n ∗ ∗ ∗ operator W : 2 → 2 with W 1 such that W V = U J . It follows from the deﬁnition of cotype and the fact that (T ) is an ideal norm that

n

1/2 2

T ei

=

i=1

n

1/2 2

T J U ui i=1 γ γ C2 (Y )(T J U ) T C2 (Y )(J U ).

(7.5)

Using Lemma 7.4, the fact that π2 is an ideal norm and Lemma 7.3(b) we get that

γ γ (J U ) T2 (X)π2 U ∗ J ∗ = T2 (X)π2 W V ∗ 1/2 n ∗

γ γ 2 xj . T2 (X)π2 V T2 (X) j =1

(7.6)

Extension of bounded linear operators

1739

Combining (7.5) and (7.6) we get that

n i=1

1/2 T ei

2

γ γ T C2 (Y )T2 (X)

n

1/2 xj

2

.

j =1

This establishes condition (7.1) and, in view of Corollary 7.2, completes the proof of Maurey’s extension theorem. The following partial converse of Maurey’s extension theorem has been recently proved in [14]. T HEOREM 7.5. Let X be a Banach space which satisﬁes one of the following properties (i) there is a constant c so that, for every n 1 and every operator T ∈ B(n2 , X), (T ) cπ1 (T ∗ ). (ii) X has the Gordon–Lewis property (in particular, X may be a Banach lattice). (iii) X is isomorphic to a subspace of a Banach lattice of ﬁnite cotype. If X satisﬁes the conclusion of Maurey’s extension theorem then X is of type 2.

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CHAPTER 41

Nonseparable Banach Spaces∗ Václav Zizler Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1, Edmonton, Canada E-mail: [email protected]

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic concepts and examples . . . . . . . . . . . . . . . . . . . Weak compact generating and projectional resolutions . . . . . Biorthogonal systems and quasicomplements . . . . . . . . . . Gâteaux smooth and rotund norms . . . . . . . . . . . . . . . . Uniformly Gâteaux smooth norms . . . . . . . . . . . . . . . . Fréchet smooth and locally uniformly rotund norms . . . . . . C k -smooth norms for k > 1 . . . . . . . . . . . . . . . . . . . Open problems and concluding remarks . . . . . . . . . . . . . A. More on special compact spaces . . . . . . . . . . . . . . . B. More on the weak topology of nonseparable Banach spaces C. More on fragmentability and σ -fragmentability . . . . . . . D. Fundamental biorthogonal systems and Mazur’s intersection E. Uniform homeomorphisms . . . . . . . . . . . . . . . . . . F. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7926-02 (Canada). HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 2 Edited by William B. Johnson and Joram Lindenstrauss © 2003 Elsevier Science B.V. All rights reserved 1743

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1. Introduction The theory of nonseparable Banach spaces is a large ﬁeld, closely related to general topology, differential calculus, descriptive set theory and inﬁnite combinatorics. In this article, we will focus on the interplay of weak topologies, smoothness and rotundity of norms, biorthogonal systems and projectional resolutions of the identity operator in nonseparable Banach spaces. We will discuss some applications in questions on special compact spaces and in smooth approximations and smooth partitions of unity on nonseparable Banach spaces. Many open problems will be mentioned. Gâteaux differentiability of norms is behind sequential and metrizability properties of weak topologies on some sets and is useful in the study of special compact spaces (Sections 2, 5). Fréchet differentiability of norms generates continuous duality mappings from Banach spaces X onto dense sets in their duals X∗ (Section 7). Local uniform rotundity of norms is related to covering properties of weak topologies in Banach spaces and is useful in questions on continuity of the identity map from the weak to norm topologies on sets in Banach spaces (Section 7). Uniform Gâteaux differentiability of norms is instrumental in embedding compact spaces into Hilbert spaces in their weak topology (Section 6). The concept of Markushevich bases provides for a good insight into problems on the Corson–Lindelöf properties of weak topologies, injections into c0 (Γ ) and quasicomplements (Section 4). Markushevich bases are related to two fundamental concepts in nonseparable Banach spaces, namely to the concept of weak compact generating and to the concept of projectional resolutions of the identity operator (Sections 2, 3, 4). In Section 8, higher-order differentiable norms on spaces of continuous functions on uncountable scattered compacts are discussed, together with applications in smooth partitions of unity on such Banach spaces. The article is ﬁnished with remarks and open problems (Section 9). An ample list of references is included (Section 10). In the end of each section, the results presented are tested on examples of Banach spaces listed in the end of Section 2. The proofs in this article will be outlined only. In order to make it short, we will say “Proof” and mean the key idea in the proof. The list of survey texts, where the reader can ﬁnd more information in this area include [6,10,20,31,25,34,57,61,60,65,64,66,67,73,80,105,126,141,142,157,177,185,195, 190,197,198,208,206,219,222,243,253,269,280,284,296,304,306,327] and [328]. Many open problems in this area are discussed, for example, in [20,57,73,80,206] and [222]. The exercises in [61,80] and [327] are accompanied with hints for their solution and discuss many “folklore” techniques as well as examples and counterexamples. We will consider real Banach spaces only and keep the standard notation as it is, e.g., in [168]. In particular, N and R will denote the set of all positive integers and reals respectively. The symbol BX will denote the unit ball of a Banach space X, i.e., BX = {x ∈ X; x 1} and SX will denote the unit sphere of a Banach space X, i.e., SX = {x ∈ X; x = 1}, where · is the norm of X. Unless stated otherwise, dual spaces X∗ are considered in their canonical dual norm f = sup{|f (x)|; x ∈ BX } for f ∈ X∗ . Compact topological spaces are assumed to be Hausdorff. If K is a compact space, C(K) will denote the Banach space of all real-valued continuous functions on K with the supremum norm f = sup{|f (t)|; t ∈ K}. The symbol p (c0 ) denotes the space p (N) (c0 (N)). Similarly, p (c) stands for p (Γ ), where card Γ is the cardinality of the

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continuum c, i.e., the cardinality of the set R. The least inﬁnite (respectively uncountable) ordinal number is denoted by ω0 (respectively ω1 ). When we think of these ordinals as cardinal numbers, we will denote them by ℵ0 and ℵ1 respectively. The Continuum Hypothesis then reads ℵ1 = c. A set S is called countable if card S ℵ0 . The cardinality of an ordinal number μ is denoted by |μ|. A Banach space X is identiﬁed with its canonical image in its second dual X∗∗ . If X is a Banach space, x ∈ X and f is in the dual space X∗ , then both f (x) and x(f ) will denote the value of f at x. The norm on a Banach space and its dual norm on X∗ will be denoted by the same symbol · . Often, we will say a space and mean a Banach space and a norm on a Banach space and mean an equivalent norm. By a subspace of a Banach space we will mean a norm closed subspace. The words differentiable and smooth have the same meaning in this article. If we say that the norm of a Banach spaces is differentiable, we mean that it is differentiable away from the origin. We will say that a real valued function f is Fréchet C 1 -smooth on a Banach space X, if the operator x → f (x), from X into X∗ , is norm to norm continuous, where f (x) is the w Fréchet derivative of f at x (see Deﬁnition 1). For a set A in a Banach space X, A and A mean the closure of A in the norm topology, respectively in the weak topology of X. If X∗ w∗ is a dual space, A is the closure of A in the weak star topology of X∗ . If K is a subset of a Banach space X, then span K denotes the closed linear hull of K in X. If K ⊂ X∗ , ∗ then span w K denotes the weak star closed linear hull of K in X∗ , while span · K denotes the norm closed linear hull of K. Similarly, conv K is the closed convex hull of K in X. The symbol χA will denote the characteristic function of the set A in the topological space T . We will say, typically, that a set K in a Banach space X is weakly compact if it is compact in the relative topology inherited from the weak topology of X. The density character or density (dens T ) of a topological space T is the minimal cardinality of a dense set in T . Unless stated otherwise, for a Banach space X, dens X is the density of X in the norm topology. A bump function on a Banach space X is a real-valued function on X with bounded non-empty support.

2. Basic concepts and examples In this section we will discuss basic concepts in the interplay of weak topologies and Gâteaux smoothness of norms in nonseparable Banach spaces. We will deﬁne projectional resolutions of identity, several types of compact spaces that will be discussed in this article and list Banach spaces that will serve as examples for testing the results presented in this article. D EFINITION 2.1. A real-valued function ϕ on a Banach space X is Gâteaux (Fréchet) differentiable at x ∈ X if there is an f ∈ X∗ such that limt →0 1t (ϕ(x + th) − ϕ(x)) = f (h) for every h ∈ SX (uniformly in h ∈ SX ). Such f is then called the Gâteaux (Fréchet) derivative of ϕ at x and is denoted by ϕ (x), i.e., f (h) = ϕ (x)(h) for every h ∈ X. If · is a norm on a Banach space X and x ∈ SX , we say that · is Gâteaux (Fréchet) differentiable at x if the real-valued function ϕ deﬁned for y ∈ X by ϕ(y) = y is Gâteaux (Fréchet) differentiable at x. In this case we write ϕ (x) = x . If the norm · is Gâteaux

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(respectively Fréchet) differentiable at each x ∈ SX , we say that · is a Gâteaux (respectively Fréchet) differentiable norm on X and call it a G (respectively an F) norm on X. The norm · on X is Gâteaux (Fréchet) differentiable at x ∈ SX if and only if limt →0 1t (x + th + x − th − 2) = 0 for every h ∈ SX (uniformly in h ∈ SX ). If the norm · is Gâteaux differentiable at x ∈ SX , then the functional x is supporting BX at x, i.e., x ∈ SX∗ and x (x) = 1 (cf., e.g., [57, p. 2], [80, Chapter 8]). Every Banach space X in its weak topology is a completely regular space (cf. e.g. [80, Chapter 3, Example] or [73, p. 56]). The following results are from [17,41] and [265]. For more in this direction we refer to Section 3 of this article and references therein. T HEOREM 2.2. Let X be a Banach space. Then X in its weak topology is a Lindelöf space if and only if it is a normal space if and only if it is a paracompact space. While inﬁnite-dimensional Banach spaces are never metrizable in their weak or weak star topologies (cf., e.g., [80, Chapter 3]), BX in its weak topology is metrizable if and only if X∗ is separable and BX∗ in its weak star topology is metrizable if and only if X is separable (cf., e.g., [80, Chapter 3]). We will often use the fact that given a compact space K, C(K) is norm separable if and only if K is metrizable (cf., e.g., [80, Chapter 3]). If X∗ is weak star separable and K is a weakly compact set in X, then K in its weak topology is metrizable by the metric ρ(x, y) = 2−i fi −1 |fi (x − y)| for x, y ∈ K, where {fi } is weak star dense in X∗ (cf., e.g., [80, Chapter 3]). In this case, K in its weak topology is separable and span K is norm separable (cf. Mazur’s theorem (e.g., [80, Chapter 3]) and so is K. A set K in a Banach space X is compact in the weak topology of X if and only if K is weakly sequentially compact in X (i.e., every sequence in K has a subsequence converging in K in its weak topology) (Eberlein, Šmulyan) (cf., e.g., [80, Chapter 4]). A topological space T is called pseudocompact if every continuous real-valued function f on T is bounded. Note that then f attains its supremum on T . The following theorem is a combination of the results of James, Krein, Preiss and Simon (cf. [10, Chapter IV.5], [99] and [98] (where a new proof is given), [80, Chapter 3], [168, 258]). T HEOREM 2.3. A subset K of a Banach space X is weakly compact if K is either weakly pseudocompact or K is closed convex and each f ∈ X∗ attains its supremum on K. The latter happens if K is a closed convex hull of a weakly compact set in X. P ROOF. Let C be a weakly pseudocompact set in a Banach space X. Then K := conv C is a weakly compact set in X by James’ weak compactness theorem as every bounded linear functional attains its supremum on K (= its supremum on C) (cf., e.g., [80, Chapter 3], w [168,99]). Assume a ∈ C \ C. Preiss and Simon proved in [258] that there is a sequence w w (Un ) of nonempty sets in C open in the relative weak topology of C and such that w (Un ) converges to a, i.e., for every neighborhood U of a in C , there is n0 ∈ N such that Un ⊂ U for every n n0 . Let fn be continuous functions on C such that the support of fn is in Un and fn (xn ) = n for some point xn ∈ Un ∩ C for each n. Then consider f := fn .

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It is a continuous function on C as it is locally a ﬁnite sum (a ∈ / C). The function f is unbounded on C. This is a contradiction with the pseudocompactness of C, showing that C is weakly closed in K and C is thus weakly compact. For more information in this direction see [10, Chapter IV.5], [305] and [80, Chapter 12] and references therein. For non-linear versions on the James theorem we refer to [15], where it is proved, among other things that if X∗ is inﬁnite-dimensional and separable, then there is a Fréchet C 1 -bump function b on X such that {b (x), x ∈ X} = X∗ . The proof of the separable version of James’ weak compactness theorem (cf., e.g., [57, p. 17], [80, Chapter 3] or [231]) gives that if X is a Banach space and B is a norm separable set in BX∗ such that conv · B = BX∗ , then there is x ∈ SX such that f (x) < 1 for all f ∈ B (Godefroy, Rodé, Simons). This is in contrast with the case of the norm nonseparable set B = {±δt , t ∈ [0, 1]} ⊂ BC[0,1]∗ , where δt is the Dirac measure corresponding to the point t ∈ [0, 1] ([80, Chapter 3]). For a compact space K, a norm bounded set A in C(K) is weakly compact in C(K) if and only if it is a compact set in the topology of pointwise convergence in C(K). This is the Grothendieck theorem (cf., e.g., [60, p. 156]) or [80, Chapter 12]. Goldstine’s theorem asserts that for a Banach space X, BX is weak star dense in BX∗∗ (cf., e.g., [80, Chapter 3]). From the proof of Theorem 2.3, we obtain that if K is a weakly compact set in a Banach w space, A ⊂ K and a ∈ A , then there is a sequence (an ) ⊂ A that weakly converges to a, i.e., every weakly compact set K in a Banach space in its weak topology is an angelic compact. However, if {en } is the sequence of the standard unit vectors in 2 , then one checks w √ directly (cf.,√e.g., [6, p. 110] or [80, Chapter 3, Example]) that 0 ∈ { nen } and no subsequence of { nen } converges weakly to 0 (by the Banach–Steinhaus uniform boundedness principle (cf., e.g., [80, Chapter 3])). Every Banach space in its weak topology has countable tightness, i.e., if A is a subset w w in a Banach space X and a ∈ A , then there is a countable C ⊂ A such that a ∈ C (cf., e.g., [80, Chapter 4]). From the countability of the supports of elements of c0 (Γ ), we can see that B∞ (Γ ) in its weak star topology does not have countable tightness if Γ is uncountable. w Like for every inﬁnite-dimensional Banach space, 0 ∈ S1 . However, no sequence in in S1 weakly converges to 0 (Schur, cf., e.g., [80, Chapter 5]). The dual ball B∗∗ 1 its weak star topology does not have countable tightness ([41]). On the other hand, if a separable Banach space X does not contain any isomorphic copy of 1 , then BX∗∗ in its weak star topology is an angelic compact ([272,273,30]). Note that this implies that if a separable Banach space does not contain any isomorphic copy of 1 , then every element of BX∗∗ is a weak star limit of a sequence from BX , BX∗∗ is weak star sequentially compact and for every bounded sequence {xn } in X, there is a subsequence {xni } such that {f (xni )} is convergent for every f ∈ X∗ . Finally, in this case, card X∗∗ = card X. The latter properties characterize spaces not containing any isomorphic copy of 1 among separable spaces. These are the results of Odell and Rosenthal ([228]) and Rosenthal ([272], cf., e.g., [197, p. 101], [57, p. 115]). If X∗ is separable, then all these properties follow immediately as

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BX∗∗ in its weak star topology is a metrizable compact space. One of the examples of separable spaces X with nonseparable dual such that X does not contain any isomorphic copy of 1 is the space J T discussed below in this section. The following result is the Josefson–Nissenzweig theorem ([171,225], cf., e.g., [61, p. 219], [18]). T HEOREM 2.4. Let X be an inﬁnite-dimensional Banach space and f ∈ BX∗ . Then there is a sequence fn ∈ SX∗ such that fn → f in the weak star topology of X∗ . It is proved in [26] that the statement in Theorem 2.4 is equivalent with the statement that for every inﬁnite-dimensional Banach space X there is a convex continuous function on X that is somewhere Gâteaux but not Fréchet differentiable. Related to Theorem 2.4 is also the following remark: let X be an inﬁnite-dimensional Banach space, fn ∈ X∗ , fn = 1/n for each n. Write each fn = limk fn,k , in the weak star topology, fn,k = n for each n, k ∈ N. Then by the Banach–Steinhaus uniform boundedness principle, no sequence in {fn,k } converges weakly to 0. Compare this with the fact that the uniform limit of a sequence of Baire functions is a Baire one function (i.e., a pointwise limit of a sequence of continuous functions, see [221, Chapter XV.1]). For more applications of Theorem 2.4 in non-linear analysis, we refer to [27] and references therein. In particular, as a corollary of Theorem 2.4, we get that in every inﬁnitedimensional Banach space X, there is a continuous convex function deﬁned on X that is unbounded on BX (cf., e.g., [80, Chapter 8, Example]). The Rainwater–Simons theorem reads as follows: assume that X is a Banach space and B is a subset of BX∗ such that for every x ∈ SX , there is b ∈ B such that b(x) = 1. Then a bounded sequence {xn } ⊂ X converges weakly to x ∈ X whenever b(xn − x) → 0 for every b ∈ B (cf., e.g., [99], [80, Chapter 3]). Note that we can take the set of all extreme points of BX∗ for B by the Krein–Milman theorem (cf., e.g., [80, Chapter 3]). It is the result in [236] (an extension of a related result of Borwein) that a separable Banach space X does not contain an isomorphic copy of 1 if and only if each sequence {fn } in X∗ converges in the norm topology to 0 whenever it converges to zero uniformly on all weakly compact subsets of X (cf., e.g., [26]). Note that if we allow nets instead of sequences above, we get the reﬂexivity of X by the Mackey–Arens–Katˇetov theorem (cf., e.g., [80, Chapter 4]). The following is Šmulyan’s classical result (cf., e.g., [57, p. 3] or [80, Chapter 8]). We will call it Šmulyan’s lemma. T HEOREM 2.5. The norm · of a Banach space X is Gâteaux (Fréchet) differentiable at x ∈ SX if and only if fn − gn → 0 in the weak star topology (norm topology) of X∗ whenever fn , gn ∈ SX∗ are such that fn (x) → 1 and gn (x) → 1. It follows from Theorem 2.5 that · is Gâteaux differentiable at x ∈ SX if and only if there is a unique supporting functional to BX at x, i.e., a unique f ∈ BX∗ with f (x) = 1, namely f = x . Also, if · is a Fréchet differentiable norm, then the map x → x from SX into SX∗ is norm-to-norm continuous. This map is usually called the duality mapping. Finally, it follows from Theorem 2.5 that Gâteaux and Fréchet differentiability of norms coincide in ﬁnite-dimensional spaces.

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Assume that the norm · of X is Gâteaux differentiable at x ∈ SX . As x is continuous on BX∗ in its weak star topology and attains its supremum on BX∗ exactly at x , we get that x is a Gδ point of BX∗ in its weak star topology. This simple fact is behind many sequential properties of weak star and weak topologies on Banach spaces. A point p in a topological space T is called a Gδ point of T whenever p is the intersection of a countable collection of open sets in T . If T is a compact space, this means that T has a countable neighborhood base at p. The following statement is from [75]. It shows the interplay of smoothness, the Bishop– Phelps theorem and the Josefson–Nissenzweig theorem. The Bishop–Phelps theorem asserts that for a Banach space X, those elements of X∗ that attain its supremum on BX form a norm dense set in X∗ (cf., e.g., [168], [57, p. 13] or [80, Chapter 3]). If f ∈ X∗ attains its supremum on BX , we say f attains its norm. T HEOREM 2.6. Assume that the norm of a Banach space X is Gâteaux differentiable and ∗ that S ⊂ BX∗ is such that conv w S = BX∗ . If f ∈ BX∗ , then there is a countable C ⊂ S ∗ w such that f ∈ C . P ROOF. If x ∈ SX and f := x , there is {fn } ⊂ S such that fn (x) → 1. From Theorem 2.5, we have fn → f in the weak star topology of X∗ . Hence the statement holds for f = x . By the Bishop–Phelps theorem, the statement holds for every f ∈ SX∗ . Theorem 2.4 can be used to ﬁnish the proof of Theorem 2.6. One of the non-linear versions of the Bishop-Phelps theorem is the following statement. If b is a non-negative Gâteaux smooth continuous function on a Banach space X with bounded non-empty support, then the cone generated by the set {b (x); x ∈ X} is norm dense in X∗ (see, e.g., the text preceding Theorem 7.2). Related to the notion of countable tightness is the following result of Pol ([251], cf., e.g., [80, Chapter 12]). T HEOREM 2.7. Let X be a Banach space. Then the following conditions (i) and (ii) are equivalent. ∗ w∗ (i) If A ⊂ BX∗ and f ∈ A , then there is a countable C ⊂ A such that f ∈ conv w C. (ii) If a family A of convex closed sets in X has empty intersection, then some countable subfamily B of A has empty intersection. D EFINITION 2.8. If the conditions in Theorem 2.7 are satisﬁed, we say that X has property C. Assuming the Continuum Hypothesis, there is a compact space K that has countable tightness and C(K) does not have property C (Haydon, Pol, see [251]). A compact space K has countable tightness if C(K) is Lindelöf in its pointwise topology ([41]). From applications of property C we mention the following result of Pol [251] that is an extension of the former result of Grothendieck (cf. [195]).

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T HEOREM 2.9. Assume that K is a compact space such that C(K) has property C. Then every ﬁnite positive regular measure on K has separable support. The support of a measure is the complement of the set of points that have neighborhoods of measure 0. P ROOF. Assume that the measure μ is supported exactly on K and let us show that K is separable. Fix i ∈ N. For x ∈ K put Cx = {f ∈ C(K); K f dμ 1/i and f (x) = 0}. If f ∈ C , then f = 0 identically on K and thus x∈K x K f = 0, a contradiction. Hence C = ∅. As C is closed and convex for every x ∈ K and C(K) has property C, x x∈K x x∈Ai Cx = ∅ for some countable set Ai ⊂ K. We claim that K = i Ai . Indeed, assume that there is f ∈ C(K) such that f = 0 on i Ai and K f dμ > 0. Find i ∈ N such that x∈Ai Cx , a contradiction. K f dμ > 1/i. Then f ∈ In [251] (cf., e.g., [80, Chapter 12]), Pol proved the following result. T HEOREM 2.10. Property C is a three space property, i.e., a Banach space X has property C, whenever there is a subspace Y of X such that both Y and X/Y have property C. The Banach–Dieudonné theorem asserts that a subspace D in a dual space X∗ is weak star closed if D ∩ BX∗ is weak star closed (cf., e.g., [80, Chapter 4]). The second part of the following result is the Corson–Lindenstrauss result from [43], the ﬁrst part is in [217] and [21]. For a simple proof of a version of this statement we refer to [83]. T HEOREM 2.11. Let K be a weakly compact set in a Banach space X. Consider K in its weak topology. Then K contains a subset S that is Gδ dense in K and such that the weak and norm topologies on K coincide at each point of S. The set S in its topology from K is metrizable by a complete metric. Thus, in particular, the Gδ points of K form a dense set in K. P ROOF. Namioka proved the ﬁrst part by applying Baire category arguments to the identity map from the weak topology into the norm topology on K ([217], see Section 9). Earlier, Corson and Lindenstrauss used renorming theory and Gâteaux smoothness of norms to prove the last part of the statement ([43]). The set S is metrizable by a complete metric as it is Gδ in K in its norm topology ([275, p. 164]). Each point of S is clearly a Gδ point of K as S is metrizable and dense in K. Let us illustrate the statement in Theorem 2.11 on the case K := B2 (Γ ) , where Γ is uncountable. In this case, we can put S = S2 (Γ ) , as on the unit sphere of a Hilbert space the norm and weak topologies coincide (see, e.g., Section 7). Moreover, S2 (Γ ) is a Gδ set in B2 (Γ ) in its weak topology as S2 (Γ ) = n (B2 (Γ ) \ (1 − n1 )B2 (Γ ) ). The set S2 (Γ ) is dense in B2 (Γ ) in its weak topology (cf., e.g., [80, Chapter 3, Example]). The Gδ points of B2 (Γ ) in its weak topology are exactly the points of S2 (Γ ) ([195, p. 255]).

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There is a weakly compact set K in a Banach space, such that K in its weak topology is non-metrizable and every closed metrizable subset of K is Gδ in K ([21]). There exists a weakly compact set K in a Banach space such that K in its weak topology is nonmetrizable, convex and all points of K are Gδ points of K ([195, p. 269]). However, if K is a nonseparable, weakly compact convex and symmetric set in a Banach space in its weak topology, then this cannot happen as K contains a one point compactiﬁcation of an uncountable discrete set ([195], cf., e.g., the proof of Theorem 4.2). We can ﬁnd by a standard argument that the ball B∞ (Γ ) in its weak star topology has no Gδ points if Γ is uncountable. Next result in this direction is the following theorem, which is a combination of the results in [40,257] and [56] (cf., e.g., [73, Chapter II.2]). Before stating the result, we note that Lipschitz Gâteaux differentiable bump functions can easily be constructed from Gâteaux differentiable norms. T HEOREM 2.12. Let K be a compact space such that C(K) admits a Lipschitz bump function that is Gâteaux differentiable. Then K is sequentially compact and K contains a dense Gδ set that is metrizable by a complete metric. P ROOF. The compact space K is sequentially compact by Theorem 2.13 below. Assuming that a Banach space X admits a Gâteaux differentiable norm, it is proved in [257] that every continuous convex function on X is Gâteaux differentiable on a Gδ dense set in X (cf., e.g., [73, p. 72], i.e., X is then a weak Asplund space. The same conclusion holds if X admits a Lipschitz Gâteaux differentiable bump function ([56,100]). In particular, under the assumptions in Theorem 2.12, the supremum norm of C(K) is Gâteaux differentiable on a dense Gδ set in C(K). It is proved in [40] (cf., e.g., [73, p. 45] that then the conclusion in Theorem 2.12 follows. For further results in this direction see Theorem 7.17, the text preceding Theorem 2.15, the text following Theorem 5.2 and Section 9. In separable Banach spaces X, the set of all points of Gâteaux differentiability of a convex continuous function on X is Gδ dense in X (Mazur, cf., e.g., [80, Chapter 8], [243, p. 12]). We will see in Section 5 that the standard norm of 1 (Γ ) is nowhere Gâteaux differentiable if Γ is uncountable. The set of all points of Gâteaux differentiability of the supremum norm of the space D (see below in this section) is a dense but not a residual set in D (cf., e.g., [73, p. 49]). Assuming the Continuum Hypothesis, Argyros and Mercourakis showed in [13] that 1 (c) admits an equivalent norm, the Gâteaux differentiability points of which form a set that is dense but not residual in 1 (c). It is shown in [149] that in a nonseparable Hilbert space H there exists a continuous convex function on H , the set of all Gâteaux differentiability points of which is not Gδ (even not Borel) in H , though it is residual in H (cf., e.g., [57, Chapter I] or [243, Chapter II]). This is in contrast with the points of Fréchet differentiability (cf., e.g., [80, Chapter 8], [168] or [243, p. 14]). The following result follows from the smooth variational principle (cf., e.g., [57, p. 9] or [53]) and from the results in [130,174] and [289] (cf., e.g., [80, Chapter 10]).

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T HEOREM 2.13. Let X admit a Lipschitz Gâteaux differentiable bump function. Then BX∗ in its weak star topology is sequentially compact and norm-dens X∗ card X. If X admits a Lipschitz Fréchet C 1 -smooth bump function, then norm-dens X∗ = dens X. w∗ P ROOF. Let {fn } be a sequence in BX∗ . For n ∈ N put An = {fn }j n and A = n An . Deﬁne the function p on X by p(x) = sup{f (x); f ∈ A}. From the smooth variational principle (cf., e.g., [57, p. 9], [80, Chapter 10] or [53]) it follows that p is Gâteaux differentiable at some x0 ∈ X. From the Gâteaux differentiability of p at x0 , it then follows that p (x0 ) is the weak star limit of a subsequence of {fn } (Theorem 2.5) (cf., e.g., [73, p. 38], [80, Chapter 8] or [57, Chapter II]). If the norm of X is Gâteaux differentiable, then the mapping x → x maps SX onto a norm dense set in SX∗ by the Bishop–Phelps theorem. This gives the second part in the statement. If the norm of X is Fréchet differentiable, then by Theorem 2.5, the mapping x → x is norm to norm continuous. Thus dens X∗ dens X. The reverse inequality holds true for all Banach spaces (cf., e.g., [80, Chapter 3]). The following argument provides for an alternative way of proving the ﬁrst ˇ part of the statement if we assume the Continuum Hypothesis. First, Cech and Pospíšil showed in [36] that the cardinality of every compact space that is not sequentially compact is greater than or equal to 2ℵ1 . Thus by using the second part of the statement, assuming that the density character of X is c and that BX∗ is not weak star sequentially compact, we get card X∗ (dens X∗ )ℵ0 (card X)ℵ0 ((dens X)ℵ0 )ℵ0 = (cℵ0 )ℵ0 = cℵ0 = c = ℵ1 < 2ℵ1 card BX∗ card X∗ , a contradiction. Theorem 2.13 implies that some spaces, the dual ball of which is not weak star sequentially compact, do not admit Lipschitz Gâteaux differentiable bump functions. This applies for instance to ∞ , as we cannot extract a weak star convergent subsequence from the sequence {fn } ⊂ B∗∞ deﬁned for {ai } ∈ ∞ by fn ({ai }) = an . Indeed, if n1 < n2 . . . , deﬁne a ∈ ∞ by ani = (−1)i and aj = 0 otherwise. Then fni (a) = (−1)i . Alternatively, one can show that ∞ does not admit any equivalent Gâteaux differentiable norm as follows: as 1 (c) is isometric to a subspace of C[0, 1]∗ , 1 (c) is isometric to a subspace of ∞ , since every space dual to a separable space is isometric to a subspace of ∞ . Hence dens ∗∞ dens 1 (c)∗ = dens ∞ (c) = 2c > c = card ∞ . Haydon showed that c∞ (ℵ1 ) does not admit any continuous (not necessarily equivalent) Gâteaux differentiable norm (cf., e.g., [57, p. 89]). The non-equivalent norm deﬁned on ∞ by |||x|||2 = 2−i xi2 is continuous and Fréchet differentiable on ∞ . For the use of nonequivalent smooth norms in analysis on Banach spaces we refer, e.g., to [16] and [15] and references therein. We will now deﬁne several types of compacts that will be discussed in this article. D EFINITION 2.14. Let K be a compact space. Then (i) K is an Eberlein compact if it is homeomorphic to a weakly compact set in c0 (Γ ) considered in its weak topology for some set Γ . (ii) K is a uniform Eberlein compact if K is homeomorphic to a weakly compact set in 2 (Γ ) considered in its weak topology for some set Γ .

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(iii) K is a Corson compact if for some set Γ , K is homeomorphic to a subset S of [−1, +1]Γ considered in its product topology such that all points of S are countably supported (i.e., if f ∈ S, then card{α ∈ Γ ; f (α) = 0} ℵ0 ). The compact space K is scattered if each subset of K has a relatively isolated point. A tree is a partially ordered set (T , ) such that for every t ∈ T , the set {s ∈ T ; s t} is well ordered. We introduce two elements 0 and ∞ which are not in T , such that 0 < t < ∞ for every t ∈ T . If s, t ∈ T then (s, t] = {u ∈ T ; s < u t}, (0, s] = {u ∈ T ; u s}. For each t ∈ T , r(t) is the unique ordinal which has the same order type as (0, t). The height h(T ) is deﬁned by h(T ) = sup{r(t) + 1; t ∈ T }. We will assume that T is Hausdorff, i.e., if (0, t) = (0, t ) and r(t) = r(t ) is a limit ordinal, then t = t . If t ∈ T , then T + is the set of immediate successors of t, i.e., t + = {u ∈ T ; s < u if and only if s t}. We equip T with the weakest topology τ for which all intervals (0, t] are open and closed. We will identify T = T ∪ {∞} with the Alexandrov compactiﬁcation of T and denote by C0 (T ) the space of continuous functions f on T such that f (∞) = 0. The space T is a compact scattered space. The full uncountably branching tree of height ω1 is α<ω1 ω1α , where the ordering β

is given by s t if s ∈ ω1α , t ∈ ω1 with α β and the restriction of t to α equals to s. Any pointwise compact subset in c0 (Γ ) in its pointwise topology is homeomorphic to a weakly compact subset of c0 (Γ ) (consider the map Φ(f )(γ ) = λ(γ )f (γ ) for suitable positive numbers λ(γ ) to make the set bounded). Similarly, a compact set K is a Corson compact if and only if K is homeomorphic to a subset S of RΓ formed by countably supported elements. If K is a compact metric space, then K is a uniform Eberlein compact as then C(K) is separable and thus there is a bounded linear one-to-one map from C(K)∗ into 2 that is weak star to weak continuous. The unit ball of 2 (c) in its weak topology is an example of a uniform Eberlein compact that is not metrizable. The one point compactiﬁcation of a locally compact metric space is an Eberlein compact ([195]). Every Eberlein compact is evidently a Corson compact. For an example of a Corson compact that is not Eberlein, see, e.g., [73, Chapter 7.3]. The ﬁrst example of an Eberlein compact that is not a uniform Eberlein compact appeared in [22] (see, e.g., [80, p. 419]. A compact space K is scattered if and only if the Cantor derived set K (α) = ∅ for some ordinal α (cf., e.g., [57, Chapter VI]). The Cantor derived set K (1) is deﬁned as the set of all accumulation points of K and for higher ordinals, K (α) is deﬁned inductively. For a (β) limit ordinal β, K = α<β K (α) . A compact space K is scattered if and only if C(K) is an Asplund space, i.e., each separable subspace of C(K) has separable dual (cf., e.g., [57, p. 258], [80, Chapter 12]). Every scattered Corson compact is Eberlein ([4]). A compact space is a scattered Eberlein compact if and only if it is homeomorphic to a weakly compact set S in c0 (Γ ) for some Γ in its weak topology and such that S is formed by characteristic functions of ﬁnite sets ([21,4]) – the so called strong Eberlein compacts. For an example of a strong Eberlein compact that is not a uniform Eberlein compact see [22] (cf., e.g., [57, Chapter VI], [80, Chapter 12, Example], or [12]). If K is a scattered compact space, then K has countable tightness if and only if C(K) has property C ([251]). Every Corson compact is angelic (cf., e.g., [80, Chapter 12, Example]). A Corson compact is metrizable if it is separable (cf., e.g., [80, Chapter 12, Example]). Any Corson compact K contains a dense subset formed by Gδ points of K (Šapirovskii,

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cf., e.g., [80, Chapter 12]), but need not in general contain a dense metrizable subset (see, e.g., the text following Theorem 5.2 below). The following results are from [21,207,127] see also [223,319] and [57, Chapter V]. For a short proof of the ﬁrst part of the following result we refer also to [83,84] and for a short proof of the second part we refer to the note after Corollary 6.7 and to [316]. T HEOREM 2.15. A continuous image of an Eberlein compact is an Eberlein compact, a continuous image of a uniform Eberlein compact is a uniform Eberlein compact, a continuous image of a Corson compact is a Corson compact and a continuous image of a scattered compact is a scattered compact. A family F of sets Fα is point-ﬁnite in a set K, if every point of K lies in at most ﬁnite number of the sets Fα . A family is σ -point-ﬁnite if it is a countable union of point-ﬁnite families. The family F weakly separates the points of K if for x, y ∈ K, x = y, there is F ∈ F such that χF (x) = χF (y). The part (i) in the following statement is due to Rosenthal [271], cf., e.g., [80, Chapter 12]. The part (ii) is in [21]. T HEOREM 2.16. (i) A compact space K is an Eberlein compact if and only if it contains a σ -point ﬁnite family F of open Fσ -sets such that F weakly separates points of K. (ii) A compact space K is a uniform Eberlein compact if and only if K contains a points of K and such that there is a function family F of open Fσ sets that weakly separate κ : N → N and a decomposition F = Fn so that for every n ∈ N and x ∈ K, card{U ∈ Fn ; x ∈ U } κ(n). P ROOF. (i) Let K ⊂ c0 (Γ ) in its weak topology. For n ∈ N, j ∈ ±N, |j | 2 and α ∈ Γ , n = {x ∈ K; (j − 1)/n < x(α) < (j + 1)/n}. Then for n ∈ N put U = {U n ; α ∈ put Uα,j n α,j Γ, j ∈ ±N, 2 |j | n}. Note thatfor every inﬁnite sequence of distinct αi we have n = {x ∈ K; |x(α)| > 1/n}. Thus U is x(αi ) → 0 for all x ∈ K and that 2|j |n Uα,j n n } point ﬁnite. If x1 , x2 ∈ K, x1 = x2 , then (x1 )(α) = (x2 )(α) for some α ∈ Γ and thus {Uα,j weakly separates x1 and x2 . We refer to, e.g., [80, Chapter 12] for the proof of the reverse implication. (ii) We refer to [21] or, e.g., to [80, Chapter 12] for the proof. Theorem 2.16 should be compared with the result of Corson and Michael that a compact space K is metrizable if and only if it contains a σ -point-ﬁnite family F of open Fσ sets such that F separates points of K, i.e., given x1 , x2 ∈ K, x1 = x2 , there is G ∈ F such that / G (cf., e.g., [6, p. 157], [21] or [271]). The difference here is due to the x1 ∈ G and x2 ∈ special rôle of zero in the deﬁnition of Eberlein compacts. In the statement of Theorem 2.16, the word “Fσ ” cannot in general be dropped ([282]). T HEOREM 2.17. Let K be a compact set. Then (i) K is an Eberlein compact if and only if BC(K)∗ in its weak star topology is an Eberlein compact.

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(ii) K is a uniform Eberlein compact if and only if BC(K)∗ in its weak star topology is a uniform Eberlein compact. P ROOF. (i) follows from Theorems 4.6 and 4.8 and (ii) is in [21].

Assuming the Continuum Hypothesis, there is a Corson compact K such that BC(K)∗ in its weak star topology is not a Corson compact ([14]). We will test the results presented in this article on the spaces p (Γ ), c0 (Γ ), ∞ (Γ ) (cf., e.g., [168]) and on the following spaces. If Γ is a set, the space c∞ (Γ ) is a subspace of ∞ (Γ ) formed by all countably supported vectors in ∞ (Γ ), i.e., x ∈ ∞ (Γ ) belongs to c∞ (Γ ) if and only if {γ ∈ Γ ; x(γ ) = 0} is countable. The space C[0, ω1 ] is the Banach space of all continuous real-valued functions with the supremum norm on the ordinal segment [0, ω1 ] equipped with its usual order topology (cf., e.g., [63, p. 66] or [69, p. 59]). This space is isomorphic to its hyperplane C0 [0, ω1 ] formed by all elements in C[0, ω1 ] that vanish at ω1 . The ordinal segment is a scattered compact space and thus C[0, ω1 ] is an Asplund space. If c0 [0, ω1 ] denotes the space c0 (Γ ), where Γ = [0, ω1 ], then c0 [0, ω1 ]∗ is not weak star separable (consider the countability of the supports of elements of c0 [0, ω1 ]∗ ). The space c0 [0, ω1 ] embeds into C0 [0, ω1 ] (elementary). Thus C[0, ω1 ]∗ is not weak star separable. The space C0 [0, ω1 ] does not have property C ([41]). Indeed, for α ∈ [0, ω1 ), let Kα = {x ∈ C0 [0, ω1 ]; x − χ[0,α] 1/2}. If 1 αi < ω1 for i = 1, 2, . . . and β > sup αi , then χ ∈ Kαi . On the other hand, assume [0,β] 2 that for some x ∈ C0 [0, ω1 ], we have x ∈ α<ω1 Kα . Then x(α) = 1/2 for all 0 < α < ω1 , a contradiction. As C0 [0, ω1 ] is a pointwise closed hyperplane in C[0, ω1 ], C[0, ω1 ] is thus not Lindelöf in the topology of pointwise convergence on [0, ω1 ]. However, C[0, ω1 ] is Lindelöf in the topology of pointwise convergence on [0, ω1 ) ([233]). The Banach space D of all left continuous real-valued functions on [0, 1] that have ﬁnite right limits with the supremum norm has the property that D/C[0, 1] is isomorphic to c0 [0, 1] ([41], cf., e.g., [73, Chapter 2.3] or [80, Chapter 12]). Indeed, the operator T : D → c0 [0, 1) deﬁned by Tf = (f (x) − f (x + ))x∈[0,1), where f (x + ) denotes the right limit of f at x, is an onto map ([41] or [73, p. 46] and its kernel equals to C[0, 1]. Thus D/C[0, 1] is isomorphic to c0 [0, 1] by the Banach open mapping theorem (cf., e.g., [80, Chapter 2]). The space D is linearly isometric with C(K), where the compact set K (called “two arrows space”, “double arrow space” or “split interval”) is deﬁned as the lexicographically ordered product [0, 1] × {0, 1} of the unit interval and the two-element ordering {0, 1} with the order topology, i.e., K = {x ∈ [0, 1]} ∪ {x + ; x ∈ [0, 1]} and x ≺ x + ≺ y whenever x < y in [0, 1]. The basis of the topology is given by the intervals (z, x] and [x + , y). The projection p from K onto [0, 1] is deﬁned by p({x, x + }) = x. The isometry T from D onto C(K) is deﬁned for t ∈ [0, 1] by Tf (t) = f (t) and Tf (t + ) = lims→t + f (s) if t < 1 and Tf (1+ ) = f (1). Each point of K is a Gδ point of K. By considering the projection p, we can show that the space K is hereditarily Lindelöf and hereditarily separable (i.e., each subspace is such) (cf., e.g., [175]), any metrizable subspace of K is countable (cf., e.g., [73, p. 47] and by the Baire category theorem, K contains no dense Gδ metrizable subspace (cf., e.g., [73, p. 47]). In

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order to see that D in its weak topology is not normal ([41]), let S be the set of all characteristic functions of the intervals [0, t], 0 t < 1. The set S with its relative weak topology is weakly closed and discrete (every convex combination of these functions is 1 on some interval (0, t)). Thus there are 2c continuous functions on S. If D were normal in the weak topology, by Tietze’s extension theorem, there would be 2c different continuous functions on D in its weak topology. Since the cardinality of the set of all weakly continuous functions on a Banach space X equals the cardinality of X∗ ([41], see, e.g., [80, Chapter 12]), we would have that the cardinality of D ∗ is at least 2c . This is impossible as the cardinality of both C[0, 1]∗ and c0 [0, 1]∗ is c (for c0 [0, 1]∗ we use the fact that its elements have countable supports and for C[0, 1]∗ we use the separability of C[0, 1]). The dual space D ∗ is weak star separable (use the rational points in [0, 1]). Identify the set of all real numbers with the collection of sequences of rational numbers that are in turn identiﬁed with sequences of natural numbers. In this way we get continuum of inﬁnite sets Nγ in N such that the intersection of any two different of them is at most ﬁnite (almost disjoint sets) (Sierpi´nski). Assume without loss of generality that {Nγ } is a maximal family of almost disjoint sets that contain {Nγ }. The Johnson–Lindenstrauss subspace JL0 ([167, p. 224]) of ∞ is generated by c0 and the characteristic functions of χNγ of the sets Nγ in N. If x = ni=1 λi χNγi , we can ﬁnd y ∈ c0 such that x − y = maxni=1 |λi |. This implies that JL0 /c0 is isometric to c0 (c). If we put A = JL0 ⊕ R, then we get a subalgebra in ∞ . It follows that A is in fact C(K), where K is homeomorphic to the one point compactiﬁcation of a disjoint union of N and Γ , where card Γ = c and N ∪ Γ is topologized as follows: each point of N is isolated, while a set S ⊂ N ∪ Γ is a neighborhood of γ ∈ Γ if γ ∈ S and Nγ \ S is ﬁnite. The third Cantor derived set K (3) = ∅ (cf., e.g., [329]). Thus K is scattered. As K is uncountable and scattered, it cannot be metrizable (see, e.g., [80, Chapter 12] or [57, Chapter VI]). Since K is separable (use the points of N), BJL∗0 is weak star separable and thus JL∗0 is weak star separable. However, JL0 is nonseparable (as it factors to nonseparable c0 (c)). Pol showed that if K (ω1 ) = ∅, K is separable and C(K) is weakly Lindelöf, then K is countable ([249]). As K is uncountable, JL0 is not thus Lindelöf in the weak topology. The space JL0 is Lipschitz homeomorphic to c0 (c) ([2]), while JL0 it is not isomorphic to c0 (c) as c0 (c)∗ is not weak star separable. The space JL0 has property C as c0 (Γ ) is Lindelöf in its weak topology (Theorem 3.8), c0 is norm Lindelöf and property C is a three space property (Theorem 2.10). The space JL0 can be used to produce an equivalent dual norm on ∗∞ , the unit ball of which is not weak star separable, unlike the standard bidual norm (Goldstine’s theorem) (cf., e.g., [96], [80, Chapter 12]). The Banach space JL2 of Johnson and Lindenstrauss ([167, p. 222]) is deﬁned as follows. Let U0 be the algebraic span in ∞ of c0 ∪ {χNγ }, where Nγ ’s are as in JL0 . The norm on U0 is deﬁned by ||| ki=1 aγi χγi + y||| = max( ki=1 aγi χγi + y∞ , ( ki=1 |aγi |2 )1/2 ), where · ∞ is the usual supremum norm in ∞ , y ∈ c0 and γi = γj if i = j . The space JL2 is the completion of U0 in the norm ||| · |||. Then JL2 /c0 is isometric to 2 (c) and from the lifting property of 1 (c) ([197, p. 104]), we get that JL∗2 is isometric to 1 ⊕ 2 (c). It follows that JL∗2 is weak star separable. However, bounded sets in JL∗1 are not weak star separable ([167]). Thus JL2 is not isomorphic to a subspace of ∞ . The space 2 (c) is a quotient of JL2 and thus JL2 is not separable. The space JL2 has property C as property C is a three space property. Every nonseparable subspace Y of JL2 contains an isomorphic copy

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of c0 because otherwise (as JL2 admits a C ∞ smooth norm – Section 8), Y would admit a uniformly Fréchet smooth norm ([168,111], cf., e.g., [57, Chapter V]) and thus would be in particular reﬂexive ([57, Chapter IV], [80, Chapter 9] or [111]), which is not the case (see Section 4). The space JL2 in its weak topology is not Lindelöf ([65]). The dual ball of JL∗2 in its weak star topology is not angelic, however every weak star sequentially continuous linear functional on JL∗2 is weak star continuous ([65]). The space JL∗2 is not weak star sequentially separable, i.e., there is no countable set C ⊂ JL∗2 such that each f ∈ JL∗2 is the weak star limit of a sequence in C. Indeed, otherwise, by the Banach–Steinhaus uniform boundedness principle (cf., e.g., [80, Chapter 3]) and by Baire’s category theorem, JL2 would be isomorphic to a subspace of ∞ , which is not the case ([167]). Assuming the Continuum Hypothesis, Kunen (cf. [222]) constructed an uncountable scattered separable compact space K such that C(K) is hereditarily weakly Lindelöf, C(K)∗ is hereditarily weakly star separable ([149,158]) and such that if A is an uncountable set in C(K), then there is a ∈ A such that a ∈ conv{A \ {a}}. Note that C(K) is nonseparable as K is not metrizable (cf., e.g., [80, Chapter 3], [57, p. 258]). Every subspace of C(K) is a countable intersection of hyperplanes (cf. [222,96,149], [80, Chapter 12, Example]). From Pol’s result mentioned in the discussion on JL0 , it follows that for Kunen’s space K, the Cantor derivation number must be strictly greater than ω1 . This follows also from Theorem 8.8 below, as Kunen’s C(K) does not admit any Fréchet differentiable norm by the result in [158]. In order to see the latter, note that the dual sphere of a Fréchet differentiable norm would otherwise be norm separable. Indeed, it is weak star separable and at all of its points that attain their norm, the weak star and norm topologies coincide by Theorem 2.5. Hence C(K)∗ would be norm separable, a contradiction. We will call this C(K) space the Kunen’s C(K) space. Next space that we will test the results on is the James tree space JT ([150,196], see, e.g., [94], [80, Chapter 6, Example]). It is a separable space with nonseparable dual that does not contain any isomorphic copy of 1 . The space JT is the dual to JT ∗ , JT ∗ /JT ∗ is isomorphic to 2 (c) and JT ∗∗ is isomorphic to JT ⊕ 2 (c). The space JT ∗ is not Lindelöf in the weak topology. Indeed, Stegall showed that in nonseparable duals of separable spaces, there is an uncountable weakly discrete set ([288], see, e.g., [65], [80, Chapter 9]). The bidual ball BJT ∗∗ is angelic ([272,273,30]) but not Corson in its weak star topology, as a separable Corson compact is metrizable, which would mean that JT ∗ is separable (cf., e.g., [80, Chapter 3]). Ciesielski and Pol constructed in [38] (cf. [57, Chapter VI.8]) a Banach space that we will denote by CP and call it the Ciesielski–Pol CP space. It is a C(K) space, where the third derived set of the compact space K is empty and such that there is no bounded linear one-to-one operator from C(K) into any c0 (Γ ) and yet, C(K) contains a subspace Y such that both Y and C(K)/Y are isomorphic to some c0 (Γ ) s. D EFINITION 2.18. Let X be a Banach space with density character ℵ and μ be the minimal ordinal of cardinality ℵ. A transﬁnite sequence of bounded linear projections {Pα ; 0 α μ} of X is called a projectional resolution of identity (PRI) on X if (i) Pα = 1 for all α > 0, (ii) Pα Pβ = Pβ Pα = Pmin{α,β} , P0 = 0, Pμ = Identity, (iii) the density character of Pα X is less than or equal to max{ℵ0 , |α|} for all α, and

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(iv) the map α → Pα x is continuous from the ordinal segment [0, μ] in its order topology into X in its norm topology for every x ∈ X. If the density character of X is ℵ1 , then (Pα+1 − Pα )(X) is separable for all α < μ. This makes the situation easier. For many spaces, we can decompose (Pα+1 − Pα )(X) inductively and achieve that they are separable. The price we pay for this is that we lose norm 1 on projections. This is shown in [308], cf., e.g., [57, Chapter VII]. If {Pα ; α < μ} is a PRI on X, then Pβ x ∈ span{(Pα+1 −Pα )x; α < β} for each x ∈ X and each β μ. If {Pα ; α ω1 } is a PRI for X, then by transﬁnite induction, each x ∈ X lies in some Pα X for α < ω1 . Indeed, given x ∈ X, the function α → (I − Pα )x is continuous all α some β, where β < ω1 . Note and equals zero at ω1 . Thus this function is zero for that (iv) is usually achieved by ensuring that Pβ X = α<β Pα+1 X. If {xα , fα }0α<μ is an unconditional basis for a Banach space X, then the natural projections Pβ (x) = αβ fα (x)xα form a PRI on X. A biorthogonal system {xα , fα }α∈Γ in a Banach space X is an unconditional basis for X if for each x ∈ X, x = fα (x)xα , where the summation is meant in the sense that for a given ε > 0 there is a ﬁnite set A ⊂ Γ such that x − α∈B fα (x)xα ε whenever B is a ﬁnite subset of Γ that contains A. If X is a separable space, then a PRI on X is formed for instance by the projections P0 = 0, Pn = Identity, for 1 n ω0 . In C[0, μ], deﬁne the projections Pα , for 0 α μ by P0 = 0 and for x ∈ C[0, μ] and β μ, Pβ x = x on [0, β] and Pβ x = x(β) on [β, μ]. Then {Pα } is PRI on C[0, μ]. The Banach space C0 [0, ω1 ] does not admit any PRI in its supremum norm (cf., e.g., [57, p. 259], or [92]). However, the projections Pα x(β) = x(α) for β α and Pα x(β) = 0 elsewhere, satisfy all the properties needed for PRI but (iv). Moreover, ∗ ∗ ∗ α<ω1 Pα (C0 [0, ω1 ] ) = C0 [0, ω1 ] . An example of a Banach space that has no separable inﬁnite-dimensional complemented subspaces is ∞ as every complemented inﬁnite-dimensional subspace of ∞ is isomorphic to ∞ (Lindenstrauss, see, e.g., [197, p. 57]). In this direction, we refer also to [248]. Shelah and Stepr¯ans proved in [281] that there is a nonseparable Banach space X for which every bounded linear operator from X into X has the form S + ρI , where S is an operator with separable range, I is the identity operator and ρ is a real number. Argyros showed in [11] that there is a nonseparable Banach space X such that X does not contain any couple of inﬁnite-dimensional subspaces Y and Z with Y ∩ Z = {0} and Y + Z closed in X (hereditarily indecomposable space). His space X has moreover the following properties: ﬁrst, every bounded linear operator T from X into X has the form T = λI + S, where I is the identity operator and S is a weakly compact operator with separable range. Second, X∗ = W ⊕ 1 (Γ ) with W separable. Note that X∗ is then necessarily weak star separable. Indeed, if X∗ is not weak star separable, we can ﬁnd a subspace Y of density ℵ1 in X that has a PRI by the technique of the construction of a Schauder basic sequence in spaces ([23], cf., e.g., [197, p. 4]). Earlier, Odell showed in [227] that there is a nonseparable Banach space that contains no isomorphic image of c0 or p , p ∈ [1, ∞). The following simple statement is a version of Theorem 2.6. It shows the rôle of the Gâteaux smoothness in this area.

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T HEOREM 2.19. Assume that a Banach space X with density character ℵ1 admits a PRI {Pα ; 0 α ω1 } in its Gâteaux differentiable norm. Then α<ω1 Pα∗ (X∗ ) = X∗ and BX∗ in its weak star topology is a Corson compact. P ROOF. Given x ∈ SX , put f = x and let α < ω1 be such that x ∈ Pα X. Then Pα∗ f (x) = 1, Pα∗ f 1 and from the uniqueness of the support functional at x to BX , we have Pα∗ f = f , i.e., f ∈ Pα∗ (X∗ ). Given f ∈ SX∗ , by the Bishop–Phelps theorem, take fn ∈ SX∗ that attain their norms and fn − f → 0. If fn ∈ Pα∗n (X∗ ) with αn < ω1 and α = sup{αn }, then α < ω1 and f ∈ Pα∗ (X∗ ). For each α < ω1 , choose {xnα } dense in the ball of (Pα+1 − Pα )(X). Given f ∈ X∗ , let β < ω1 be such that f ∈ Pβ∗ (X∗ ). Then f = Pβ∗ f is supported on a countable {xnα ; α < β}. The proof can easily be ﬁnished by using evaluations of f ∈ X∗ on all {xnα }.

3. Weak compact generating and projectional resolutions A fundamental concept in nonseparable Banach space theory is the concept of weakly compactly generated spaces, introduced and studied ﬁrst by Corson and Lindenstrauss (cf. [42, 195,191,192] and references therein). This concept allows for splitting many nonseparable Banach spaces into separable ranges of projections and generates nice geometric and topological properties of such spaces. D EFINITION 3.1. (i) A Banach space X is weakly compactly generated (WCG) if there is a weakly compact set K ⊂ X such that X = span K. (ii) A Banach space X is weakly countably determined or a Vašák space (WCD) if there is a sequence of weak star compact sets Kn ⊂ X∗∗ such that for x ∈ X and u ∈ X∗∗ \ X there is n such that x ∈ Kn and u ∈ / Kn . (iii) A Banach space X is weakly Lindelöf determined (WLD) if BX∗ in its weak star topology is a Corson compact. (iv) A Banach space X is weakly Lindelöf (WL) if X in its weak topology is a Lindelöf space. (v) A Banach space X is said to have the density property or to be a DENS space if the weak star density of X∗ equals the density of X. Baire’s space Σ := NN will be considered in its product topology. It can be identiﬁed with the irrationals with their topology inherited from R (by means of partial fractions for instance, cf., e.g., [57, p. 248]). If {Kn } is the sequence in Deﬁnition 3.1(ii), note that {Kn ; x ∈ Kn } ⊂ X for every x ∈ X. Denote by Σ = σ = (ni ) ∈ Σ; ∅ = Kni ⊂ X .

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Deﬁne the multivalued mapping ϕ from Σ into X by ϕ(σ ) =

Kni .

Then ϕ is an usco map for the weak topology in X, it is upper semicontinuous and ϕ(σ ) is a weakly compact set in X. The upper semicontinuity of ϕ means that for every weakly open set U in X, {σ ∈ Σ ; ϕ(σ ) ⊂ U } is open in Σ. If Σ can be chosen Σ, then X is called weakly K-analytic (cf., e.g., [57, p. 250], [73, p. 67], [269]). Before proceeding, note that for every Banach space X, the weak star density of X∗ is less than or equal to the norm density of X (cf., e.g., [80, Chapter 11]). The part (i) of the following result is in [46] (cf., e.g., [80, Chapter 11] or [73, p. 12]). The part (ii) is from [13]. T HEOREM 3.2. (i) A Banach space X is WCG if and only if there is a reﬂexive space Y with an unconditional basis and a bounded linear one-to-one operator T from Y onto a dense set in X. (ii) A Banach space X is WLD if and only if there is a WLD space Y with an unconditional basis and a bounded linear operator T from Y onto a dense set in X. P ROOF. (i) If Y is reﬂexive and T is a bounded linear operator from Y onto a dense subset of a Banach space X, then T (BY ) is weakly compact and X = span T (BY ). Thus X is then WCG. The main idea of the proof of the reverse implication is the following interpolation technique ([46], cf., e.g., [73, p. 12] or [80, Chapter 11]). Assuming that K is convex, symmetric and weakly compact such that X = span K, forx ∈ X and n ∈ N, put xn the Minkowski functional of (2n K + 2−n BX ), |||x||| = ( x2n )1/2 , Y = {x ∈ X; |||x||| < ∞}, Xn = (X, xn ), Z = (Xn )2 , T be the inclusion map of the Banach space Y into X and ϕ be a map from Y into Z deﬁned by ϕ(y) = (T y, T y, . . .). If x ∈ K, then xn 2−n for each n and thus |||x||| 1. Hence K ⊂ T (BY ). Furthermore, if x ∈ T (BY ), then xn 1 for each n and x ∈ Un for each n. Thus for each n w∗

w∗

w∗

⊂ 2n K + 2−n BX ⊂ 2n K + 2−n BX∗∗ = 2n K + 2−n BX∗∗ . Hence we have T (B Y) ∗∗ n T (BY ∗∗ ) ⊂ n (2 K + 2−n BX∗∗ ) ⊂ n (X + 2−n BX∗∗ ) = X. Thus Y ∗∗ ⊂ (T ∗∗ )−1 (X) ⊂ Y and Y is reﬂexive. We refer to [13] for the proof of (ii). T HEOREM 3.3. Every weakly compact set in a Banach space in its weak topology is linearly homeomorphic to a weakly compact set in a reﬂexive space considered in its weak topology. P ROOF. Use T −1 from the proof of Theorem 3.2.

T HEOREM 3.4. The dual ball of a WCG space in its weak star topology is linearly homeomorphic to a weakly compact set in a reﬂexive space considered in its weak topology. P ROOF. Use T ∗ from the proof of Theorem 3.2.

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The following theorem is the result of Amir and Lindenstrauss in [7] (cf., e.g., [80, Chapter 11]). More general result is contained in Theorem 3.8. T HEOREM 3.5. Let X be a WCG space. Then X is a DENS space. P ROOF. The weak star density of X∗ is always less than or equal to the density of X (cf., e.g., [80, Chapter 11]). If S is a countable weak star dense set in X∗ and T is a bounded linear operator from a reﬂexive space Y onto a dense set in X, then T ∗ (S) is weakly dense in Y ∗ and thus Y ∗ is weakly separable and thus norm separable (cf., e.g., [80, Chapter 3]. Hence Y is separable and so is X. The general case follows similarly. Alternatively, if X∗ is weak star separable and X = span K, for a weakly compact set K, then K is norm separable and so is X (cf. Section 2). C OROLLARY 3.6. If X∗ is WCG, then X is an Asplund space. P ROOF. Let Y be a separable subspace of X. We need to show that Y ∗ is separable. The space Y ∗ is isomorphic to X∗ /Y ⊥ and a quotient of a WCG space is WCG. Thus Y ∗ is WCG. From Goldstine’s theorem, Y ∗∗ is weak star separable. From Theorem 3.5 we thus get that Y ∗ is separable. If μ is a ﬁnite measure, by using the “identity” map from L2 (μ) into L1 (μ), we can see that L1 (μ) is WCG if μ is a ﬁnite measure. The following is a result of Rosenthal ([271], cf., e.g., [60, p. 190]). T HEOREM 3.7. There is a ﬁnite measure μ such that L1 (μ) contains a subspace with unconditional basis that is not WCG. Thus a subspace of WCG space need not in general be WCG. For further examples of this, cf., e.g., [73, p. 29]). A Banach space X is a subspace of a WCG space if and only if there is a sequence {Kn } of weak star compact convex symmetric sets in X∗∗ such that for every x ∈ X and every ε > 0, there is n such that x ∈ Kn ⊂ X + εBX∗∗ [84]. It is not difﬁcult to see that a subspace of a WCD space is WCD (cf., e.g., [57, Chapter VI]). From Theorem 2.15 it follows that a subspace of a WLD space is WLD. T HEOREM 3.8. Every WCG Banach space is WCD, every WCD space is WLD, every WLD space is WL, every WLD space admits a PRI, every WLD is DENS and every WL space has property C. No couple of these classes of spaces coincide. P ROOF. Let X be WCG and K be a weakly compact convex symmetric set such that X = span K. Then {nK; n ∈ N} is dense in X and it is easy to see that the sets Kn,m := nK + m1 BX∗∗ satisfy the requirements to get that X is WCD. We postpone the proof of the fact that every WCD space is WLD till Section 4. Alster and Pol and independently Gul’ko showed in [5] and [127] that any WLD space is a WL space (cf., e.g., [80, Chapter 12], where Orihuela’s functional analytic proof of this result from [233] is

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presented). In fact, they showed that a C(K) space is pointwise Lindelöf if K is a Corson compact. Valdivia ([316,318,319]) and independently Argyros and Mercourakis ([13]) proved that WLD implies the existence of a PRI. In the end of the present proof we will outline the direct proof of Amir and Lindenstrauss in [7] (a modiﬁcation of their original proof, as it is in [160]) that WCG implies the existence of PRI. The proof that WLD implies DENS is postponed until Section 4. Every WL space clearly has property C. The subspace of L1 (μ) in Theorem 3.7 is an example of WCD that is not WCG. Under the Continuum Hypothesis, Ciesielski and Pol have an example of a WL space that does not admit any one-to-one bounded linear operator into any c0 (Γ ) and thus their space is not WLD (see Section 4). Kunen’s space C(K) is an example of a WL space that is not DENS. For the rest of examples needed to show that no couple of these classes of spaces coincide we refer to [222,206,14,13] and references there and in [10], [57, Chapter VI], [73] and [80, Chapter 12]. The Preiss–Talagrand original proof that any WCD space is a WL space (see [294], cf., e.g., [57, Chapter VI] and [80, Chapter 12]) follows the argument explained in th text following Deﬁnition 3.1 (see, e.g., [57, p. 250]). We will show now the result of Amir and Lindenstrauss in [7] that every WCG space admits a PRI. This is one of the most important results in nonseparable Banach space theory. L EMMA 3.9. Let K be a convex and symmetric weakly compact set such that nonseparable X = span K. Deﬁne the norm on X∗ (non-equivalent in general) by |f | = supK f . Let F be a ﬁnite-dimensional subspace of X∗ , n ∈ N and x1 , . . . , xl ∈ X. Then there is an ℵ0 -dimensional (non-closed) subspace Z of X∗ containing F such that for every subspace of X∗ containing F as a subspace of codimension n and every ε > 0 there is a linear operator T : V → Z satisfying T (f ) = f for every f ∈ F , T 1 + ε, |T | 1 + ε and |v(xi ) − T v(xi )| εv for every v ∈ V and i = 1, . . . , l. P ROOF. The space Z is constructed as follows: choosing a ﬁnite net {vh } for F and a ﬁnite net {λj } for Sn1 , consider the map Φ from (X∗ )n into RN for a large enough N j deﬁned for (f1 , . . . , fn ) ∈ (X∗ )n by Φ(f1 , . . . , fn ) := (fi , |fi |, vh + ni=1 λi fi , |vh + n j N i=1 λi fi |, fi (xk )). Using in R the metric ρ of maximum coordinate distance we see that we can ﬁnd a sequence {Φ(f t )}t that is ρ-dense in Φ((X∗ )n ). From this the space Z is constructed. L EMMA 3.10. In the notation as above, for every ﬁnite-dimensional subspace F of X∗ and sequence {xi } ⊂ X, there is a bounded linear operator T from X∗ into a weak star separable subspace of Z of X∗ , such that T = |T | = 1, T v = v for every v ∈ F , T ∗ (xi ) = xi for every i. Moreover, T ∗ (X∗ ) is weak star separable and thus T (X) is norm separable (note that X is WCG). P ROOF. This lemma is proven by using Tychonoff theorem in (2BX∗ )BX∗ , where in 2BX∗ we consider the weak star topology. The weak-star to weak-star continuity of T is ensured by |T | = T = 1 and by the bipolar theorem, since T ∗ (K) ⊂ K as K is weakly compact and thus T ∗ (X) ⊂ X as T = 1.

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L EMMA 3.11. In the notation as above, let ℵ be an inﬁnite cardinal, F be a subspace of X∗ of the weak star density ℵ and G be a subspace of X of density ℵ. Then there is a bounded linear weak-star to weak-star continuous projection P in X∗ such that |P | = P = 1, Pf = f for every f ∈ F , P ∗ x = x for every x ∈ G and dens P (X) = w∗ dens P ∗ (X∗ ) ℵ. P ROOF. This lemma is proven by using the preceding lemma and Mazur’s exhaustion argument, i.e., the operators are taken to be the identity on larger and larger subspaces in Zn from the preceding lemma and taking the weak star closure W of their linear span. The limit operator is then the identity on W as it is weak star to weak star continuous. Moreover, it maps X∗ into W . Thus it is a projection onto W . The ﬁnal result is then achieved by transﬁnite induction. Additional work is needed to ensure that the projections preserve a given Y ⊥ ⊂ X∗ . One way of doing this is to work in countably many norms whose unit balls are (BY ⊥ + n1 BX∗ ). Then we preserve Y ⊥ . By using this and the properties of the set deﬁning the WCD, we can show that every WCD space admits a PRI. This is the result of Vašák in [325] (see, e.g., [57, Chapter VI]). The construction of PRI’s was extended to spaces with certain biorthogonal systems by Plichko in [245]. Then, independently, the so called projectional generator was introduced in [235]. We refer to, e.g., [73, p. 106] and [57, Chapter VI] for this topic. The space c0 (Γ ) for any Γ is WCG (use the “identity” operator from 2 (Γ ) into c0 (Γ )). If X is a separable Banach space, {xn } is dense in SX and the operator T from X∗ into 2 is deﬁned by Tf (i) = f (xi )/2i , then the dual operator T ∗ maps 2 onto a dense set in X. Thus any separable Banach space is WCG (Theorem 3.2). The spaces ∞ and 1 (Γ ) for uncountable Γ are not WCG, as they are not separable and weakly compact sets in them are norm separable (cf., e.g., [80, Chapter 3, Example]). The space 1 (c) is isometric with a subspace of ∞ (cf., e.g., Section 2). As B∗∞ is weak star separable by Goldstine’s theorem, 1 (c)∗ is weak star separable and 1 (c) is thus not DENS. The space 1 (c) does not satisfy Pol’s dual formulation of the property C (Theorem 2.7). Thus ∞ does not have property C. The space C[0, ω1 ] does not have property C (Section 2) and thus is not WLD. The space C[0, ω1 ] is DENS as it contains a subspace isomorphic to c0 [0, ω1 ], which is DENS. The space JL0 is not WL (Section 2) and it is not DENS. Thus it is not WCG by Theorem 3.8. However, JL0 /c0 is isometric to the WCG space c0 (Γ ). The space c0 is not complemented in JL0 , since otherwise JL0 would be WCG as the direct sum of two WCG spaces is WCG. Hence c0 is not complemented in ∞ (Phillips, cf., e.g., [80, Chapter 5]). The nonseparable space JL2 is not DENS (as JL∗1 is weak star separable) and JL∗2 is WCG. Kunen’s C(K) space is not WCG as it is not DENS. The space JL2 is not WL ([65]). The space D is not WL (Section 2). Let us ﬁnish this section with mentioning that for, say, nonseparable reﬂexive Banach spaces X, PRI can be constructed by using the fact that the standard “up and down” argument leads the existence of closed subspaces Y of X and W of X∗ that mutually one norm each other. Then Y + W⊥ = X and again the method from p. 4 in [197] shows that Y is complemented in X. This method originated in the works of Plichko (in spaces with biorthogonal systems) and independently in the works of Gul’ko, Orihuela and Valdivia. It

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is explained in [57, Chapter VI] and in [73, Chapter 6], see also [80, Chapter 11, Example]. It works for WCD spaces as well [57, Chapter VI] and as well in WLD spaces ([73]). In the direction of PRI’s, we refer also to the recent result in [248].

4. Biorthogonal systems and quasicomplements A biorthogonal system in a Banach space X is a collection {xα , fα }α∈Γ ⊂ (X × X∗ ) such that fβ (xα ) = 1 if α = β and fβ (xα ) = 0 otherwise. D EFINITION 4.1. A biorthogonal system {xα , fα }α∈Γ in a Banach space X is called a Markushevich basis or simply an M-basis for X if span{xα }α∈Γ = X and span w∗ {fα }α∈Γ = X∗ . If an M-basis {xα , fα }α∈Γ has the property that span · {fα }α∈Γ = X∗ , then {xα , fα }α∈Γ is called shrinking. A Markushevich basis {xα , fα }α∈Γ will be called a weakly compact M-basis, respectively weakly Lindelöf M-basis, if the set {xα }α∈Γ ∪ {0} in its relative weak topology is compact respectively Lindelöf. Any Schauder basis for a separable Banach space (cf., e.g., [168] or [197, p. 1]) and any unconditional basis for a Banach space (cf., e.g., Section 2) is a Markushevich basis. For an example of a Markushevich basis for a separable space that is not a Schauder basis in any ordering the indexes of the basis see, e.g., [197, p. 43]. Any shrinking M-basis {xα , fα } of a Banach space X with {xα } bounded is a weakly compact M-basis and thus X is then WCG. Indeed, given a sequence {yn } ⊂ {xα } of distinct points, for each fα we have fα (yn ) → 0 and thus yn → 0 weakly as span · {fα } = X∗ and {xα } is bounded. Hence {xα } ∪ {0} is weakly compact by the Eberlein–Šmulyan theorem. If {xi , fi } is a Markushevich basis for a separable Banach space X, then {i −1 xi −1 xi , fi } is a norm compact Markushevich basis for X. There is a normalized unconditional basis of a separable Banach space (i.e., xn = 1 for all n) that is weakly compact but not shrinking ([242]). The following theorem summarizes equivalent characterizations of several classes of spaces discussed in Section 3 in terms of Markushevich bases. T HEOREM 4.2. (i) A Banach space X is WCG if and only if it admits a weakly compact M-basis. If {xα , fα } is a weakly compact Markushevich basis for a Banach space X, then there is a reﬂexive Banach space Y with an unconditional basis {yα , gα } and a bounded linear operator T from Y into X such that T ({yα }) = {xα }. (ii) A Banach space X is a subspace of a WCG space if and only if there is a Markushevich basis {xαε , fα }α∈Γ of X such that for every ε > 0, the set Γ can be decomposed into Γ = ∞ i=1 Γi so that for every f ∈ BX ∗ and every i, card γ ∈ Γiε ; f (xγ ) > ε < ∞. Any M-basis in a subspace of a WCG space has this property.

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N (iii) A Banach space X is WCD if and only if for some a Marku Σ ⊂ N , X admits shevich basis {xα , fα }α∈Γ such that {xα }α∈Γ ∪ {0} = {LK ; K ∈ K(Σ )}, where LK is weakly compact in X and K1 ⊂ K2 implies LK1 ⊂ LK2 . Here K(Σ ) denotes the collection of all compact subsets of Σ . (iv) A Banach space X is WLD if and only if it admits a weakly Lindelöf M-basis. If X is WLD and {xα , fα } is an M-basis for X, then {xα , fα } is a weakly Lindelöf M-basis and every f ∈ X∗ is countably supported on {xα }. (v) Let X be a Banach space. Then X∗ is WL if and only if X∗ is WLD. (vi) Assume that the density of a Banach space X is ℵ1 . Then X is WLD if and only if X admits a PRI in every equivalent norm. (vii) Assume that the density of a WLD Banach space X is ℵ1 and that {Pα }, α ω1 is a PRI for X. Then α<ω1 Pα∗ (X∗ ) = X∗ . If {Qα }, α ω1 is another PRI for X, then for every α < ω1 , there is β > α, β < ω1 such that Pβ = Qβ . (viii) If K is a Corson compact, then C(K) admits a PRI formed by projections that are pointwise continuous and admits an M-basis that is Lindelöf in the pointwise topology. (ix) For separable X, X∗ has property C if and only if X does not contain an isomorphic copy of 1 . (x) A Banach space X is WCD if and only if there is an M-basis {xγ , fγ }γ ∈Γ for X such that Γ can be split into Γ = ∞ i=1 Γn in such a way that given γ ∈ Γ , given f ∈ BX ∗ and given ε > 0, there is n such that γ ∈ Γn and

card γ ∈ Γn ; f (xγ ) > ε < ∞. P ROOF. (i) Let a weakly compact convex and symmetric set K generate a nonseparable WCG space X. We construct a PRI of X such that Pα (K) ⊂ K. Then we use the proof of Markushevich’s result that every separable Banach space admits an M-basis ([197, p. 43]), the construction of appropriate M-bases in (Pα+1 − Pα )X and transﬁnite induction to get that every WCG space X admits an M-basis that is contained in 2K (cf., e.g., [80, Chapter 11]). The rest of (i) is in [46]. (ii) follows from (iv) and from the results in [93]. For a short proof we refer to [83]. (iii) was proved in [324], based on [205]. (iv) ([233,319,324]) Assume that {xα , fα } is a weakly Lindelöf M-basis for X. Let f ∈ X∗ and n ∈ N be given. Let Un = {x ∈ {xα } ∪ {0}; |f (x)| < 1/n} and Uα = {x ∈ {xα } ∪ {0}, fα (x) > 0}. Then {Un } ∪ {Uα } is an open cover of {xα } ∪ {0} in its weak topology. / Uβ if α = β, From the Lindelöf property, this cover has a countable subcover. Since xα ∈ all but countably many xα are in Un . Hence f is countably supported on {xα }. Thus X is WLD. If X is WLD, then X is WL and has a PRI (Theorem 3.8). From this we have that X has an M-basis {xα , fα }. As X is WL, its weakly closed subset {xα } ∪ {0} is weakly Lindelöf. (v) If X∗ is weakly Lindelöf, then X is Asplund. Indeed, otherwise X∗ contains an uncountable weakly discrete set by Stegall’s result ([288], see [65]). Hence X∗ admits an M-basis by Theorem 7.13. Thus X∗ is WLD by Theorem 4.4 below. (vi) (Kalenda [177]) If dens X = ℵ1 , X admits a PRI for every equivalent norm and BX∗ is not Corson in its weak star topology, then it is proven in [54] that BX∗ in its weak star topology contains a copy D of the segment [0, ω1 ]. It follows from the result in [75]

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(which originated in [95,97] and [124]) that X admits an equivalent norm · such that D ⊂ conv · E, where E is formed by supporting functionals to the ball of · in X at all points of Gâteaux smoothness of · . Since X in · admits a PRI, similarly as in the proof of Theorem 2.19 we get that the dual ball is Corson in its weak star topology, a contradiction. (vii) The ﬁrst part follows from the fact that BX∗ in its weak star topology is angelic. For the second part we refer to [245]. (viii) We refer to, e.g., [57, p. 254] for a proof, originally in [14] and [316,319] where ideas independently obtained also by Gul’ko and Plichko can be found. (ix) If X contains an isomorphic copy of 1 , then X∗ contains an isomorphic copy of 1 (c) by the result of Pełczy´nski in [241]. As 1 (c) does not have property C, neither does X∗ . If X does not contain an isomorphic copy of 1 , BX∗∗ in its weak star topology is an angelic space ([30]) and thus X∗ has property C by Theorem 2.7. (x) This uses Sokolov’s characterization of WCD spaces ([286], see also [73, p. 130]) and is proved in [85]. For spaces with unconditional bases there are the following results of Johnson (see [271]) and Argyros and Mercourakis [13]. T HEOREM 4.3. Assume that a Banach space X admits an unconditional basis. Then (i) If X is WCG, then every unconditional basis in X is σ -weakly compact. (ii) X is WLD if and only if it does not contain an isomorphic copy of 1 (ℵ1 ). P ROOF. (i) We refer to [271]. (ii) The space 1 (ℵ1 ) is not WLD (Section 3) and thus it cannot have a copy in X if X is WLD by Theorem 3.8. If X is not WLD, then X contains an isomorphic copy of 1 (ℵ1 ) (the proof is similar to that a separable space X with unconditional basis contains a copy of 1 if X∗ is nonseparable. (James, see, e.g., [197, p. 21] or [73, p. 49]). The following result can be found in [233] and [324]. T HEOREM 4.4. Let a Banach space X admit an M-basis. Then X has property C if and only if X is WL if and only if X is WLD. P ROOF. Obviously, WL implies property C and WLD implies WL by Theorem 3.8. It remains to show that in our case, property C implies X is WLD. To this end, let S denote the collection of all elements of X∗ that are countably supported on {xα }. From Theorem 2.7 it follows that S ∩ BX∗ is weak star closed. Then by the Banach–Dieudonné theorem, S is weak star closed in X∗ as S is a subspace of X∗ . Since S contains all fα , we have S = X∗ . It is easy to see that then BX∗ in its weak star topology is a Corson compact (by considering the evaluation map of the elements of BX∗ on {xα }). If X and Y are separable inﬁnite-dimensional Banach spaces, then there is a bounded linear one-to-one operator from X onto a dense set in Y . Indeed, if {xi , fi } and {yi , gi } are Markushevich bases in X and Y respectively with {fi } and {yi } bounded (cf., e.g., [197,

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−i p. 43]), put for x ∈ X, T x = ∞ i=1 2 fi (x)yi . Then T is a bounded linear operator from X into Y and if T x = 0, then for each i, 0 = gi (T x) = 2−i fi (x) and thus x = 0, as {xi , fi } is a Markushevich basis in X. Moreover, for each i, T (xi ) = 2−i yi and thus T (X) is dense in Y . For nonseparable Banach spaces, the existence of linear injections has a profound impact on the structure of spaces involved. For example, from the proof of Pitt’s theorem (cf., e.g., [197, p. 76]) we obtain that if Γ is uncountable, then there is no bounded linear one-to-one operator from p (Γ ) into q (Γ ) if q < p. Also, if Γ is uncountable, then there is no bounded linear one-to-one operator T from c0 (Γ ) into any reﬂexive Banach space X. Indeed, otherwise, T ∗ (BX∗ ) would be a weakly compact set in c0 (Γ )∗ , which would be norm compact by Schur’s theorem. Thus T ∗ (X∗ ) would be norm separable and c0 (Γ )∗ would be weak star separable as T ∗ maps X∗ onto a weak star dense set in c0 (Γ )∗ . However, c0 (Γ )∗ is not weak star separable if Γ is uncountable. If Γ is uncountable, then there is no bounded linear operator from c0 (Γ ) into ∞ . Indeed, as ∗∞ is weak star separable, we would otherwise obtain that so is c0 (Γ )∗ , which is not the case. Thus the following theorem is a useful result in this direction. T HEOREM 4.5. Assume that a Banach space X admits an M-basis. Then there is a bounded linear one-to-one operator from X into c0 (Γ ) for some Γ . If X is WCG, then there is a bounded linear one-to-one operator from X∗ into c0 (Γ ) for some Γ that is weak star to weak continuous. P ROOF. If {xα , fα }α∈Γ with {fα } bounded is an M-basis for X, deﬁne a map T from X into ∞ (Γ ) by T x(α) = fα (x) for α ∈ Γ . Then T maps span {xα } one-to-one into c0 (Γ ) and from the continuity of T and the closedness of c0 (Γ ) in ∞ (Γ ) we get that T maps X into c0 (Γ ). If {xα , fα }α∈Γ is a weakly compact M-basis for X, deﬁne a bounded linear operator T from X∗ into ∞ (Γ ) by Tf (α) = f (xα ). Let {yn } be a sequence of distinct points in {xα }. Then fα (yn ) → 0 for every α and since {xα } ∪ {0} is weakly compact, we have yn → 0 weakly in X. From this it follows that T maps X∗ into c0 (Γ ). The operator T is oneto-one and it is weak star to weak continuous on BX∗ as on bounded sets in c0 (Γ ) the weak and pointwise topologies coincide. Hence T is weak star to weak continuous by the Banach–Dieudonné theorem. T HEOREM 4.6. Every weakly compact set in a Banach space is an Eberlein compact. If X is a WCG Banach space, then BX∗ in its weak star topology is an Eberlein compact. P ROOF. If K is a weakly compact set in a Banach space X, then Z := span K is WCG, Z admits an M-basis and thus there is a bounded linear operator T from Z into some c0 (Γ ) (Theorems 4.2, 4.5). Then T is a weak homeomorphism of K onto a weakly compact subset of c0 (Γ ). Similarly one can prove the second part of the statement by using Theorem 4.2(i). By using Theorem 4.6, Theorem 3.3 and [80, Chapter 12], we can state

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T HEOREM 4.7. Let K be a compact space. Then the following are equivalent. (i) K is an Eberlein compact. (ii) K is homeomorphic to a weakly compact set in a Banach space in its weak topology. (iii) K is homeomorphic to a weakly compact set in a reﬂexive Banach space in its weak topology. (iv) K is homeomorphic to a pointwise compact space in C(L) for some compact space L in its pointwise topology. The following result can be found, e.g., in [80, Chapter 11]). T HEOREM 4.8. A compact set K is an Eberlein compact if and only if C(K) is WCG if and only if C(K) contains a weakly compact set that separates points of K. that a weakly compact set K ⊂ 12 Bc0 (Γ ) . Let A be the family of all ﬁnite P ROOF. Assume *n products i=1 x(γi ), x ∈ K, γi ∈ Γ , joined with the function 1. Then every sequence of distinct elements of A converges pointwise and thus weakly to zero (Grothendieck’s theorem). Thus A ∪ {0} is a weakly compact set in C(K) and span A is an algebra in C(K) that separates points of K. Thus span A = C(K) by the Stone–Weierstrass theorem. Hence C(K) is WCG if K is an Eberlein compact. If C(K) is WCG, then BC ∗ (K) is an Eberlein compact in the weak star topology (Theorem 4.6) and thus K ⊂ BC ∗ (K) is an Eberlein compact. We refer to [80, Chapter 11] for the rest of the proof. T HEOREM 4.9. (i) A Banach space X is a subspace of a WCG Banach space if and only if BX∗ in its weak star topology is an Eberlein compact if and only if C(BX∗ ) is WCG, where BX∗ is considered in its weak star topology. (ii) A Banach space X is WCD if and only if C(BX∗ ) is WCD, where BX∗ is considered in its weak star topology. P ROOF. (i) If X is a subspace of WCG space, then BX∗ is a continuous image of an Eberlein compact and thus BX∗ in its weak star topology is an Eberlein compact by Theorem 2.15. If BX∗ in its weak star topology is an Eberlein compact, then X is a subspace of the WCG space C(BX∗ ) (Theorem 4.8), where BX∗ is considered in its weak star topology. Thus (i) follows by Theorem 4.8. (ii) We refer to [73, pp. 121, 123] for the proof. Note that the operator T x = {2−i xi } maps ∞ one-to-one into c0 and yet, ∞ does not admit any M-basis. Indeed, we have the following results of Johnson in [165] and Plichko in [246] (cf., e.g., [80, Chapter 6]). T HEOREM 4.10. The space ∞ admits no M-basis but it is a complemented subspace of a Banach space Z with an M-basis. P ROOF. We will prove the ﬁrst part of the statement. Assume that {xα , fα } is an M-basis in X = ∞ . Then put Y := span · {fα }. Let {yn } be a sequence in Y ∩ BX∗ . As all yn have

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countable support on {xα } we can extract a subsequence {ynk } such that ynk → y ∈ BX∗ in the weak star topology. As X∗ is a Grothendieck space ([61, p. 103]), we have ynk → y weakly and thus y ∈ Y . Hence Y is a reﬂexive space. Consequently Y ∩ BX∗ is weakly compact and thus weak star closed. Hence Y is weak star closed by the Banach–Dieudonné theorem. As Y contains {fα } which is weak star dense in X∗ , Y = X∗ and X is reﬂexive, a contradiction. Let Γ be a set and Σ be a subset of the space NN . The Banach space c1 (Σ × Γ ) is deﬁned as the subspace of ∞ (Σ × Γ ) formed by all functions whose restrictions to K × Γ belong to c0 (K × Γ ) for all compact sets K of Σ . T HEOREM 4.11. (i) A Banach space X is WCG if and only if there is a bounded linear one-to-one operator from X∗ into some c0 (Γ ), that is weak star to weak continuous. (ii) A Banach space X is WCD if and only if there is a set Σ ⊂ NN , a set Γ and a bounded linear one-to-one operator of X∗ into c1 (Σ × Γ ) that is weak star to pointwise continuous. (iii) X is WLD if and only if there is a bounded linear one-to-one operator from X∗ into some c∞ (Γ ) that is weak star to pointwise continuous. P ROOF. (i) If X is WCG, then the statement follows from Theorem 4.5. If such T exists, then T ∗ (B1 (Γ ) ) ⊂ X is weakly compact and generates X. (ii) was proved in [205]. (iii) If such T exists, then, clearly, BX∗ in its weak star topology is a Corson compact. On the other hand, let {xα , fα }α∈Γ be a weakly Lindelöf M-basis for X (Theorem 4.2). By Theorem 4.2(ii), each element of X∗ is countably supported on {xα }α∈Γ . Assuming that {xα } is bounded, the operator T from X∗ into c∞ (Γ ) deﬁned by Tf (α) = f (xα ), α ∈ Γ , satisﬁes the requirements. We will now show the missing parts in the proof of Theorem 3.8. First of all, if X is WCD, then it has PRI and an M-basis ([325]). By the proof of Theorem 3.8, X is WL. Thus it follows from Theorem 4.4 that WCD implies WLD. We will now show that X is DENS if X is WLD. For simplicity, let us prove that X is separable if X is WLD and X∗ is weak star separable. Let a countable S ⊂ X∗ be weak star dense in X∗ . Let {xα , fα }α∈Γ ∗ be an M-basis for X. By Theorem 4.2(ii), each element of X is ∗countably supported on {xα }. We note that Γ = f ∈S {α ∈ Γ ; f (xα ) = 0}. As each f ∈ X is countably supported on {xα } and S is countable, we get that Γ is countable. Thus X is separable. Before proceeding, we recall that by a subspace of a Banach space we mean a closed subspace. D EFINITION 4.12. A subspace Y of a Banach space X is said to be quasicomplemented in X if there is a subspace Z of X such that Y ∩ Z = {0} and Y + Z = X. The subspace Z is called a quasicomplement of Y in X. Murray showed in [216] that any subspace of a reﬂexive separable Banach space X is quasicomplemented in X. This result was extended to all separable spaces X by Mackey

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in [199]. The following result in [319] and [323] is a variant of the result of Lindenstrauss in [193]. T HEOREM 4.13. Assume that X is a WLD space and Y is a subspace of X. Then Y is quasicomplemented in X. P ROOF. If X is separable and {xi , fi } is a Markushevich basis of Y , then fi can be extended to X and {zi } ∈ X and gi ∈ X∗ added such that {xi , zi , fi , gi } is a Markushevich basis of X ([128], cf., e.g., [80, Chapter 11]). Then Z := span{zi } is a quasicomplement of Y in X. Indeed, if w ∈ Y ∩ Z, then fi (w) = 0 = gi (w) for all i. Thus w = 0 as {fi } ∪ {gi } separate the points of X. Having this proved, we proceed by transﬁnite induction, using projectional resolution of identity {Pα } of X such that Pα Y ⊂ Y . Indeed, if Zα is a quasicomplement of (Pα+1 − Pα )Y in (Pα+1 − Pα )X, then span( Zα ) is a quasicomplement of Y in X. If Z is a Banach space, a sequence (yn ) ⊂ Z ∗ is called w∗ -basic provided that ∗ there exists (xn ) ⊂ Z biorthogonal to (yn ) such that for each y ∈ span w {yn }, y = n ∗ limn i=1 y(xi )yi in the weak star topology of Z ([197, p. 10]). The following is a result of Lindenstrauss and Rosenthal (cf. [166]). T HEOREM 4.14. Assume that Y is a subspace of a Banach space X such that Y ∗ is weak star separable and X/Y has an inﬁnite-dimensional separable quotient. Then Y is quasicomplemented in X. P ROOF. Since X/Y has a separable quotient, there exists a biorthogonal sequence (xn , xn∗ ) in X with (xn∗ ) ⊂ Y ⊥ , (xn∗ ) w∗ -basic and such that xn = 1 for each n ([197, p. 11]). As Y ∗ is w∗ -separable, a biorthogonalization argument gives that there exists a biorthogonal sequence (yn , yn∗ ) for Y with (yn∗ ) ⊂ X∗ , Y ∩ (y ∗ )⊥ = {0} and normalized so that yn∗ = 1 for every n ([197, p. 43]). Deﬁne an operator T from X into X by T x = ∞ −n−1 y ∗ (x)x . Then T 1/2 and hence I + T is an isomorphism on X. Thus n n n=1 2 (I + T )∗ is a weak star isomorphism on X∗ . Therefore (xn∗ + T ∗ xn∗ ) is a weak star basic sequence weak star equivalent to (xn∗ ). We have T ∗ xn∗ = 2−n−1 yn∗ for every n. We claim that (xn∗ + 2−n−1 yn∗ )⊥ is a quasicomplement of Y in X. In order to see this, let ∗ x ∗ ∈ Y ⊥ ∩ span w {xn∗ + 2−n−1 yn∗ }. Then x ∗ = limn ( ni=1 αi xi∗ + ni=1 2−n−1 αi yi∗ ) in the weak star topology, for some sequence (αi ) of scalars. As x ∗ ∈ Y ⊥ , for each n, x ∗ (yn ) = 2−n−1 αn = 0. Thus x ∗ = 0, showing that Y + (xn∗ + 2−n−1 yn∗ )⊥ is dense in X. Let now y ∈ Y ∩ (xn∗ + 2−n−1 yn∗ )⊥ . Then xn∗ (y) = 0 for each n as y ∈ Y . Hence yn∗ (y) = 0 for each n. This shows y ∈ (yn∗ )⊥ ∩ Y = {0}. As we mentioned in Section 4, c0 is not complemented in ∞ . This should be compared with the following result of Rosenthal [270]. T HEOREM 4.15. The space c0 is quasicomplemented in ∞ .

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ˇ P ROOF. Let βN denote the Stone–Cech compactiﬁcation of the discrete space N. Then βN \ N is a perfect compact space. Hence there is a continuous map ϕ from βN \ N onto [0, 1] (cf., e.g., [185, p. 29]). From this it follows ([270]) that L1 [0, 1] is isometric to a subset of C(βN \ N)∗ = c0⊥ ⊂ ∗∞ . Hence c0⊥ contains an isomorphic copy of 2 by Kintchine’s inequality (see, e.g., [80, Chapter 6]). As this copy is weak star closed (2 is reﬂexive), we get that 2 is isomorphic to a quotient of ∞ /c0 . Hence Theorem 4.15 follows from Theorem 4.14. The following result extends Johnson’s result ([166]) and is from [161]. T HEOREM 4.16. Let X be a WCG Asplund space and Y ⊂ X∗ be a WCG subspace of X∗ . Then Y has a weak star closed quasicomplement in X∗ . The following result of Lindenstrauss is from [194]. T HEOREM 4.17. If Γ is uncountable, then c0 (Γ ) is not quasicomplemented in ∞ (Γ ). P ROOF. Assume that Z is a quasicomplement of c0 (Γ ) in ∞ (Γ ). Let π be the quotient map of ∞ (Γ ) onto ∞ (Γ )/Z. Consider the restriction of π to c0 (Γ ) and call it T . Then T (c0 (Γ )) is dense in ∞ (Γ )/Z and thus ∞ (Γ )/Z is WCG. Hence B(∞ /Z)∗ in its weak star topology is sequentially compact (Theorem 4.6 and the Eberlein–Šmulyan theorem). If yn := π ∗ (xn ), xn ∈ B(∞ (Γ )/Z)∗ , let {xnk } be a weak star convergent subsequence of {xn }. Then {ynk } is weak star convergent in ∞ (Γ )∗ . As ∞ (Γ ) has the Grothendieck property, {ynk } is weakly convergent. Hence π ∗ is a weakly compact operator and so is π . The same is true for T . Hence T ∗ : (∞ (Γ )/Z)∗ → 1 (Γ ) is a weakly compact operator and thus norm compact operator as 1 (Γ ) has the Schur property. Thus T ∗ (∞ (Γ )/Z)∗ is a norm separable subset of 1 (Γ ). As T is one-to-one (use the deﬁnition of the quasicomplement), T ∗ maps ∞ (Γ )/Z onto a weak star dense set in c0 (Γ ). This means that c0 (Γ )∗ is weak star separable, which is not the case. The standard unit vector basis of c0 (Γ ) is a shrinking M-basis for c0 (Γ ) for every Γ . From the PRI in C[0, ω1 ] discussed in Section 2 we easily construct an M-basis for this space. None of the spaces D, JL0 and JL2 contain a nonseparable subspace with an M-basis. Indeed, such subspace Y then has property C and thus is WLD (Theorem 4.4). Therefore its dual Y ∗ is not weak star separable (Theorem 3.8). However, Y ∗ is a quotient of one of the weak star separable spaces D ∗ , JL∗0 , or JL∗2 , a contradiction. The spaces JL0 and JL2 admit C ∞ smooth norms (Section 8). However, they do not contain any of p (Γ ), p > 1, or c0 (Γ ) for Γ uncountable. Thus Deville’s theorem on containment of p or c0 in C ∞ -smooth spaces ([50], cf., e.g., [57, Chapter V] or [111]) has no nonseparable analogue. The space JL0 , being a subspace of ∞ , is a subspace of a space with an M-basis (Theorem 4.10). Kunen’s C(K) space does not admit any Mbasis (Section 2). The space JT ∗ does not admit any M-basis as it has property C and is not WLD since it is nonseparable with weak star separable dual. Ciesielski–Pol space CP admits no M-basis as it does not inject into any c0 (Γ ) (Theorem 4.5). The subspace c0

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is quasicomplemented in JL2 and also in JL0 , by Theorem 4.14 as both c0 (c) and 2 (c) contain inﬁnite-dimensional separable complemented subspaces. The subspace C[0, 1] is quasicomplemented in D for the same reason. It follows from the recent results in [172] that if X is a nonseparable WCG subspace of ∞ (Γ ) such that X does not contain an isomorphic copy of 1 , then X is not quasicomplemented in ∞ (Γ ).

5. Gâteaux smooth and rotund norms D EFINITION 5.1. The norm · of a Banach space X is strictly convex or rotund (R) if x = y for every x, y ∈ X such that 2x2 + 2y2 − x + y2 = 0. Thus · is R if and only if its unit sphere contains no non-degenerate line segments. Every separable Banach space admits Gâteaux differentiable and rotund norms (see Theorems 5.2, 5.3 below). This is no longer the case for nonseparable spaces. Most of the renormings by Gâteaux differentiable norms are done by renorming the dual spaces by dual rotund norms. This is because majority of C 1 smooth renormings use the fact that the sum of two convex bodies in a ﬁnite-dimensional space is C 1 smooth if one of them is. In inﬁnite-dimensional spaces such sums need not be closed. It is the strict convexity of the dual norm that removes this problem and gives a Gâteaux differentiable norm on spaces. Indeed, let B1 and B2 be the unit balls of the norms · 1 and · 2 on X. Assume that the dual norm of · 2 is rotund. Let · 3 be the norm the unit ball of which is B1 + B2 . Then the dual norm of · 3 is the sum of the dual norms of · 1 and · 2 as f 3 = sup{f (x); x ∈ B1 + B2 } = sup{f (x); x ∈ B1 } + sup{f (x); x ∈ B2 } = f 1 + f 2 . It is easy to see that the sum of two norms is strictly convex if one of them is (cf. Theorem 5.2 below). Thus the dual norm · 3 is then strictly convex, though the algebraic sum of B1 and B2 need not in general be closed. We use the fact that the norm is Gâteaux smooth if the dual norm is strictly convex (cf., e.g., [168]). Indeed, using the remark following Theorem 2.5, if x ∈ SX and f, g ∈ SX∗ are such that f (x) = g(x) = 1, then 2 f + g (f + g)(x) = 2 and from the rotundity of the dual norm we get f = g. Before proceeding, we note that in renorming theory, a new norm ||| · ||| for a Banach space X with the original norm · is constructed such that for some constant C > 0, Cx |||x||| C1 x for every x ∈ X. Many constructions are such that the constant C can be chosen arbitrarily close to 1. We will not explicitly mention this fact in the constructions in this article. On the other hand, please see Problem 5 in Section 9. The following is Mercourakis’ result in [205], cf., e.g., [57, p. 288]. T HEOREM 5.2. Let X be a WCD space. Then X admits an equivalent norm whose dual norm is rotund. In particular, every WCD space admits an equivalent Gâteaux differentiable norm.

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P ROOF. First, we will describe a strictly convex norm on c0 (Γ ) for any Γ . An example of such norm is Day’s norm ([47], cf., e.g., [57, p. 69]) deﬁned for x = {xγ } ∈ c0 (Γ ) by x = sup

n

1/2 2

x (γk )/4

k

,

k=1

where the supremum is taken over all n ∈ N and all ordered n-tuples (γ1 , . . . , γn ) of distinct elements of Γ . We need to show that Day’s norm is rotund. To this end, we ﬁrst make the following simple observation: > n and |a| < |b|, then a 2 /n2 + b 2 /m2 < b2 /n2 + ∞ 2 if jm1/2 2 2 a /m . Thus x = ( j =1 xγj /4 ) , where γj are distinct and such that |xγ1 | |xγ2 | · · ·. Calling such a sequence {γj } an appropriate sequence for x, we have that if x + y = x + y, x = y = 1 and {γj } is an appropriate sequence for x + y, then 2 = x + y =

∞ (x + y)2 γj

<

∞ x2 ρj j =1

4j

1/2

+

∞ x2 γj j =1

∞ y2 γj j =1

4j

j =1

1/2

4j

4j

1/2

+

∞ y2 γj j =1

1/2

4j

1/2 = 2,

where {ρj } is an appropriate sequence for x, a contradiction, unless {γj } is an appropriate sequence for x. Similarly we argue for y. Thus γj is an appropriate sequence for both x and y as well. By the parallelogram equality we then get x = y. Hence Day’s norm is strictly convex. If X is WCG, there is a bounded linear one-to-one operator T from X∗ into c0 (Γ ) for some Γ that is weak star to weak continuous (Theorem 4.11(i)). Let · D denote Day’s norm on c0 (Γ ), let · be the original standard dual norm of X∗ and deﬁne an equivalent norm ||| · ||| on X∗ by |||f |||2 = f 2 + Tf 2D . Then the norm ||| · ||| is weak star lower semicontinuous and thus by the bipolar theorem, it is a dual norm on X∗ (cf., e.g., [57, p. 27] or [80, Chapter 4]). Let us show now that it is strictly convex. Let 2|||f |||2 + 2|||g|||2 − |||f + g|||2 = 0. Since this expression is the sum of the two corresponding expressions for · and T (·)D and such expressions are always non-negative (from the convexity of norms), we get that both corresponding expressions must be zero. Thus in particular 2Tf 2D + 2T g2D − T (f + g)2D = 0. From the rotundity of Day’s norm we have Tf = T g. As T is one-to-one, we get f = g, showing the strict convexity of ||| · |||. A similar situation is with WCD spaces, where we use Theorem 4.11(ii) and the result in [205] that c1 (Σ × Γ ) admits an equivalent strictly convex norm that is pointwise lower semicontinuous (a variant of Day’s norm). Assuming the Continuum Hypothesis, it is shown in [14] that there is a nonseparable compact set K with BC(K)∗ Corson in the weak star topology, such that K satisﬁes the C.C.C. (countable chain condition), i.e., there is no uncountable family of pairwise disjoint open sets in K. Such space C(K) is a WLD space that admits no Lipschitz Gâteaux

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differentiable bump function (in particular no Gâteaux differentiable norm). Indeed, otherwise, by Theorem 2.12, K would contain a nonseparable metrizable set C dense Gδ in K that would contain thus an uncountable family of pairwise disjoint open sets Cα as C is α ∩ C for each α, where C α is open in K, then nonseparable and metrizable. If Cα = C α } would be an uncountable family of pairwise disjoint open sets in K as C is dense {C in K. This would contradict property C.C.C. of K. Note that we have just proved in particular that no nonseparable Eberlein compact K satisﬁes the C.C.C., as C(K) then admits a Gâteaux differentiable norm by Theorem 5.2. Note that every Banach space in its weak topology satisﬁes the C.C.C. ([41]). This follows by considering when two basic neighborhoods in the weak topology are disjoint, together with the lemma on the root of the Δ-system from inﬁnite combinatorics (cf., e.g., [57, p. 262]. For details, see, e.g., [80, Chapter 12]). The canonical norm of 1 (Γ ) is nowhere Gâteaux differentiable if Γ is uncountable. Indeed, if x ∈ S1 (Γ ) , x(α) = 0 and eα is the standard unit vector in 1 (Γ ), then x + teα + x − teα − 2 = 2t for all t > 0. Thus it is not true that limt →0 1t (x + teα + x − teα − 2) = 0, which means that the norm is not Gâteaux differentiable at x in the direction eα . From this and from the smooth variational principle ([58], cf., e.g., [57, p. 9], [80, Chapter 10], [53]) it follows that if Γ is uncountable, then 1 (Γ ) admits no Lipschitz Gâteaux smooth bump function. The same conclusion holds for ∞ and ∞ /c0 as the lim sup function produces a convex function that is nowhere Gâteaux differentiable (see, e.g., [57, Chapter I] or [80, Chapter 8, Example]). There is a WLD space X such that the dual norm of X∗ is rotund and yet, X is not WCD ([14]). Note that in the proof of Theorem 5.2 we proved the following result, which goes back to Clarkson [39] and Day [47], cf., e.g., [57, p. 46]. T HEOREM 5.3. Let X and Y be Banach spaces such that there is a bounded linear oneto-one operator from X into Y . Assume that the norm of Y is rotund. Then X admits an equivalent rotund norm. In particular, X admits an equivalent rotund norm whenever there is a bounded linear one-to-one operator from X into c0 (Γ ) for some Γ . The latter happens if X∗ is weak star separable or if X has an M-basis. P ROOF. If {fn } is weak star dense in X∗ , then the bounded linear operator T from X 1 into c0 deﬁned by T x(n) = 2n f fn (x) is one-to-one. If X has an M-basis, we use Theon rem 4.5. The ﬁrst example of a Banach space X that admits a rotund norm but does not admit any bounded linear one-to-one operator into any c0 (Γ ) was constructed in [44]. One of the spaces of this type constructed in [44] consists of the subspace of ∞ [0, 1] formed by all functions f in ∞ [0, 1] such that for every ε > 0, the second derived set of the set {t ∈ [0, 1]; |f (t)| ε} is empty. The part (i) of the following result was proved by Talagrand in [295]. The part (ii) is due to Partington [238] (cf., e.g., [57, Chapter II.7]). Both results belong to the area of the so called distorted norms, see, e.g., [197, p. 97], [230].

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T HEOREM 5.4. (i) Let ||| · ||| be an equivalent norm on ∞ and · be the standard supremum norm on ∞ . Then there is δ > 0 such that for every n ∈ N, there is a subspace Xn of ∞ isomorphic to ∞ and such that on Xn ,

δ − 2−n · ||| · ||| δ + 2−n · . (ii) If Γ is uncountable, then ∞ (Γ ) in any equivalent norm contains an isometric copy of ∞ in the supremum norm. Thus ∞ (Γ ) admits no strictly convex norm if Γ is uncountable. In particular, there is no bounded linear one-to-one operator from ∞ (Γ ) into c0 (Γ ) if Γ is uncountable. Bourgain showed in [29] that ∞ /c0 admits no equivalent R-norm. We will see in Theorem 8.3 that C[0, ω1 ] has an equivalent C ∞ -smooth norm. However, Talagrand showed in [297] (cf., e.g., [57, p. 313]) the following result. T HEOREM 5.5. There is no equivalent norm on C[0, ω1 ] such that its dual is a rotund norm. P ROOF. Assume that ||| · ||| is a dual rotund norm on C[0, ω1 ]∗ . For α ∈ [0, ω1 ], let δα be the Dirac measure corresponding to α. Then the function α → |||δα ||| is lower semicontinuous on [0, ω1 ]. Thus it is constant, equal to, say, a on a closed coﬁnal subset A of [0, ω1 ). For α ∈ A, let α be the successor of α in A. The map α → |||(δα + δα )/2||| is lower semicontinuous on A and thus equal to say b on a closed coﬁnal subset B of A. Let (αn ) be a strictly increasing sequence in B such that α = lim αn . Using the facts that δα is a weak star limit of (δαn + δαn )/2 and that ||| · ||| is weak star lower semicontinuous, we get that lim |||(δαn + δαn )/2||| |||δα |||. Hence b a. On the other hand, |||(δα + δα )/2||| |||δα |||/2 + |||δα |||/2 = |||δα |||. Hence a = b and for every α ∈ B,

(δα + δα )/2 = |||δα ||| + |||δα ||| /2 = a. The following result of Haydon ([143]) contains a key argument in some questions on renorming spaces of continuous functions on trees by smooth norms. The result says that there is no “too ﬂat” Gâteaux smooth norm on C0 [0, ω1 ]. This is then used in Theorem 5.7. T HEOREM 5.6. Let · be an equivalent norm on C0 [0, ω1 ] that satisﬁes (∗) x + λχ(β,γ ] = x whenever supp x ⊂ [0, β], β < γ < ω1 and 0 λ x(β). Then · is not a Gâteaux differentiable norm on C0 [0, ω1 ]. P ROOF. In fact, assuming that a Gâteaux differentiable norm · · ∞ on C0 [0, ω1 ] satisﬁes (∗), we construct, by transﬁnite induction, xα , α < ω1 in C0 [0, ω1 ] such that supp xα ⊂ [0, α]; if β α < γ , then xγ (β) = xα (β); x∞ = xα (α); (xα ∞ ) is strictly increasing and xα − x0 12 (xα ∞ − x0 ∞ ). This is a contradiction as there is no strictly increasing function on [0, ω1 ). Theorem 5.6 can be compared with the result in [79] that C0 [0, ω1 ] admits no lattice Gâteaux renorming and with the result in [81] that C[0, ω1 ] admits no equivalent Gâteaux

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differentiable norm that would be lower semicontinuous in the topology of pointwise convergence on [0, ω1 ). The following is Haydon’s result in [143]. T HEOREM 5.7. Let T be a full uncountably branching tree of the height ω1 and let · be an equivalent norm on C0 (T ). Then there is a subspace of C0 (T ) which is isometric to C0 [0, ω1 ] equipped with a norm that satisﬁes property (∗) from Theorem 5.6 (cf., e.g., [57, p. 323]). Thus C0 (T ) does not admit any equivalent Gâteaux smooth norm though C0 (T ) is an Asplund space. For a similar reason, the space C0 (T ) does not admit any equivalent rotund norm either. P ROOF. Any norm on C0 [0, ω1 ] that satisﬁes (∗) is not rotund.

Preiss proved in [256] the following result. T HEOREM 5.8. Every Lipschitz function is Gâteaux (Fréchet) differentiable on a dense set in X if X admits a Gâteaux (Fréchet) differentiable norm. For every Γ , the space c0 (Γ ), C[0, ω1 ], JL0 and JL2 all admit C ∞ smooth norms. This will be discussed in Section 8. However, the space C[0, ω1 ] admits no equivalent Gâteaux differentiable norm that would be pointwise lower semicontinuous for points in [0, ω1 ) ([81]). The space D = C(K) does not admit Lipschitz Gâteaux differentiable bump function, in particular it admits no Gâteaux differentiable norm ([297], cf., e.g., [57, p. 303], [73, p. 46]). This follows from the fact that, otherwise, by Theorem 2.12, the compact space K would contain a dense Gδ completely metrizable set, which is not the case (see Section 2). The space c0 (Γ ) admits a rotund norm by the proof of Theorem 5.2. The spaces JL0 , JL2 and D all admit rotund norms by Theorem 5.3 as their duals are weak star separable. The space C[0, ω1 ] admits a rotund norm by Theorem 7.3 below. The space JT ∗ admits a dual Gâteaux differentiable norm ([133]). The paper [133] contains the study on separable spaces that admit norms whose second dual norm is rotund. Note that X is reﬂexive if the fourth dual norm of X is rotund (Dixmier). In fact, X is reﬂexive if the third dual norm of X is Gâteaux differentiable (Giles, Kadets, Phelps, cf., e.g., [80, Chapter 8, Example]). For information on spaces that admit norms whose third conjugate is rotund we refer to, e.g., [283] and [285]. It was proved in [82] that a WLD space X of density ℵ1 is WCG if and only if there is a bounded set S in X with span S = X and there is an equivalent Gâteaux differentiable norm · on X such that the derivative of · at each point of the unit sphere is uniform in the directions in S. 6. Uniformly Gâteaux smooth norms D EFINITION 6.1. The norm · of a Banach space X is uniformly Gâteaux differentiable (UG) if for each h ∈ SX , lim

t →0

1 x + th + x − th − 2 = 0 t

uniformly in x ∈ SX .

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The following result can be found, e.g., in [57, p. 65]. T HEOREM 6.2. If a Banach space Y has a uniformly Gâteaux differentiable norm and a bounded linear operator T maps Y onto a dense set in a Banach space X, then X admits a uniformly Gâteaux differentiable norm. P ROOF. The result follows from Šmulyan’s characterization of UG norms which says that · of X is UG if and only if its dual norm is weak star uniformly rotund (W ∗ UR), i.e., whenever fn , gn ∈ X∗ are such that {fn } is bounded and 2fn 2 + 2gn 2 − fn + gn 2 → 0, then fn − gn → 0 in the weak star topology of X∗ (cf., e.g., [57, p. 63]). Indeed, deﬁne the norm on X∗ by |||f |||2 = f 2 + T ∗ f 21 , where · is the original norm of X∗ and · 1 is a weak star uniformly rotund norm on Y ∗ . Like in the proof of Theorem 5.2, we get that ||| · ||| is W∗ UR. If X is separable and {xn } is dense in SX , then the operator T : X∗ → 2 deﬁned for f ∈ X∗ by Tf (i) = 2−i f (xi ) is dual to an operator that maps 2 onto a dense set in X as T is one-to-one. Thus we get Šmulyan’s classical result: C OROLLARY 6.3. Every separable Banach space admits an equivalent UG norm. The space c0 (Γ ) admits an equivalent UG norm for any Γ (use the “identity”operator of 2 (Γ ) into c0 (Γ ) and the fact that the norm of Hilbert space is UG (Theorem 6.2)). The following theorem is the result of Troyanski in [311]. T HEOREM 6.4. Assume that {eα , fα }α∈Γ is a normalized unconditional basis for a Banach space X. Then X admits an equivalent UG norm if and only if for every ε > 0, the set Γ can be decomposed into Γ = Γiε such that for each i and for distinct {γj }ij =1 ⊂ Γiε , we have ij =1 eγj εi. The following result is in [76]. T HEOREM 6.5. Let X be a Banach space with an equivalent uniformly Gâteaux smooth norm. Then X is a Kσ δ subset of (X∗∗ , w∗ ), i.e., X = n1 m1 Km,n , where Km,n are some weak∗ compact sets in X∗∗ . In particular, X is then a WCD space. P ROOF. Assume that · is an arbitrary equivalent norm on X. Pick any G ∈ X∗∗ \X. Let H = G−1 (0) be the subspace of X∗ consisting of all the elements of X∗ that vanish at G. The space H is a norming subspace of X∗ , that is, there is δ > 0 such that for all x ∈ SX , sup{|f (x)|; f ∈ H, f 1} δ (cf., e.g., [80, Chapter 3, Example]). Using the idea in [311] and [209], for any equivalent norm · on X, we deﬁne for all n, p ∈ N the subsets Sn,p ( · ) in X as follows:

Sn,p · = x ∈ X; f (x) − g(x) 1/p whenever f, g ∈ X∗ , f 1, g 1 and f + g > 2 − 2/n .

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We now assume that · is an equivalent uniformly Gâteaux smooth norm on X. Thus for any p ∈ N, one has

Sn,p · = X.

(1)

n1

We will show that X=

∗ Sn,p · .

p1 n1

It follows from (1) that it sufﬁces to prove that for any G ∈ X∗∗ \X there is a p ∈ N such that G∈ /

∗ Sn,p · .

(2)

n1

We set H = G−1 (0), and deﬁne an equivalent norm q on X by the formula q(x) = sup f (x); f ∈ H, f 1 . We claim that

Sn,p · ⊂ Sn,p (q). In order to prove this claim, we observe the bipolar theorem (cf., e.g., [80, Chapter 4]) implies that the dual unit ball Bq ∗ satisﬁes ∗ Bq ∗ = f ∈ X∗ ; q ∗ (f ) 1 = f ∈ H, f 1 .

(3)

Therefore if q ∗ (f ) 1, q ∗ (g) 1 and q ∗ (f + g) > 2 − 2/n, there are nets (fα ) and (gα ) in H , weak* convergent to f and g respectively, such that fα 1 and gα 1 for all α. Since the norm q ∗ is weak*-lower semicontinuous, one has q ∗ (fα + gα ) > 2 − 2/n when α is large enough. As · and q ∗ coincide on H , fα + gα > 2 − 2/n for large α. If now x ∈ Sn,p ( · ), we have |fα (x) − gα (x)| 1/p for α large enough. It thus follows that x ∈ Sn,p (q). This shows our claim. In order to prove (2), it therefore sufﬁces to show that one has G∈ /

Sn,p (q)

∗

n1

for p ∈ N large enough. To this end, choose p ∈ N such that p > 1/q ∗∗ (G). Fix n ∈ N and ∗ set for simplicity Sn,p (q) = S. We need to show that G ∈ /S .

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Pick f ∈ Bq ∗ such that G(f ) > 1/p, and let x ∈ X be such that q(x) 1 and f (x) > 1 − 1/n. By (3), there is g ∈ H with q ∗ (g) 1 and g(x) > 1 − 1/n. We have then q ∗ (f + g) (f + g)(x) > 2 − 2/n. ∗

From the deﬁnition of S, for all z ∈ S one has (f − g)(z) 1/p. Thus if G ∈ S , then G(f − g) 1/p. This contradicts the fact that G(f − g) = G(f ) > 1/p. ∗ For every p ∈ N, let the family {Kp,q }q be equal to the reindexed family {S n,p ∩ ∗∗ mBX∗∗ }n,m . The family {Kp,q } can be used to verify that X is Kσ δ in X in its weak∗ topology. Theorem 6.5 is proved. The proof of Theorem 6.5 gives the following result ([77]): Assume that M is a bounded set in a Banach space X whose norm satisﬁes the following property: supx∈M (fn − gn )(x) → 0, whenever fn , gn ∈ SX∗ are such that fn + gn → 2. Then M is weakly relatively compact. UG norms are related to uniform Eberlein compacts. We summarize some of the results in this direction in the following theorem. T HEOREM 6.6. Let X be a Banach space. Then the following are equivalent. (i) The space X admits an equivalent UG norm. (ii) The dual ball BX∗ in its weak star topology is a uniform Eberlein compact. (iii) There is a set Γ and a bounded linear operator from 2 (Γ ) onto a dense set in C(BX∗ ), where BX∗ is considered in its weak star topology. (iv) There is a Markushevich basis {xα , fα }α∈Γ of X such that for every ε > 0, there is a partition Γ = i∈N Γiε and there are integers mεi such that for every i ∈ N and every f ∈ BX∗ , one has card γ ∈ Γiε ; f (xγ ) > ε mεi . P ROOF. The implication (i) ⇒ (ii) was proved in [78] under the additional assumption that X has a PRI in its UG norm. This assumption is redundant due to Theorem 6.5 and Theorem 3.8. The implication (ii) ⇒ (iii) is proved in [21] (cf., e.g., [80, Chapter 12]). The implication (iii) ⇒ (i) follows from Theorem 6.1. (ii) ⇒ (iv) This follows from Theorem 4.2 and Theorem 2.9 in [93]. (iv) ⇒ (ii) This follows from Theorem 2.9 in [93]. The following result is in [76]. C OROLLARY 6.7. Let K be a compact space. Then K is a uniform Eberlein compact if and only if C(K) admits an equivalent UG norm. P ROOF. Let K be a uniform Eberlein compact. Then BC(K)∗ in its weak star topology is a uniform Eberlein compact [21]. Then C(K) admits an equivalent UG norm by Theorem 6.6.

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On the other hand, if C(K) admits a UG norm, then BC(K)∗ in its weak star topology is a uniform Eberlein compact by Theorem 6.6 and so is its closed subspace K. Note that Corollary 6.7 gives a short proof to the result that a continuous image of a uniform Eberlein compact is a uniform Eberlein compact (see Theorem 2.15). It also gives an alternative proof to the result in [184] on the existence of reﬂexive spaces with no equivalent UG norm. For further results in this area we refer to, e.g., [77]. It is proved there the following result: for a Banach space X of density ℵ1 , the following are equivalent: (1) X admits an equivalent norm that is UG uniformly on a set of directions h in a bounded set M with span M = X. (2) There is a bounded linear weak star to weak continuous operator T from X∗ into some c0 (Γ ) such that for every ε > 0 there is a natural number k such that card{γ ∈ Γ ; |Tf (γ )| > ε} k for every f ∈ BX∗ . There also a smoothness characterization is given of spaces X such that 2 (Γ ) can be mapped onto a dense subset in X (Hilbert generated spaces). The following is Hájek’s result [133], which solved the problem of Troyanski posed in one Frolík’s Winter School in the Czech Republic in the 70’s. For a simple proof of it we refer to [77]. C OROLLARY 6.8. If the norm of a Banach space X is weakly uniformly rotund, then X is an Asplund space. A norm · on X is weakly uniformly rotund if xn − yn → 0 weakly in X whenever xn , yn ∈ SX are such that xn + yn → 2. P ROOF OF C OROLLARY 6.8. If X is a separable space with weakly uniformly rotund norm, then X∗ has a UG norm by the Šmulyan lemma. Then X∗ is a subspace of a WCG space by Theorem 6.6 and Theorem 4.9. Thus X∗ is a DENS space by Theorem 3.8. This means that X∗ is separable. The norm · of a Banach space X is uniformly rotund in every direction (URED) if xn − yn → 0 whenever {xn }, {yn } are bounded sequences in X, 2xn 2 + 2yn 2 − xn + yn 2 → 0 and for some z ∈ SX and λn ∈ R, xn − yn = λn z. The standard norm of every Hilbert space is URED. The proof of Theorem 5.2 gives that X admits an equivalent URED norm if there is a bounded linear one-to-one operator from X into a Hilbert space. For necessary and sufﬁcient conditions for renorming spaces with unconditional bases by URED norms we refer to [311]. If Γ is uncountable, then c0 (Γ ) does not admit any equivalent URED norm. Indeed, let | · | be an equivalent norm on c0 (Γ ), denote by · the supremum norm of c0 (Γ ) and put M = sup{|x|; x 1}. Let un 1 be such that |un | → M. Choose z = 1 such that its support is disjoint from the union of the supports of all un . Then put xn := un + 12 z, yn := un − 12 z. Then xn 1 and yn 1 for all n and thus |xn | M and |yn | M for all n. As | 12 (xn + yn )| → M, we get |xn | → M and |yn | → M. This gives that | · | cannot be URED. The URED norms have been used in the ﬁxed point theory for non-expansive mappings and in the study of uniform Eberlein compacts. We refer to [57, p. 67], [12] and references therein for more information in this

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direction. For applications of URED norms in geometry of spaces with symmetric bases we refer to [310]. Rychtáˇr showed in [276] that a Banach space X with unconditional basis admits an equivalent UG norm if X∗ admits an equivalent URED norm. However, he showed in [277] that for example JL2 admits an equivalent norm whose dual is URED. The following examples due to Hájek, Kutzarova and Troyanski can be found in [133, 131,77] and [184]. E XAMPLES . (i) There is a reﬂexive Banach space that does not admit any equivalent UG norm. (ii) There is a reﬂexive Banach space X with a UG norm such that there is no bounded linear operator from any Hilbert space onto a dense subset of X. (iii) There is an equivalent norm on JT space the second dual of which is URED, though there is no UG norm on JT ∗ . (iv) There is a non-reﬂexive separable Banach space with a norm the bidual norm of which is UG. I do not know of any Banach space X the dual of which would admit a Gâteaux differentiable norm but admits no dual Gâteaux differentiable norm. Unlike the situation with Gâteaux or Fréchet differentiable bumps (see Section 8), any space that admits a uniformly Gâteaux differentiable bump function admits an equivalent UG norm ([299], cf., e.g., [91]).

7. Fréchet smooth and locally uniformly rotund norms D EFINITION 7.1. (i) The norm · of a Banach space X is locally uniformly rotund (LUR) if xn − x → 0 whenever xn , x ∈ X are such that 2x2 + 2xn 2 − xn + x2 → 0. (ii) The norm · of a Banach space X is a Kadets–Klee norm or has the Kadets–Klee property if the norm and weak topologies coincide on the unit sphere of · . If · is LUR, then it is a Kadets–Klee norm. Indeed, if xα , x ∈ SX , xα → x weakly and f ∈ SX∗ is such that f (x) = 1, then 2 lim sup xα + x lim inf xα + x lim inf f (xα + x) = 2. Thus xα − x → 0 as · is LUR. It is not difﬁcult to show that on the unit sphere of the standard norm of 1 (Γ ), the weak star and norm topology coincide (cf., e.g., [57, p. 72]). Šmulyan’s lemma (Theorem 2.5) implies that the norm is Fréchet differentiable if its dual norm is LUR. Indeed, let the dual norm be LUR and x ∈ SX be given. Let fn , gn ∈ SX∗ and f0 ∈ SX∗ be such that f0 (x) = 1 and fn (x) → 1, gn (x) → 1. Then 2 fn + f0 (fn + f0 )(x) → 2 and from LUR we get fn − f0 → 0. Similarly we get gn − f0 → 0. Thus fn − gn → 0, which, by Theorem 2.5, means that the norm of X is Fréchet differentiable at x. If X is a separable Banach space, then X admits an equivalent locally uniformly rotund norm. If X∗ is separable, then X admits an equivalent norm the dual of which is locally uniformly rotund. Both of these fundamental results are due to Kadets (cf., e.g., [111], [57, p. 48], [80, Chapter 8]). If X is separable, then X admits a Fréchet smooth norm if and only if X∗ is separable (cf., e.g., [111], [57, p. 51], [80, Chapter 8]). This follows from

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Kadets’ result above and from the Bishop–Phelps theorem. We will assume these results in the sequel. Therefore we will not discuss the proofs of some separable versions of the results, whenever no misunderstanding can occur. If · is a Fréchet differentiable norm on X, then the map Φ : x → x is Fréchet 1 C -smooth on SX by Šmulyan’s lemma. By the Bishop–Phelps theorem, Φ maps SX onto a dense set in SX∗ . A similar result is obtained by using Fréchet C 1 -smooth bump functions. Instead of the Bishop–Phelps theorem one can use here the following argument. Let b be a C 1 -smooth bump function on X and ϕ = b−2 , where b = 0 and ϕ = +∞ elsewhere. If f ∈ X∗ , then ψ = ϕ − f satisﬁes the assumptions of the smooth variational principle (cf., e.g., [57, Chapter I], [53]) and thus there is Fréchet C 1 -smooth function h on X and a point x0 in X such that ψ − h attains its minimum on X and the norm of the ﬁrst derivative of h at all points of X is smaller than a given number ε. Then (ϕ − f − h )(x0 ) = 0 and hence we get that {ϕ (x); x ∈ X} is norm dense in X∗ . Thus we obtain that X is an Asplund space if X admits a Lipschitz Fréchet C 1 -smooth bump function (cf., e.g., [243, p. 66]). Indeed, if X admits a Lipschitz Fréchet C 1 -smooth bump function b, we can suppose that b(0) = 0 and then, given a separable subspace Y of X, the restriction of b to Y is a Lipschitz Fréchet C 1 -smooth bump function on Y and thus {ϕ (xn )} is norm dense in Y ∗ whenever ϕ is constructed as above from b and {xn } is a dense sequence in Y . In fact, the Lipschitz property of b is not needed in showing that X is Asplund, namely it follows that X is an Asplund space if X admits a Fréchet differentiable bump function (cf., e.g., [57, p. 58]). No example of an Asplund space is known that does not admit a (Lipschitz) Fréchet C 1 -smooth bump function (Problem 1 below, see, e.g., [57, p. 89]). The following is the result in [259] (cf., e.g., [57, p. 69]). T HEOREM 7.2. Day’s norm on c0 (Γ ) (deﬁned in the proof of Theorem 5.2) is LUR for every Γ . P ROOF. The proof consists of a qualitative variant of the argument that we used in the proof of Theorem 5.2 (cf., e.g., [57, p. 69]). The following is a variant of the result of Troyanski in [308] and is from [330] (cf., e.g., [57, p. 284].) T HEOREM 7.3. If a Banach space X admits a PRI {Pα }, α μ, and each (Pα+1 − Pα )(X), α < μ, admits a LUR norm, then so does X. P ROOF. We will outline the key idea in the proof of this result for a Banach space X with a transﬁnite Schauder basis {eα , fα }, i.e., if X admits a PRI {Pα }, α μ, such that dim(Pα+1 − Pα )(X) = 1 for all α < μ. Put Γ = [0, μ). Let An be the family of all ﬁnite subsets of Γ with no more thann elements. For x ∈ X and A ∈ An , let EnA (x) = dist(x, span{eα }α∈A ) and FnA (x) = α∈A |fα (x)|. Put Gn (x) = sup{E A (x) + nF A (x); A ∈ An } for n ∈ N and x ∈ X. Finally, put G0 (x) = x0 for x ∈ X, where · 0 is the original norm of X. Let Δ = {0, −1, −2, . . .} ∪ Γ and deﬁne Φ : X → c0 (Δ) by

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Φ(x)(−n) = 2−n Gn (x) for −n ∈ {0, −1, −2, . . .}, Φ(x)(α) = |fα (x)| for α ∈ Γ . Deﬁne an equivalent norm x on X by x = Φ(x)D , where · is Day’s norm on c0 (Δ). Let us brieﬂy outline the main idea of the proof that · is a LUR norm on X. Let xn , x ∈ X be such that 2xn 2 + 2x2 − x + xn 2 → 0. Let ε > 0. Find n ∈ N and A ∈ An such

that EnA (x) < ε. Assume without loss of generality that / A < min fα (x); α ∈ A . sup fα (x); α ∈

(∗)

Due to the term n in the deﬁnition of G n s, if m is big enough and A ∈ Am is so chosen A (x) + mF A (x)) < ε with A ∈ A , then, necessarily, A ⊃ A and thus that Gm (x) − (Em m m A the LUR property of Day’s norm on Em (x) < ε. This is because {fα (x)} ∈ c0 (Γ ). From c0 (Δ) it follows that FnA (xk ) → FnA (x) for all A ∈ An and that Gn (xk ) → Gn (x) for all n. As the topology of the coordinatewise convergence in X is Hausdorff, in order to prove that xk − x → 0 it sufﬁces to show that {xk } is relatively norm compact in X. The A (x ) G (x ) − mF A (x ) G (x) + ε − latter is seen from the fact that for large k, Em k m k m m k A A mFm (x) + ε Em (x) + 3ε 4ε. The fact that the set A above is enlarged to ensure that the relation in (∗) holds true can be avoided by adding more parameters ([330]). Overall, the main thing with the above result is that the set A in the deﬁnition of EnA was chosen so that EnA (x) is such that FnA (x) = 0, and the supremum in the deﬁnition of Gn is “uniquely located” (see the “rigidity” condition in [145]). This Troyanski’s phenomenon explicitly or implicitly appears again in many results in this area, including the results on smooth partitions of unity or recent results of Haydon, Talagrand and others on higher-order smoothness. In particular, a Banach space X admits a LUR norm if it has a Markushevich basis {xα , fα } such that x ∈ span{xα ; fα (x) = 0} for every x ∈ X. This is a special condition on a Markushevich basis. Every separable Banach space has such bM-basis ([300–302], see also [326]). T HEOREM 7.4. Any WLD space admits a LUR norm. If K is a Corson compact, then C(K) admits an LUR norm that is pointwise lower semicontinuous. Any Banach space with a shrinking M-basis admits a norm whose dual norm is LUR. There is a Banach space with a Markushevich basis that admits no LUR norm. P ROOF. We use Theorems 7.3 and 3.8 to get that every WLD admits a LUR norm ([13, 316]). We refer to, e.g., [57, p. 286] for a proof of the result in [14] and [316] that C(K) admits a pointwise lower semicontinuous LUR norm if K is a Corson compact. We refer to [309] for the rest of the result. As ∞ does not admit any LUR norm (see the text following Theorem 7.19), the space Z from Theorem 4.10 admits no LUR norm. Similarly, like we used the attachment of the term nFnA in the construction of an LUR norm above, we can prove the following result from [122] and [158] (cf., e.g., [57, p. 299]).

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T HEOREM 7.5. (i) Let Y be a subspace of a Banach space X such that X/Y admits an equivalent LUR norm. Assume that Y admits an equivalent LUR (respectively R) norm. Then X admits an equivalent LUR (respectively R) norm. (ii) Let Y be a subspace of a Banach space X. Assume that both Y and X/Y admit norms whose dual norms are LUR. Then X admits an equivalent norm whose dual is LUR. P ROOF. As in Theorem 7.3, we attach the balls in the cosets, centered by using the the Bartle–Graves selection, to the ball of the quotient. If T is a bounded linear operator from a Banach space X onto a Banach space Y , then the Bartle–Graves selector is a continuous map from Y into X such that for y ∈ Y , T (b(y)) = y (cf., e.g., [57, p. 299]). There is a subspace Y of a Banach space X such that Y admits a LUR norm, X/Y admits an R-norm and X does not admit any R-norm ([145]). This can happen because of the fact that we no longer can properly use the continuity of the Bartle–Graves selector. If X∗ is WCG, then we attach to the map of X∗∗ into c0 (Γ ) ﬁnite-dimensional subspaces of X (using the angelicity of the second dual ball) to obtain that if X∗ is WCG, then X admits a LUR norm. A different approach for getting this result is in [121] (cf., e.g., [57, p. 296]). The following result is in [86,121], cf., e.g., [57, p. 296]. T HEOREM 7.6. A Banach space X admits a LUR norm if X∗ is WCD. Another method of constructing locally uniformly rotund norms is Godefroy’s “transfer” method ([108,120,71], cf., e.g., [57, pp. 44, 289] or [80, Chapter 11]). This method, based on weak star compactness and a geometric argument, allows to construct dual LUR norms in some dual spaces that do not admit a PRI formed by dual projections. The following is Fabian’s result in [71]. It uses Godefroy’s transfer method. T HEOREM 7.7. Assume that X∗ is WCD. Then X admits an equivalent norm whose dual norm is LUR. P ROOF. We will present here Fabian’s proof of this result in the case X∗ is WCG ([120]). There is a set Γ and a weak star to weak continuous bounded linear operator T from c0∗ (Γ ) onto a norm dense set in X∗ . Indeed, let U be a weak star to weak continuous oneto-one operator from X∗∗ into c0 (Γ ) for some Γ and put T = U ∗ . Then T maps c0∗ (Γ ) onto a norm dense set in X∗ as U is one-to-one and weak star to weak continuous. We construct a dual LUR norm on X∗ as follows. Let · be the original norm of X∗ and let | · | be an equivalent dual LUR norm on c0∗ (Γ ). Such norm exists as the standard basis of c0 (Γ ) is shrinking (Theorem 7.4). For n ∈ N and f ∈ X∗ put |f |2n = inf{f − T g2 + ∞ 1 2 ∗ 2 −n 2 ∗ n=1 2 |f |n . This is an equivalent norm on X which n |g| ; g ∈ c0 (Γ )} and |||f ||| = ∗ is weak star lower semicontinuous and thus it is a dual norm. As T (c0 (Γ )) is norm dense in X∗ , it follows that |f |n → 0 for each f ∈ X∗ . Let us now show that ||| · ||| is LUR. To this end assume that f, fj ∈ X∗ are such that 2|||fj |||2 + 2|||f |||2 − |||f + fj |||2 → 0. Then for every n, 2|f |2n + |fj |2n − |f + fj |2n → 0 with j → ∞. Find g, gj ∈ c0∗ (Γ ) such that |f |2n = f − g2 + n1 |g|2 , |fj |2n = fj − T gj 2 + n1 |gj |2 .

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Then 2|f |2n + 2|fj |2n − |f + fj |2n 2 2 2f − T g2 + |y|2 + 2fj − T gj 2 + |gj |2 n n 2 1 − f + fj − T (g + gj ) − |g + gj |2 n

2 1 f − T g − fj − T gj + 2g2 + 2|gj |2 − |g + gj |2 . n This implies that fj − T gj → f − T g and 2|g|2 + 2|gj |2 − |g + gj |2 → 0. As | · | is LUR, we have |g − gj | → 0. Thus lim sup f − fj lim sup(f − T g + T (g − gj ) + fj − T gj ) = 2f − T g 2|f |n . As this holds for each n and |f |n → 0, we get f − fj → 0 and thus ||| · ||| is LUR. The following is an alternative proof that every WCG space admits a LUR norm (which fact follows from Theorem 7.4). Let the dual norm on 1 (Γ ) be deﬁned by |||x|||2 = x21 + x22 , where xi is the norm of i (Γ ) for i = 1, 2. Let T be a bounded linear one-to-one operator of X∗ into c0 (Γ ) that is weak star to weak continuous (Theorem 4.5). The norm ||| · ||| is LUR (cf., e.g., [57, p. 72]) and we can use T ∗ to apply the proof of Theorem 7.7. There is a Banach space that does not admit any Gâteaux differentiable or rotund norms and whose dual is WLD ([145]). We will now discuss how to combine norms with various smoothness and rotundity properties. This procedure is called the Asplund averaging procedure. One method in this direction is the following approach ([89], cf., e.g., [57, p. 52]). T HEOREM 7.8. Assume that X is a Banach space that admits an equivalent LUR norm. Then the set of all equivalent LUR norms on X is residual in the space P of all equivalent norms on X with the metric of uniform convergence on the original ball of X. The metric space P is a Baire space, i.e., such a topological space T that the intersection of any countable family of open dense sets in T is dense in T . P ROOF OF T HEOREM 7.8. Let r0 be a LUR norm on X. For p ∈ P and j ∈ N, put G(p, j ) = {q ∈ P ; sup{|p2 (x) + j −1 r02 (x) − q 2 (x)|; x ∈ BX } < j −2 }. For k ∈ N, put Gk = {G(p, j ); p ∈ P ; j k} and ﬁnally put G = ∞ k=1 Gk . Then Gk is open and dense in P for all k and to ﬁnish the proof, by the Baire category theorem, it is enough to show that each element of G is LUR. This is done by using the numbers j −1 and j −2 in the deﬁnition of G(p, j ). Similarly, the set of all equivalent norms on a given Banach space X such that their dual norm is LUR is residual in P , provided there is at least one such norm on a space.

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Assume that a Banach space X admits an equivalent LUR norm and also admits an equivalent norm whose dual is LUR. By the Baire category theorem, the set of norms that have both these properties is residual in P . From Theorems 7.4, 7.7 and 7.8, we thus obtain the following result. T HEOREM 7.9. A Banach space X admits a LUR norm whose dual is LUR if X∗ is WCD. The method works for some other types of rotundity and smoothness of the ﬁrst order. If X∗ is WCD, to get a LUR Fréchet differentiable norm on X we can proceed as follows ([164]). Let · be a LUR norm on X and let · n be a sequence of norms whose dual norms are LUR and so that · n → · uniformly on bounded sets. This can be 2done by using the proofs of Theorems 7.6 and 7.7. Then put for x ∈ X, |||x|||2 = ∞ n=1 xn . By using the standard differentiability rules we can see that ||| · ||| is a Fréchet smooth norm. Assuming that 2|||xn |||2 + 2|||xn |||2 − |||xn + x|||2 → 0, we get the same convergence for each · n and thus the same convergence for · by Osgood’s uniform convergence theorem. Thus xn − x → 0 as · is LUR. Bossard, Godefroy and Kaufman proved in [28] that the set of all Fréchet differentiable norms on every inﬁnite-dimensional separable Asplund Banach space is not a Borel set in P (P is deﬁned in Theorem 7.8). This compares with the result of Mazurkiewicz [203] that the set of all differentiable real valued functions on [0, 1] is not Borel in C[0, 1]. The same holds for the set of all LUR norms on every inﬁnite-dimensional separable Banach space ([28]). The norm ||| · ||| on 1 deﬁned by |||x||| = x1 + x2 is strictly convex and a dual norm. Moreover, on its unit sphere the weak star and norm topologies coincide. Its predual norm on c0 is thus Fréchet differentiable (Theorem 2.5). The norm ||| · ||| is not LUR on 1 (by inspection). There are spaces that admit Kadets–Klee norms and do not admit any strictly convex norms ([145]). However, the following result of Troyanski in [313,314] holds true (cf., e.g., [57, Chapter IV]). T HEOREM 7.10. Assume that a Banach space X admits a norm that is a Kadets–Klee norm and that X admits a norm that is strictly convex. Then X admits an equivalent LUR norm. P ROOF (Raja [261]). There is a norm on X that shares both properties in question (the sum of the norms works). Then each point of the new unit sphere is an extreme point of the new unit ball of BX∗∗ (Lin, Troyanski, cf., e.g., [80, Chapter 3, Example]). Thus the slices form a neighborhood system of any element of the unit sphere of X in the weak topology and thus in the norm topology (cf., e.g., [80, Chapter 3, Example]). A slice is an intersection of BX with a halfspace in X. For m ∈ N, put Am = {x ∈ BX ; diam(BX ∩ H ) 1/m for every halfspace H containing x}. Then Am is closed convex and symmetric and 0 ∈ Int Am . Let · m be the Minkowski functional of the set Am . Let am > 0 be such that am x2m 2−m x2 for all x ∈ X, where · is the original norm of X. Put for x ∈ X,

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am x2m + x2 . We will show that | · | is LUR. To this end let x, xk ∈ X be such 2|x|2 + 2|xk |2 − |x + xk |2 → 0.

(∗)

Assume without loss of generality that x = xk = 1. Given ε > 0, ﬁx m ∈ N with m > 2/ε and x ∈ / Am . As Am is closed, xm > 1. From (∗), 12 (x + xk )m > 1 for large k. 1 Hence 2 (x + xk ) ∈ / Am for large k and thus for every large k there is a halfspace H such 1 that 2 (x + xk ) ∈ H and diam(BX ∩ H ) 1/m < ε/2. Now, either x or xk is in H and since the diameter of BX ∩ H is less then ε we get that x − xk ε. This means that xk → x in the norm and the proof is ﬁnished. Assume that a Banach space X has a PRI {Pα } in its Fréchet smooth norm. Then {Pα∗ } is automatically a PRI for X∗ . For simplicity, let us show this for the case that the dual norm of X∗ is locally uniformly rotund. To this end, observe that if the dual norm is LUR, then the weak star and norm topologies of BX∗ coincide at each point of SX∗ . If f ∈ SX∗ , then Pα∗ f → Pβ∗ f in the weak star topology if α → β. As Pα∗ f = Pα∗ (Pβ f ) Pβ∗ f we have from the coincidence of the weak star and norm topologies on SX∗ that Pα∗ f → Pβ∗ f in the norm. In the case of Fréchet differentiable norms we use the Bishop–Phelps theorem. By transﬁnite induction, this gives that if a WLD Banach space X admits an equivalent Fréchet differentiable norm, then X and all its subspaces admit a shrinking M-basis. Thus the property of admitting a shrinking M-basis is a hereditary property on Banach spaces ([160]). In particular, any subspace of c0 (Γ ) is WCG for any Γ . Moreover, by “forced” inclusion of the derivatives of a Fréchet C 1 -smooth function derived from the C 1 smooth bump into the ranges of the dual projections (cf. the subspace F in Lemma 3.11), we can show, similarly, that a WCG space that admits a Lipschitz, Fréchet C 1 -smooth bump function necessarily admits a shrinking M-basis. The duality mapping of a Banach space is a multivalued map D from X into the subsets of X∗ deﬁned by D(x) = {f ∈ X∗ ; x2 = f 2 = f (x)}. The following is a result in [156]. T HEOREM 7.11. If X is an Asplund space, then the duality map D admits a selector that is a pointwise limit in the norm of X∗ of a sequence of norm-to-norm continuous mappings from X into X∗ , i.e., there is a (Jayne–Rogers) selector J of D and a sequence of normto-norm continuous mappings Jn from X into X∗ such that Jn x − J x → 0 for every x ∈ X. The fact that J is a selector of D means that J (x) ∈ D(x) for x ∈ X. If X is WLD, then by “forced” inclusion of ranges of Jn into the ranges of the dual projections in Valdivia’s construction in [319] and by using Simons’ inequality (cf., e.g., [57, Chapter I] or [80, Chapter 3]) (instead of the Bishop–Phelps theorem) we get the following result in [319,309]. For WCD spaces, this result was proved earlier in [70] and [309]. T HEOREM 7.12. Let X be an Asplund space. Then X admits a shrinking M-basis if and only if it is WLD.

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Hence every WCG Asplund space is hereditarily WCG. An example of a nonseparable hereditarily WCG space that is not Asplund is 2 (c) ⊕ 1 (cf., e.g., [35]). If we assume only that X is an Asplund space, then a PRI on X∗ in its canonical supremum norm can be constructed by using the Jayne–Rogers selector (the projections cannot in general be dual maps). Thus we obtain the result in [74] (cf., e.g., [57, Chapter VI], [73, p. 150]). T HEOREM 7.13. X∗ admits a PRI (non-dual projections in general), an equivalent LUR norm (non-dual in general) and an M-basis if X is an Asplund space. Theorem 7.13 solves for dual spaces the longstanding problem (still open in its full generality) whether every space with the Radon–Nikodým property admits a LUR norm. For the deﬁnition, see, e.g., [73, p. 33]. One of the pioneering results in this direction was Tacon’s paper [291]. This paper motivated [160] and many other papers in this area in this period of time. The following result is a combination of the results of Deville, Haydon and Rogers ([49, 146], cf., e.g., [57, p. 311]). T HEOREM 7.14. Let K be a compact space such that the Cantor derived set K (ω1 ) = ∅. Then C(K) admits an equivalent LUR norm whose dual norm LUR. The following results characterize Asplund spaces in terms of some topological properties ([233,67,31]): T HEOREM 7.15. A Banach space X is an Asplund space if and only if X∗ is Lindelöf in the topology of uniform convergence on all separable bounded sets in X, if and only if each separable subset of BX in its weak topology is metrizable. Orihuela’s proof in [233] of the ﬁrst part of the statement gives also the proof of Alster– Gul’ko–Pol result that X is WL if BX∗ is Corson in the weak star topology (see Section 4, cf., e.g., [80, Chapter 12]). For further results in this direction we refer to [219] and references therein. The following Preiss’ version [255] of the separable reduction argument can be used in the proof that X is an Asplund space if and only if every continuous convex function on X is Fréchet differentiable on a dense set in X (cf., e.g., [57, Chapter I] or [243, Chapter 2]). T HEOREM 7.16. Assume that f is a continuous function on a Banach space X. Then for every separable subspace Z of X, there is a separable subspace W of X such that Z ⊂ W and f is Fréchet differentiable as a function on X at every point of W at which the restriction of f to W is Fréchet differentiable. The following result is related to Theorem 2.12 and in a more general form can be found in Ribarska’s theorem from [266] (presented in [73, p. 88]) together with the results in [57, p. 26] or [73, p. 90].

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T HEOREM 7.17. Assume that X is an Asplund space and K is a weak star compact set in X∗ . Then there is a subset S of K that is dense Gδ in K in its weak star topology and such that the weak star and the norm topologies on K coincide at each point of S. The set S is thus metrizable by a complete metric and all the points of S are Gδ points of K in its weak star topology. The following theorem is a counterpart of the results of Kadets in the separable case that we mentioned in the beginning of this section. It summarizes some results in this section. T HEOREM 7.18. Assume that a Banach space X is WLD. Then the following are equivalent. (i) X admits an equivalent norm whose dual norm is LUR. (ii) X admits an equivalent Fréchet smooth norm. (iii) X admits a Fréchet differentiable bump function. (iv) X is an Asplund space. P ROOF. (i) implies (ii): see the text in the beginning of this section. (ii) implies (iii) is standard. (iii) implies (iv): see the text in the beginning of this section. (v) implies (i): see Theorem 7.12 and Theorem 7.4.

We will now discuss the interplay of covering properties of Banach spaces and renormings by LUR and Kadets–Klee norms. Much progress in this area has been recently achieved ([209,211–213,261,262,260,263]). The following result is in [65]. T HEOREM 7.19. Assume that the norm of a Banach space X has the Kadets–Klee property. Then the norm and weak Borel sets coincide in X and X is a Borel set in X∗∗ in its weak star topology. P ROOF (Schachermayer [65]). If we consider on SX and X \ {0} the weak topology, then the map (t, x) → tx of (0, ∞) ⊕ SX into X \ {0} is a Borel homeomorphism, i.e., together with its inverse, it maps Borel sets onto Borel sets. This follows from the fact that in the inverse function y → (y, y/y), the ﬁrst coordinate is weak lower semicontinuous and thus Borel and the second coordinate is a composition of x → (x, x/x), which is Borel and a continuous function (x, x/x) → x/x. Moreover, as SX in its weak topology is a completely metrizable space if the norm has the Kadets–Klee property, SX is Gδ in its closure which contains SX∗∗ by Goldstine’s theorem. Thus X in its weak topology is then a Borel set in X∗∗ in its weak star topology. It is known that the norm and weak Borel structures in ∞ do not coincide ([293]). As a corollary of Theorem 7.19, we thus have that ∞ does not admit any Kadets–Klee norm, in particular any LUR norm ([195,309], cf., e.g., [57, p. 74]). In as much as the “method of covers” has been essential in characterizing metrizability in topological spaces, it is quite natural that this method has been helpful in characterizing Banach spaces that admit equivalent LUR or Kadets–Klee norms.

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The ﬁrst characterization in this direction is in [312] (see also [311] and [314]), where martingales were used. Several properties of coverings related to Kadets–Klee norms were studied in [141,153,152,186] and [187]. The main contribution in [210] consists of replacing topological and probabilistic conditions by linear topological properties of spaces. The following notion is in [262] and originated in [64,65] and [292]. Let Σ1 and Σ2 be families of subsets of a given set S. We say that S has property P (Σ1 , Σ2 ), if there is a sequence (An ) of subsets of S such that for every x ∈ S and every V ∈ Σ1 with x ∈ V there are n ∈ N and U ∈ Σ2 such that x ∈ An ∩ U ⊂ V . If Σ1 is a topology, this deﬁnition means that the family of sets {An ∩ Σ2 } is a network for Σ1 . For a Banach space X property P(norm topology, weak topology) (P( · , w) for short) is used in the following result in [262]. T HEOREM 7.20. Let X be a Banach space. Property P( · , w) is equivalent to the existence of a symmetric homogeneous weakly lower semicontinuous function F on X with · F 3 · such that the norm and the weak topologies coincide on the set S = {x ∈ X; F (x) = 1}. It is not known if P( · , w) characterizes the Kadets–Klee renormings. For spaces of continuous functions on trees this is true [145]. On the other hand, P( · , w) implies that the Banach space X is a Borel subset of X∗∗ in its weak star topology ([141,262,232]) and thus X isεthen σ -fragmentable ([153]), i.e., for every ε > 0, there isε a decomposition X= ∞ n=1 Xn such that for every n ∈ N and every non-empty A ⊂ Xn , there is a weak open set U such that A ∩ U = ∅ and diam(A ∩ U ) < ε. For LUR renormings we have the following result from [210]. T HEOREM 7.21. A Banach space X admits LUR norm if and only if for an equivalent ε such that for every n ∈ N and every every ε > 0 there is a decomposition X = ∞ X n=1 n x ∈ Xnε , there is an open halfspace H such that x ∈ H and diam (H ∩ Xnε ) < ε. An open halfspace in X is f −1 (a, +∞) for some f ∈ X∗ \ {0} and a ∈ R. A new transfer technique has been developed by using the covering techniques. A bounded linear one-to-one operator T from a Banach space X into a Banach space Y is called an SLD map if X has property P( · , T −1 (norm-open sets in Y)). The main result here is the following theorem. T HEOREM 7.22. Let T be an SLD map from a Banach space X into a Banach space Y . If Y admits an LUR norm, then X has an equivalent LUR norm. A bounded linear operator T is SLD whenever T −1 is a pointwise limit point of a sequence of norm to norm continuous functions, in particular when T −1 is a Baire 1 map (i.e., a pointwise limit of a sequence of continuous maps). This is the case for instance when the dual operator has norm dense range [210]. In this way we obtain, as a particular case, that X admits an equivalent LUR norm if there is a bounded linear operator T that maps X into c0 (Γ ) for some Γ and is such that T ∗ has a norm dense range in X∗ ([121], cf., e.g., [57, Chapter VII]).

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The only point, where the linearity of T is used is transferring the slices from Y into X. For the covering notions, the only condition needed is the weak to weak continuity, so they are invariant for weak homeomorphisms ([141,232,211,220,148]). Indeed, a weak homeomorphism is Baire one in both directions for the norms ([287]). As an application of this method, let us mention the following result in [211] and [212]. T HEOREM 7.23. Let X be a Banach space that is weakly locally uniformly rotund. Then X has an equivalent LUR norm. The norm is weakly locally uniformly rotund if lim(xn − x) = 0 in the weak topology of X whenever xn , x ∈ X are such that lim(2xn 2 + 2x2 − x + xn 2 ) = 0. In the case of dual norms the following is the result in [261]. T HEOREM 7.24. Let the weak and weak star topologies coincide on the dual sphere of X∗ . Then X∗ admits an equivalent dual LUR norm. In many results in this area of coverings, the fundamental construction of the norm is the one in the proof of Theorem 7.10. D EFINITION 7.25. A compact space K has the Namioka property if for every Baire space E and every continuous map ϕ from E into C(K) endowed with the pointwise topology, there is a dense Gδ subset Ω of E such that ϕ : E → (C(K), · ∞ ) is continuous at every point of Ω. The following result from [54] can be found, e.g., in [57, p. 329]. T HEOREM 7.26. Assume that K is a compact space such that C(K) admits an equivalent LUR norm · that is pointwise lower semicontinuous. Then K has the Namioka property. In particular, any Corson compact has the Namioka property. Any scattered compact space K such hat C(K) admits an equivalent LUR norm has the Namioka property. P ROOF. Let B and S denote the unit ball and the unit sphere of · respectively. It follows from the LUR property and from the pointwise lower semicontinuity of · that the identity map I from B endowed with the pointwise topology into B endowed with the norm topology is continuous at every point of S. Now, let E be a Baire space and let ϕ be a continuous map from E into C(K) in its pointwise topology. The map ψ(x) := x is pointwise lower semicontinuous. Hence there is a dense Gδ subset Ω of E such that ψ is continuous at every point of Ω. As I is continuous at every point of S, we get that any point of Ω is a point of continuity of ϕ : E → (C(K), · ). We can use Theorem 7.4 to ﬁnish the second part of the proof. If K is scattered, then the norm closed linear hull of the Dirac measures in C(K)∗ equals to C(K)∗ by Rudin’s theorem (cf., e.g., [80, Chapter 12]). Thus any equivalent norm on C(K) is pointwise lower semicontinuous if K is a scattered compact. An example of a compact set that does not have the Namioka property is B∗∞ in its weak do not have the star topology ([49]). There are trees T such that their compactiﬁcations T

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Namioka property ([132]). It is an open problem if there is a Baire space E, a compact set K and a separately continuous function f : E × K → R with no points of joint continuity. Recall that a family F of subsets of a topological space T is σ discrete if F = n Fn , where each Fn is a discrete family, i.e., for each n, each point of T has a neighborhood that meets at most one member of Fn . By the Nagata–Smirnov theorem (cf., e.g., [69, Chapter 4]), every metrizable space has a σ -discrete basis of its topology. It is apparently an open problem if every Banach space admits a σ -discrete basis for its norm topology formed by convex sets. The following result can be found in [260]. T HEOREM 7.27. Assume that a Banach space X admits an equivalent LUR norm. Then the norm topology of X has a σ -discrete basis formed by convex sets. P ROOF. Given ε > 0, deﬁne by transﬁnite induction a family of convex sets {Bα } as follows: B0 = BX , Bα+1 = Bα \ x ∈ X; xα∗ (x) > aα , where xα∗ ∈ SX∗ and aα ∈ R are such that diam BX ∩ x ∈ X; xα∗ (x) > aα < ε. The process ends when Bγ is in the open unit ball of X. For δ > 0 then deﬁne convex sets C(α, ε, δ) = Bα ∩ x ∈ X; xα∗ (x) aα + δ . The sets C(α, ε, δ) + B(0, 1/n) and their rational multiples are then used to produce the proof of the result. A family {Hγ ; γ ∈ Γ } issaid to be isolated if for every γ0 ∈ Γ , Hγ0 ∩ γ =γ0 Hγ = ∅. If Γ can be split into Γ = Γn with each family {Hγ ; γ ∈ Γn } being isolated, we say that the family {Hγ ; γ ∈ Γ } is σ -isolated. A family A of subsets of a topological space T is a network in T , if every open subset in T is a union of some members of A. A compact space is called descriptive if its topology has a σ -isolated network. A norm · on X∗ is weak star locally uniformly rotund if fn − f → 0 in the weak star topology whenever fn = f = 1 and fn + f → 2. Raja proved in [263] the following theorem. T HEOREM 7.28. If X is a Banach space, then X∗ admits an equivalent dual weak star locally uniformly rotund norm if and only if BX∗ in its weak star topology is descriptive. In the same paper Raja showed that this gives the following result. C OROLLARY 7.29. If X is an Asplund space, then X∗ admits an equivalent dual LUR norm if and only if BX∗ in its weak star topology is a descriptive compact.

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Raja also showed in [263] that there are non-WCD Banach spaces X such that BX∗ is Corson and at the same time descriptive. From Theorem 7.5(i) we get that the spaces JL0 , JL2 , D and C[0, ω1 ] all admit LUR norms and from Theorem 7.5(ii) we get that JL0 and JL2 admit norms whose duals are LUR. Kunen’s C(K) space admits no Kadets–Klee norm. Indeed, such a unit sphere would then be norm separable as C(K) is hereditarily weakly Lindelöf. Hence C(K) would be separable, a contradiction ([220]). Thus this C(K) space is an Asplund space that admits no LUR norm. The space C[0, ω1 ] admits a LUR norm as it admits a PRI with separable ranges. The space CP of Ciesielski and Pol admits an equivalent LUR norm by Theorem 7.5. The ﬁrst example of a space with a rotund norm that cannot be mapped into c0 (Γ ) by a bounded linear one-to-one operator was constructed in [44]. The space D = C(K) space for the two arrows space K (Section 2) admits an equivalent LUR norm that is pointwise lower semicontinuous ([152]). Hence this K has the Namioka property by Theorem 7.27. The latter result was ﬁrst proved in [49].

8. C k -smooth norms for k > 1 We will discuss higher-order differentiability in the Fréchet sense only. Thus we will say that a function f is, for example, C 2 -smooth if the map Φ : x → f (x) from X into X∗ in its norm topology, is Fréchet C 1 -smooth. If we algebraically add the epigraphs of the real functions tχ[0,∞] (t) and t 2 , we get the epigraph of a convex function that is not twice differentiable at the origin. Thus we cannot use, in general, sums of convex bodies to produce C k smooth bodies if k > 1. Recall that for a real valued function f on a Banach space X, the epigraph of f is deﬁned by epi f = {(x, r) ∈ X × R; r f (x)}. D EFINITION 8.1. We say that a norm · on a Banach space X locally depends on ﬁnitely many coordinates if for every x ∈ X \ {0} there is a neighborhood U of x, f1 , f1 , . . . , fn ∈ X∗ and a continuous function ϕ on Rn such that z = ϕ(f1 (z), f2 (z), . . . , fn (z)) for all z ∈ U. A typical example of such norm is the supremum norm of c0 (Γ ), for any Γ . Let a bounded linear operator T from C0 [0, ω1 ] into c0 [0, ω1 ] be deﬁned by T x(α) = x(α + 1) − x(α) if α < ω1 and T x(ω1 ) = 0. Given x ∈ C0 [0, ω1 ] with x = 1, choose β = sup{α; |x(α)| = 1}. Then T x(α) = 0. This is an example of what is now called a Talagrand operator (it originated in [297], see [144,145]). For a compact set K, it is a bounded linear operator T from a subspace X of C(K) into c0 (K) such that for every x ∈ X of supremum norm one there is k ∈ K such that |x(k)| = 1 and T x(k) = 0. If T is a Talagrand operator on X ⊂ C(K), then the norm · 1 deﬁned for x ∈ X by x1 = supα∈K {|x(α)| + |T x(α)|} is easily seen to depend locally on ﬁnitely many coordinates. Hence C0 [0, ω1 ] admits an equivalent norm that locally depends on ﬁnitely many coordinates. As C[0, ω1 ] is isomorphic to C0 [0, ω1 ] we thus have that C[0, ω1 ] admits a norm that locally depends on ﬁnitely many coordinates.

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Norms that locally depend on ﬁnitely many coordinates are often as good as smooth norms (sometimes even better). And, last but not least, they are often much easier to construct than the smooth norms. For their connection with smooth norms see Problem 3 in Section 9. If a Banach space X admits a continuous bump function that locally depends on ﬁnitely many coordinates, then X is an Asplund space and contains an isomorphic copy of c0 ([239,90], [57, Chapter V]). For a connection of such norms to polyhedral spaces see, e.g., [132] and [99]. A Banach space X is called polyhedral if BF is the convex hull of a ﬁnite set for every ﬁnite-dimensional subspace F of X. For norms that locally depend on countably many coordinates we refer to [92]. The following result can be found in [239] (cf., e.g., [57, p. 189]). T HEOREM 8.2. For any set Γ , the space c0 (Γ ) admits an equivalent norm that is at the same time C 1 -smooth and LUR and is a limit, uniform on bounded sets, of C ∞ -smooth norms that locally depend on ﬁnitely many coordinates. P ROOF. For n ∈ N, let ϕn be even, C ∞ smooth real valued function on the reals such on (1/(n + 1), ∞). For that ϕn = 0 on [−1/(n + 1), 1/(n + 1] and ϕ > 0 and ϕ

> 0 n ∈ N, deﬁne the function Φn on c0 (Γ ) for x = (xα ) by Φn (x) = α∈Γ ϕn (xα ). Note that Φn is well deﬁned and locally C ∞ smoothly depend on ﬁnitely many coordinates. Given n, m ∈ N, put Qn,m = {x ∈ c0 (Γ ); Φn (x) m}. Let · n,m denote Minkowski’s functional of Qn,m . Finally, deﬁne the norm on c0 (Γ ) for x ∈ c0 (Γ ) by x2 = ∞ −n−m x2 . Then the norm · has the required properties. Indeed, we can n,m n,m=1 2 see that · is LUR (use the properties of the supports of ϕn ) and is C 1 as it is a sum of terms that have ﬁrst derivatives bounded. This is not the case with the higher-order deriva tives, where we can only say that · n,m are C ∞ smooth. The following result of Haydon [144] extends the result of Talagrand [297], where C 1 -smoothness was studied. T HEOREM 8.3. The space C[0, μ] admits a C ∞ smooth norm for every ordinal μ. The following result is in [118] (cf., e.g., [57, p. 194]). T HEOREM 8.4. If K is a compact space such that the Cantor derived set K (ω0 ) = ∅, then C(K) admits a C ∞ -smooth norm. As K (ω0 ) = ∅ means K (n) = ∅ for some n ∈ N, Theorem 8.4 follows by induction by using the following result in [118] (cf., e.g., [57, p. 194]). T HEOREM 8.5. Assume that k ∈ N∪{+∞}. Let X be a Banach space and Y be a subspace of X such that Y is isomorphic to c0 (Γ ) for some Γ and that X/Y admits an equivalent C k -smooth norm. Then X admits an equivalent C k -smooth norm. Theorem 8.5 can be proved by a variant of the construction in the proof of Theorem 8.2.

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Hájek recently showed in [138] that C(K) admits a C ∞ smooth norm that locally depends on ﬁnitely many coordinates if K (ω1 ) = ∅. Later on, Hájek and Haydon showed [139] that C(K) admits an equivalent C ∞ smooth norm if C(K)∗ admits an equivalent dual locally uniformly rotund norm. The following result is due to Haydon [145]. be the one point compactiﬁcation of a tree T . Then C(T) admits a T HEOREM 8.6. Let T ∞ C -smooth bump function. Haydon has examples of trees that create many counterexamples for renormings: there are trees T such that C(T) then admits a Gâteaux differentiable norm but no strictly convex norm. The full dyadic tree of the height ω1 , i.e., T = α<ω1 {0, 1}α has the property that C(T) admits a Kadets–Klee norm but no strictly convex norm. For trees T , C0 (T) spaces admit a LUR norm if and only if they admit a Fréchet differentiable norm. All of this is in [145]. The space c0 is isomorphic to a C(K) for some compact metric K if and only if K (ω0 ) = ∅ ([24], cf., e.g., [168]). This is no longer true for nonseparable c0 (Γ ). Indeed, as we saw in Section 2, for JL0 = C(K), K (ω0 ) = ∅ and yet, c0 (Γ ) is not isomorphic to C(K). However, if dens C(K) ℵ1 and K is an Eberlein compact such that K (ω0 ) = ∅, then C(K) is isomorphic to c0 (Γ ) ([115]). A one-to-one mapping ϕ from a Banach space X onto a Banach space Y is called a Lipschitz homeomorphism (uniform homeomorphism) if both ϕ and ϕ −1 are Lipschitz (uniformly continuous). T HEOREM 8.7. For a compact space K, the following are equivalent: (1) K (ω0 ) = ∅, (2) C(K) is Lipschitz homeomorphic to c0 (Γ ), (3) C(K) is uniformly homeomorphic to c0 (Γ ). P ROOF. Indeed, if K (ω0 ) = ∅, then C(K) is Lipschitz homeomorphic to c0 (Γ ) ([2,55], cf., e.g., [57, p. 264]). If C(K) is uniformly homeomorphic to c0 (Γ ), then K (ω0 ) = ∅ (Godefroy, see [169]). It is proved in [169] that if a Banach space X is uniformly homeomorphic to p (Γ ), 1 < p < ∞, then X is isomorphic to p (Γ ). Pelant recently proved in [240] that C[0, ω1 ] is not uniformly homeomorphic to any subset of c0 (Γ ). Marciszewski showed in [202] that for a compact set K, C(K) is isomorphic to a subspace of c0 (Γ ) for some Γ if and only if C(K) is isomorphic to some c0 (Γ ). Thus JL0 and c0 (Γ ) are two Lipschitz homeomorphic Banach spaces that are not isomorphic [2]. The example of two WCG spaces that are Lipschitz homeomorphic but not linearly isomorphic is in [19]. The existence of reﬂexive spaces with this property is not known. For more in this direction we refer, e.g., to [114]. It is shown in [19] that an Eberlein compact set is a uniform Eberlein if K (ω0 +1) = ∅ and that (ω0 + 1) is optimal in this respect.

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For higher-order differentiability, Asplund averaging does not work in general. For example, if · is an LUR C 2 -smooth norm on X, then we can construct on X an equivalent norm that has a uniformly continuous ﬁrst differential on the sphere, by looking at a neighborhood of a point on a supporting hyperplane. A Banach space is superreﬂexive whenever it admits a norm whose derivative is uniformly continuous on the sphere (cf., e.g., [57, p. 152] or [80, Chapter 9]). Thus we obtain the following result in [88] (cf., e.g., [57, p. 187]). T HEOREM 8.8. Any Banach space X with a norm that is LUR and C 2 -smooth at the same time is necessarily superreﬂexive. Since c0 is not superreﬂexive ([168,111], cf., e.g., [57, Chapter IV]), it admits no C 2 -smooth norm that is at the same time LUR, though it admits norms that separately have these properties (Theorem 8.2). Hence the collection of all C 2 -smooth norms on c0 is not residual in all equivalent norms in the metric space P deﬁned in Theorem 7.8. Let · 1 be a C ∞ -smooth norm on c0 (Theorem 8.2) and · 2 be the standard norm of 2 . Let T be a bounded linear one-to-one operator of c0 into 2 . Then the norm |||x||| = (x21 + T x22 )1/2 is a strictly convex C ∞ -smooth norm on c0 . Hájek proved the following result ([136,134,137]). T HEOREM 8.9. If Γ is uncountable, then there is no C 2 -smooth function on c0 (Γ ) that would attain its minimum exactly at one point. In particular, c0 (Γ ) admits no equivalent strictly convex C 2 -smooth norm if Γ is uncountable. This solved the problem posed by Jaramillo. The key argument for Theorem 8.9 is Hájek’s result ([136,134]) that for a C 2 -smooth function ϕ on c0 , the map x → ϕ (x) is locally compact, i.e., given x ∈ c0 , there is a neighborhood U of x such that ϕ (U ) is relatively compact in 1 (cf., e.g., [59] and [20, Chapter 14] for more in this area). By using functions that locally depend on ﬁnitely many coordinates, Torunczyk [307] (cf., e.g., [57, Chapter VIII]) proved the following result. T HEOREM 8.10. Let k ∈ N ∪ {+∞}. A Banach space X admits C k -smooth partitions of unity if and only if there is a set Γ and a homeomorphic embedding ϕ of X into c0 (Γ ) such that the coordinate function ϕ(·)γ is a C k smooth function on X for every γ ∈ Γ . For deﬁnition of smooth partitions of unity we refer to, e.g., [57, p. 351]. The local dependence of the functions that form smooth partitions of unity on c0 (Γ ) is crucial as it ensures smoothness of functions that form smooth partitions of unity on X only by means of coordinatewise smoothness of the map ϕ. From applications of Theorem 8.10 let us mention the following results from [132,55, 204,322], see also [57, Chapter VIII]. T HEOREM 8.11. Let k ∈ N ∪ {+∞}.

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(i) Assume that X admits a C k -smooth bump function and that X admits a PRI {Pα , α μ} such that Pα (X) admits C k -smooth partitions of unity for all α < μ. Then X admits C k -smooth partitions of unity. (ii) If T is the one-point compactiﬁcation of a tree T , then C(T) admits C ∞ -smooth partitions of unity. (iii) If X admits a LUR norm whose dual is LUR, then X admits C 1 -smooth partitions of unity. (iv) If X contains a subspace Y that is isomorphic to some c0 (Γ ) and X/Y admits C k -smooth partitions of unity, then X admits C k -smooth partitions of unity. In particular, if K (ω0 ) = ∅, then C(K) admits C ∞ -smooth partitions of unity. (v) Assume that X admits a C k -smooth bump function and that X∗ is WCG. Then X admits C k -smooth partitions of unity. P ROOF. We will outline here the proof of other result instead, namely that any WCG Banach space X that has a transﬁnite Schauder basis {xα , fα }0α<μ admits Gâteaux differentiable partitions of unity ([162]). Let Δ be a disjoint union of {−1}, N × N and [0, μ). Finally, let the homeomorphism ϕ from X into c0 (Δ) required in Theorem 8.11 be deﬁned for x ∈ X by ⎧ 2 for α = −1, ⎪ ⎨ x ϕ(x)α = 2−n−i Gn,i (x) for α = (n, i) ∈ N × N, ⎪ ⎩ fα (x) for α < μ, where the Gâteaux smooth functions Gn,i uniformly approximate the functions {Gn } in Troyanski’s proof of Theorem 7.3 and are constructed by adding weakly compact smooth sets generating X to their epigraphs. The norm x is a Gâteaux smooth norm on X (Theorem 5.2). The functions Gn,i ensure that whenever ϕ(xn ) → ϕ(x), then {xn } is norm relatively compact in X. The proof uses the techniques in the proof of Theorem 7.3. Further results on smooth partitions of unity can be found in [103] and [260]. It has recently been showed (see [139]) that C(K) space admits C k smooth partitions of unity if it admits a C k smooth bump function. The space c0 (Γ ) admits a C ∞ -smooth norm and so do the spaces JL0 and JL2 and CP (Theorem 8.5). If · is a C ∞ -smooth norm on JL0 and T is a bounded linear one-to-one operator of JL0 into 2 (JL∗0 is weak star separable), then the norm deﬁned for x ∈ JL0 by |||x||| = x + T x2 is a strictly convex C ∞ -smooth norm on JL0 . This should be compared with Theorem 8.9. Kunen’s space C(K) admits no norm that would locally depend on ﬁnitely many coordinates. This follows from a variant of Theorem 2.5 and the fact that C(K)∗ is weak star hereditarily separable (cf., e.g., [92] or [99, Proposition 6.20]). We mentioned above that CP does not admit any bounded linear one-to-one operator into any c0 (Γ ) (cf., e.g., [57, Chapter VI]). This should be compared with Theorem 8.11(iv). The spaces JL0 and JL2 both admit C ∞ -smooth partitions of unity by Theorem 8.11. The space C[0, ω1 ] admits C ∞ -smooth partitions of unity as it has a transﬁnite Schauder basis and admits a C ∞ -smooth norm ([120,144]). As C[0, ω1 ] is not WLD and does not contain

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any isomorphic copy of 1 (ℵ1 ) (since, for instance, it admits a C ∞ -smooth norm unlike 1 (ℵ1 ) (see Section 5)), C[0, ω1 ] does not admit any unconditional basis by Theorem 4.3. A real-valued function ϕ on a Banach space X is called a real analytic function on X if for every x ∈ X, there is a neighborhood U of x in X such that the Taylor expansion of ϕ at x (built of multilinear forms on X) converges uniformly to ϕ on U . If K is a compact space, then C(K) admits a real analytic norm (on X \ {0}) if and only if K is countable ([52,135]).

9. Open problems and concluding remarks We will now present some open problems in this area. 1. Assume that X is an Asplund space. Does X admit a (Lipschitz) Fréchet C 1 -smooth bump function? In particular, does Kunen’s C(K) space (see Section 2) admit a Fréchet C 1 -smooth bump function? It is not known if the property of a Banach space to admit a Fréchet C 1 -smooth bump function is a three space property. For partial results see, e.g., [55]. However, being an Asplund space is a three space property (cf., e.g., [73, p. 7], [328], [34, Section 4.11]). For more information on this problem, see [57, p. 89]. Assume X is an Asplund space. Does there exist a C 1 -smooth (not necessarily equivalent) norm on X? 2. Assume that K is a scattered compact. Does C(K) admit a C ∞ -smooth bump function? Assume that K is a scattered compact such that C(K) admits a Fréchet C 1 smooth norm. Does C(K) admit an equivalent C ∞ -smooth norm? The result holds true if C(K) admits an equivalent norm whose dual norm is LUR [139]. 3. Assume that X admits an equivalent norm that locally depends on ﬁnitely many coordinates. Does X admit an equivalent C ∞ -smooth norm? If X is separable, the answer is yes [132]. 4. Assume that k ∈ N ∪ {+∞}. Let X admit a Fréchet C k -smooth bump function. Does X admit C k -smooth partitions of unity? The answer is yes for WLD spaces, for spaces of continuous functions on trees and many other spaces (cf. Theorem 8.12). Recently, it has been showed (see [139]) that the answer is yes for every scattered compact space. Does the space c0 (Γ ) admit partitions of unity formed by uniformly Gâteaux differentiable functions? For smooth partitions of unity on nonseparable superreﬂexive spaces formed by functions that are uniformly Fréchet differentiable we refer to [159]. 5. Are Fréchet differentiable equivalent norms on C[0, ω1 ] dense (residual) in all equivalent norms on this space in the metric space P deﬁned in Theorem 7.8? The space C[0, ω1 ] admits C ∞ -smooth norm by Theorem 8.3. Does there exist a C 1 -smooth LUR norm on C[0, ω1 ]? Does C[0, ω1 ] admit a LUR norm that is a uniform limit (on bounded sets) of C ∞ -smooth norms? 6. Is every equivalent norm on 2 (c) approximable by C ∞ -smooth norms in the metric space P deﬁned in Theorem 7.8? Note that for 2 , the statement is true ([51]). 7. Let X be an Asplund space. Does X admit an equivalent LUR norm if and only if X admits a Fréchet smooth norm? This was shown to be true for spaces of continuous functions on compactiﬁcations of trees by Haydon [145]. Very recently, Haydon

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9.

10.

11.

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showed that X admits an equivalent LUR norm if X∗ admits and equivalent dual LUR norm. Assume that X is a WLD space. Is it true that every continuous convex function on X is Gâteaux differentiable on a dense set in X? There exists a WLD space X with an unconditional basis which is not a weak Asplund space ([13]). It is known that if X is a WLD space and K is a weak star compact convex subset of X∗ , then K contains an extreme point that is a Gδ point of K in its weak star topology ([13]). Note that if K = {x ∈ B2 (c) ; x(γ ) 0 for all γ }, then 0 is an extreme point of K that is not a Gδ point of K in its weak topology ([43]). Assume that X is a weak Asplund space. Is every subspace of X a weak Asplund space? Is X × R a weak Asplund space if X is a weak Asplund space? Assume that X∗∗ is WCG. Is X necessarily WCG? Assume X∗ admits a UG norm. Does there exist a bounded linear one-to-one operator of X into some c0 (Γ )? Assume that X is a WCG space with an unconditional basis. Is every subspace of X WCG? Let X be a WCG Banach space. Does there exist a Markushevich basis {xα , fα } of X such that the subspace Y := span · {fα } is norming in X? The space Y is norming if δ := infx∈SX {supf ∈BY {f (x)}} > 0). The space C[0, ω1 ] does not admit such M-basis ([3]) and 1 ⊕ 2 (c) admits a norm in which it admits no one-norming M-basis (i.e., δ = 1) (Troyanski (see [324]) and [320])). For more information in this direction we refer to [110] or [323] and references therein. Let X be a Banach space. Does there exist a subspace Y of X such that X/Y is inﬁnite-dimensional and separable? For separable spaces, this problem has a solution in the positive (Johnson, Rosenthal, cf., e.g., [197, p. 10]). Note that it is standard to show that this problem has a positive answer for every Banach space that admits a Markushevich basis. It is also known that X admits such a quotient if and only if X contains an inﬁnite-dimensional separable subspace that is quasicomplemented in X ([270], cf., e.g., [215]). This is the case if X = C(K) space (Lacey, cf., e.g., [215].) Let K be a compact space. Is it true that C(K) is isomorphic to a C(K1 ), where K1 is a totally disconnected compact space? A totally disconnected compact space K1 is such that given two different points x, y ∈ K1 , there is a clopen set in K1 (i.e., both closed and open) that contains one point but not the other. This problem has a positive answer for metrizable compacta. Indeed, for an uncountable metrizable compact set K, C(K) is isomorphic to C[0, 1] (Milutin, cf., e.g., [327, p. 160]). For countable metric compact spaces, C(K) is isomorphic to the space of continuous functions on an ordinal segment by the result of Bessaga and Pełczy´nski ([24], cf., e.g., [168]). Find a characterization of compact spaces K such that C(K) admits a LUR or R norm. This is contained in Problem 11 in [195] and is still open, though much progress has recently been done (cf. Section 7). Assume that X is a separable Banach space that does not contain any isomorphic copy of 1 . Is it true that the bidual norm on X∗∗ has points of Fréchet differentiability? For the space JT, this problem was solved in the positive by Schachermayer ([278]).

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15. Assume that X is a reﬂexive Banach space and · is an equivalent Fréchet differentiable norm on a subspace Y of X. Can · be extended to a Fréchet differentiable equivalent norm on X? There is a Gâteaux differentiable norm on a subspace of a separable Banach space X that cannot be extended to an equivalent Gâteaux differentiable norm on X (cf., e.g., [57, p. 85]). The analogous problem for uniform Fréchet differentiable norms on superreﬂexive spaces is open as well. There is a connection of this problem with Maurey’s extension results ([332]). For several notions of rotundity, the problem has a positive answer ([72,298], cf., e.g., [57, p. 82]). We will ﬁnish this article with some comments and remarks.

A. More on special compact spaces A compact space K is called a Valdivia compact if for some Γ , K is homeomorphic to a subset S of [−1, 1]Γ taken in its pointwise topology, such that the countably supported elements of S are dense in S. Examples of Valdivia compacts include Corson compacts, [0, ω1 ] (cf. the text following Deﬁnition 2.18) and for any Γ , B∞ (Γ ) in its weak star topology. B∞ (c) is separable and non-metrizable in its weak star topology (Section 2). If ω2 is the least ordinal of cardinality strictly larger than ℵ1 , then [0, ω2 ] in its usual order topology is not a Valdivia compact [54]. If K is a Valdivia compact, then C(K) admits a PRI ([14,318], cf., e.g., [57, Chapter VI]), admits a pointwise lower semicontinuous LUR norm ([318], cf., e.g., [57, Chapter VII]) and BC(K)∗ is a Valdivia compact in its weak star topology ([233]). If K is a Valdivia compact but not a Corson compact, then K contains a homeomorphic copy of [0, ω1 ] ([54]) and admits a continuous map onto a non-Valdivia compact ([177]). The ﬁrst example of a non-Valdivia compact that is a continuous image of a Valdivia compact is in [321]. It is shown in [92] that Kalenda’s result that BC0 [0,ω1 ]∗ is not a Valdivia compact ([176]) follows from the technique of the proof of Theorem 2.6. The two arrows compact space K (Section 2) is not a Corson compact as it is separable and not metrizable (C(K) is not separable). It is not a Valdivia compact either, as otherwise it would contain a homeomorphic copy of [0, ω1 ] ([54]), which contradicts the fact that K is hereditarily Lindelöf (cf., e.g., [175]). Kalenda recently showed in [178] that the fact that BX∗ is a Valdivia compact does not in general imply that X admits a PRI. We refer to, e.g., [57, Chapter VI], [75,92,177,233,321] and references therein for more information on Valdivia compacts. A compact space K is called a Rosenthal compact if K is homeomorphic to a subset of the space B1 (P ) of all Baire one functions on a Polish space P , in its pointwise topology. A topological space is called a Polish space if it is homeomorphic to a complete separable metric space. Examples of Rosenthal compacts include the space of all non-decreasing functions from [0, 1] into [0, 1] with the pointwise topology (Helly’s space, cf., e.g., [221, Chapter 15]), the two-arrows space (Section 2) (it is homeomorphic to a subset of Helly’s space by using the characteristic functions of the intervals) and BJT ∗∗ in its weak star topology (cf., e.g., [197, p. 101]). All these examples are separable and non-metrizable. If K is a Rosenthal compact, so is BC(K)∗ in its weak star topology ([107]). A separable compact K is a Rosenthal compact if and only if for every countable dense set D in K, the space

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C(K) is analytic in the topology of pointwise convergence on D (Godefroy ([107]). A set in a metric space is analytic if it is a continuous image of Baire’s space NN . In this direction, let us mention that it is not known if a compact space K is an Eberlein compact whenever for each dense set S in K, C(K) admits an equivalent Gâteaux smooth norm that is S-lower semicontinuous [81]. Every Rosenthal compact is angelic ([272,273,30]). If K is a separable non-metrizable Rosenthal compact, then there is a continuous map of K onto a non-Rosenthal compact ([119]). Any Rosenthal compact contains a metrizable dense set ([305]). As the support of every ﬁnite positive measure satisﬁes the C.C.C., every ﬁnite positive regular measure on a Rosenthal compact has thus separable support (Godefroy [107]). If K is a non-metrizable separable Rosenthal compact, then BC(K)∗ with its weak star topology contains a discrete subspace of cardinality continuum ([119]). Every non-metrizable Rosenthal compact contains either an uncountable discrete subspace or a homeomorphic copy of the two arrows space ([305]). We refer to [107,119,222,253] and [305] and references therein for more on Rosenthal compacts. For information on other types of compacts (Gul’ko, Talagrand, Radon–Nikodým, etc.) we refer the reader to [13, 73,206,219,222] and references therein.

B. More on the weak topology of nonseparable Banach spaces If {xi∗ } is norm dense in BX∗ for aBanach space X, then BX in its weak topology is xn∗ (y)|. In this metric, the completion metrizable by the metric d(x, y) = 2−n |xn∗ (x) − ∗∗ ∗∗ −n of BX is BX∗∗ under the metric d(x , y ) = 2 |x ∗∗ (xn∗ ) − y ∗∗ (xn∗ )|. Thus BX in this metric is complete if and only if X is reﬂexive. We will say that a Banach space X has Polish ball if BX in its weak topology is a Polish space. Godefroy proved in [106] that X has Polish ball if X∗∗ is separable (cf., e.g., [80, p. 414]). Edgar and Wheeler showed in [67] that X contains an inﬁnite-dimensional reﬂexive subspace if X has Polish ball. The ˇ predual of the space JT has Polish ball ([67]). A Banach space X is said to have Cech ˇ complete ball if BX in its weak topology is a Cech complete space, i.e., Gδ in some (every) ˇ of its compactiﬁcations. A metrizable space is Cech complete if and only if it is metrizable by a complete metric ([69, p. 142], [102,1]). This, together with the result in [67] that X∗ ˇ is separable whenever X is separable and has Cech complete ball, gives that a Banach ˇ space X has Polish ball if and only if X is separable and has Cech complete ball. Edgar ˇ and Wheeler proved in [67] that X has Cech complete ball if and only if X is isomorphic to a direct sum Z ⊕ W , where Z is reﬂexive and W has Polish ball. For results on (nondual) balls in dual spaces that are weak star Polish we refer to [274]. If the norm of X has the Kadets–Klee property, then BX in its weak topology is a Baire space. In order to see this, assume that X is inﬁnite-dimensional and consider BX in its weak topology. Then SX is dense in BX and as we saw in Section 2, SX is Gδ in BX . Since the norm of X has the Kadets–Klee property, SX is metrizable by the metric of the norm of X. Hence BX in its weak topology is a Baire space. Although c0 admits an equivalent norm with the Kadets–Klee property (for example, Day’s norm, see Section 7), the unit ball B∞ of c0 in the standard norm is not a Baire space in its weak topology. Indeed, B∞ = ∞ n=1 An , where each An = {x ∈ B∞ ; |xk | 1/2 for all k n} is closed and nowhere dense in B∞ in the weak topology.

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A Banach space X is called weakly realcompact if X in its weak topology is homeomorphic to a closed subset of RI in its pointwise topology for some set I . Corson showed in [41] that a Banach space X is weakly realcompact if and only if every f ∈ X∗∗ , which is weak star continuous on all weak star separable subsets of X∗ is weak star continuous on X∗ . For a connection of this concept to measure theory on Banach spaces we refer to [65] and references therein. A Banach space X is weakly realcompact if X has either property C or satisﬁes the condition of Mazur, i.e., every f ∈ X∗∗ that is sequentially continuous on BX∗ in its weak star topology is weak star continuous on X∗ . An example of a space that is not weakly realcompact is C[0, ω1 ] ([65]). An example of a weakly realcompact X that does not satisfy the condition of Mazur or has property C is ∞ (for Mazur’s condition use the Grothendieck property of ∗∞ ). If we denote by C[0, ω1 ]s the subspace of C[0, ω1 ]∗∗ of all weak star sequentially continuous functionals on C[0, ω1 ]∗ , then the codimension of C[0, ω1 ] in C[0, ω1 ]s is one ([279]). From the proof of this result it follows that C[0, ω1 ] is not isomorphic to C[0, ω1 ] ⊕ C[0, ω1 ] though the duals of these spaces are isometric ([279]). From the area of the weak compact generating, let us mention the following two results from [167] (cf., e.g., [34, p. 131] or [60, p. 154]). First, if Y is a reﬂexive subspace of a Banach space X, then X is WCG if X/Y is WCG. Second, if Y is a subspace of a Banach space X such that X/Y is separable, then X is WCG if and only if Y is. For further results on the three space problem on WCG and WCD properties, we refer to [167,317], [34, Chapter 4.10] and [35]. In [46], one can ﬁnd the following extension of the former James–Lindenstrauss theorem: if X is a WCG Banach space, then there is a Banach space Z such that X is isomorphic to Z ∗∗ /Z. If X∗∗ /X is ﬁnite-dimensional, then X is WCG ([195]).

C. More on fragmentability and σ -fragmentability Let (T , τ ) be a topological space and ρ be a metric on T not necessarily related to τ . If ε > 0 and P is a subset of T , we will say that P is fragmented by ρ down to ε if whenever A is a non-empty subset of P there is a non-empty relative τ -open subset B of A such that ρ-diameter of B less than ε. We will say that a topological space (T , τ ) is fragmented by ρ if T is fragmented by ρ down to ε for each ε > 0. We will say that a topological space is fragmented if it is fragmented by some metric. Ribarska proved in [266] that if a compact space (T , τ ) is fragmented, then T is fragmented by a complete metric that is stronger than τ . Namioka proved in [218] that if a compact space K is fragmented by a lower semicontinuous metric, then K is homeomorphic to a weak star compact set in the dual of an Asplund space. If this happens for a Corson compact K, then K is necessarily an Eberlein compact ([234,290,87], Rezniˇcenko, cf., e.g., [73, p. 155]). As we already discussed in Section 7, a topological space (T , τ ) is called σ -fragmented by ρ if for every ε > 0, T can be decomposed as T = ∞ n=1 Tn , where (Tn , τ ) is fragmented by ρ down to ε. If X is a separable Banach space, {xi } is dense in X and ε > 0, then X = (xi + εBX ), showing that X in its weak topology is σ -fragmented by the metric given by the norm of X. On the other hand, the Banach space c0 in its weak topology is not a countable union of sets fragmented by the norm as it follows from the Baire category theorem and from the fact that each relatively weakly open set in Bc0 has diameter > 1 ([154]) (cf.,

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e.g., [80, Chapter 12]). Each weakly compact set in its weak topology in a Banach space X is fragmented by the norm of X (Troyanski, cf., e.g., [219] and references therein). If X admits a Gâteaux differentiable norm, then X∗ in its weak star topology is fragmented (Ribarska, cf., e.g., [73, p. 81]). For an extension of this result for Lipschitz Gâteaux differentiable bump functions see [100]. Fragmentability of X∗ in the weak star topology (which is equivalent with BX∗ being fragmented in the weak star topology (see, e.g., [73, p. 86])) in turn implies that X is a weak Asplund space. The converse implication does not hold in general ([181]). BX∗ in its weak star topology is fragmented by the norm of X∗ if and only if X∗ in its weak star topology is σ -fragmented by the norm of X∗ if and only if X is an Asplund space (Namioka, Phelps, cf., e.g., [57, Chapter I], [219]). If X is WCG, then X in its weak topology is σ -fragmented by the norm and all Xi in the deﬁnition of σ -fragmentability can be taken weakly closed (Jayne, Namioka, Rogers [152]). If X admits a Kadets–Klee norm, then X in its weak topology is σ -fragmented by the norm and all the sets Xi can be differences of weakly closed sets (Jayne, Namioka, Rogers [152]). Let X be ∞ equipped with its weak topology. The X is not σ -fragmented by its norm. However, X is fragmented by the lower semicontinuous metric τ (x, y) = 2−i min{1, |xi − yi |}. If Γ is uncountable, then the space c∞ (Γ ) equipped with its weak topology is not fragmented by any metric nor is it σ -fragmented by any lower semicontinuous metric (all of this is in [154]). For more information on these topics we refer to [73] and [219] and references therein.

D. Fundamental biorthogonal systems and Mazur’s intersection property A biorthogonal system {xα , fα } in a Banach space X is said to be a fundamental biorthogonal system for X if span{xα } = X. Davis, Johnson and Godun ([45,123] proved that if a Banach space X has a WCG quotient space of the same density character as X, then X has a fundamental biorthogonal system. This is the case with ∞ , as 2 (c) is isomorphic to a quotient of ∞ ([270]). On the other hand, if card Γ > c, then c∞ (Γ ) does not admit any fundamental biorthogonal system (Godun, Kadets, Plichko, cf., e.g., [129, p. 238]). Kunen’s space C(K) does not have any nonseparable subspace with a fundamental biorthogonal system (Section 2). A Markushevich basis {xα , fα }α∈Γ for a Banach space X is called a bounded Markushevich basis if supα∈Γ {xα ·fα } < ∞. For bounded Markushevich bases we refer to [197, p. 44], [244,247]. A Banach space X is said to have Mazur’s intersection property if every closed bounded convex set in X is an intersection of a family of balls in X. By the results of Mazur and Phelps, this property is shared by all spaces with Fréchet differentiable norms (cf., e.g., [57, p. 55] or [80, Chapter 8]). We refer to [158], where it is proved that the non-Asplund space 1 ⊕ 2 (c) can be renormed to possess Mazur’s intersection property and that Kunen’s C(K) space in turn does not admit any norm with Mazur’s intersection property. The tool used in [158] is the concept of biorthogonal systems {xα , fα } in X such that span · {fα } = X∗ . It is shown in [158] that every space that admits such a system admits a norm with Mazur’s intersection property and that this is the case if X = C(K) for the compactiﬁcation K of any tree. I do not know of any Banach space that would admit a C 1 smooth bump function and would not have Mazur’s intersection property at the same time. Related to

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Mazur’s intersection property is the result that every weakly compact set in a Banach space is the intersection of a family of ﬁnite unions of balls. This result was proved by Corson and Lindenstrauss in [43] for weakly compact sets in reﬂexive spaces (cf., e.g., [80, Chapter 8]) and later on, in full generality, by Godefroy and Kalton in [113]. Every WLD space X can be so renormed that in the new norm each weakly compact convex set is the intersection of a family of balls ([331]). E. Uniform homeomorphisms We have discussed only a little on nonlinear classiﬁcations of Banach spaces. We refer the reader to [20,112,114–116,169] and more references given below for this topics. F. Concluding remarks We have seen in this paper that in many cases, the separable and nonseparable theories of Banach spaces differ. Let us ﬁnish this article with mentioning a few more instances when this happens. First, it is well known that separable Lp spaces admit unconditional bases if 1 < p < ∞ (Paley). This is no longer true if Lp is nonseparable and p = 2 ([68, 101]). James showed that any non-reﬂexive separable Banach space with unconditional basis contains either c0 or 1 (see, e.g., [197, p. 23] or [80, p. 185]). However, there exists a nonseparable Banach space X with symmetric basis that does not contain any subspace isomorphic to c0 (Γ ) for uncountable Γ while every inﬁnite-dimensional subspace of X contains a subspace isomorphic to c0 ([310]). This space is thus nonseparable nonreﬂexive with unconditional basis and does not contain an isomorphic copy of c0 (Γ ) or 1 (Γ ) for uncountable Γ . Lindenstrauss proved that every separable Banach space with unconditional basis is isomorphic to a complemented subspace of a space with a symmetric basis ([197, p. 123]). Troyanski showed in [310] that c0 (Γ ) × 1 (Γ ) is not isomorphic to any subspace of a space with a symmetric basis. An example of the use of [310] in the separable theory is, e.g., in [140]. Acknowledgements I would like to thank Marián Fabian, Gilles Godefroy and Kamil John for their long term collaboration with me in nonseparable Banach spaces. This chapter was prepared when I held the position at the Mathematical Institute of the Czech Academy of Sciences in Prague during the years 1998–2001. I thank this institute for providing me with excellent working conditions that enabled me to work on this chapter. I am grateful to Marián Fabian, Gilles Godefroy, Petr Hájek, William Johnson, Ondˇrej Kalenda, Joram Lindenstrauss, Anibal Molto, Matthias Neufang, José Orihuela, Jan Pelant, Jan Rychtáˇr, Stanimir Troyanski, and Vicente Montesinos, who contributed by their help, advice and/or suggestions to this chapter. Above all, I am indebted to my wife Jarmila for her creating continuing excellent conditions for my life long work in Banach spaces.

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[291] D.G. Tacon, The conjugate of a smooth Banach space, Bull. Austral. Math. Soc. 2 (1970), 415–425. [292] M. Talagrand, Sur la structure borelienne des espaces analytiques, Bull. Sci. Math. 101 (1977), 415–422. [293] M. Talagrand, Comparison des boréliens d’un espace de Banach pour topologies faibles et fortes, Indiana Math. J. 27 (1978), 1001–1004. [294] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math. 119 (1979), 407–438. [295] M. Talagrand, Sur les espaces de Banach contenant 1 (τ ), Israel J. Math. 40 (1981), 324–330. [296] M. Talagrand, Pettis Integral and Measure Theory, Mem. Amer. Math. Soc. 307 (1984). [297] M. Talagrand, Renormages de quelques C(K), Israel J. Math. 54 (1986), 327–334. [298] W.K. Tang, On the extension of rotund norms, C.R. Acad. Sci. Paris Sér. I 323 (1996), 487–490. [299] W.K. Tang, Uniformly differentiable bump functions, Arch. Math. 68 (1997), 55–59. [300] P. Terenzi, Every norming M-basis of a separable Banach space has a block perturbation which is norming strong M-basis, Extracta Math. (1990), 161–169. [301] P. Terenzi, Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis, Studia Math. 111 (1994), 207–222. [302] P. Terenzi, A positive answer to the basis problem, Israel J. Math. 104 (1998), 51–124. [303] S. Todorˇcevi´c, Trees and linearly ordered sets, Handbook of Set Theoretic Topology, K. Kunen and J. Vaughan, eds, North-Holland (1984). [304] S. Todorˇcevi´c, Topics in Topology, Lecture Notes in Math. 1652, Springer, Berlin (1997). [305] S. Todorˇcevi´c, Compact subsets of the ﬁrst Baire class, J. Amer. Math. Soc. 12 (1999), 1179–1212. [306] N. Tomczak-Jaegermann, Banach–Mazur Distances and Finite-dimensional Operator Ideals, Pitman Monographs Surveys Pure Appl. Math. 38 (1989). [307] H. Torunczyk, Smooth partitions of unity on some nonseparable Banach spaces, Studia Math. 46 (1973), 43–51. [308] S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173–180. [309] S. Troyanski, On equivalent norms and minimal systems in nonseparable Banach spaces, Studia Math. 43 (1972), 125–138. [310] S. Troyanski, On nonseparable Banach spaces with a symmetric basis, Studia Math. 53 (1975), 253–263. [311] S. Troyanski, On uniform rotundity and smoothness in every direction in nonseparable Banach spaces with an unconditional basis, C.R. Acad. Bulgare Sci. 30 (1977), 1243–1246. [312] S. Troyanski, Locally uniformly convex norms, C.R. Acad. Bulgare Sci. 32 (1979), 1167–1169. [313] S. Troyanski, On a property of the norm which is close to local uniform convexity, Math. Ann. 271 (1985), 305–313. [314] S. Troyanski, Construction of equivalent norms for certain local characteristics with rotundity and smoothness by means of martingales, Proc. 14 Spring Conference of the Union of Bulgarian Mathematicians (1985), 129–156. [315] M. Valdivia, Some more results on weak compactness, J. Funct. Anal. 24 (1977), 1–10. [316] M. Valdivia, Resolutions of the identity in certain Banach spaces, Collect. Math. 39 (1988), 127–140. [317] M. Valdivia, Some properties of weakly countably determined Banach spaces, Studia Math. 93 (1989), 137–144. [318] M. Valdivia, Projective resolutions of identity in C(K) spaces, Arch. Math. 54 (1990), 493–498. [319] M. Valdivia, Simultaneous resolutions of the identity operator in normed spaces, Collect. Math. 42 (1991), 265–284. [320] M. Valdivia, On certain classes of Markushevich basis, Arch. Math. 62 (1994), 445–458. [321] M. Valdivia, On certain topological spaces, Revista Mat. 10 (1997), 81–84. [322] J. Vanderwerff, Smooth approximations in Banach spaces, Proc. Amer. Math. Soc. 115 (1992), 113–120. [323] J. Vanderwerff, Extensions of Markuševiˇc bases, Math. Z. 219 (1995), 21–30. [324] J. Vanderwerff, J.H.M. Whitﬁeld and V. Zizler, Markuševiˇc bases and Corson compacta in duality, Canad. J. Math. 46 (1994), 200–211. [325] L. Vašák, On a generalization of weakly compactly generated Banach spaces, Studia Math. 70 (1981), 11–19. [326] R. Vershynin, On constructions of strong a uniformly minimal M-bases in Banach spaces, Arch. Math. 74 (2000), 50–60.

1816 [327] [328] [329] [330]

V. Zizler

P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Stud. Adv. Math. 25 (1991). D. Yost, Asplund spaces for beginners, Acta Univ. Carolin. 34 (1993), 159–177. D. Yost, The Johnson–Lindenstrauss space, Extracta Math. 12 (1997), 185–192. V. Zizler, Locally uniformly rotund renorming and decomposition of Banach spaces, Bull. Austral. Math. Soc. 29 (1984), 259–265. [331] V. Zizler, Renormings concerning the Mazur intersection property of balls for weakly compact convex sets, Math. Ann. 276 (1986), 61–66. [332] V. Zizler, Smooth extension of norms and complementability of subspaces, Arch. Math. 53 (1989), 585– 589.

Addenda and Corrigenda to Chapter 7, Approximation Properties by Peter G. Casazza 1. p. 309, Proposition 8.8 contains an example of a Banach space with the approximation property (AP) but failing the bounded compact AP. The ﬁrst such example was due to Reinov [4a]. 2. p. 281, Theorem 2.5. Recently [1a] the classiﬁcation of Banach spaces with the approximation property has been extended to coverings of compact sets in X by operator ranges from a universal Banach space with quite strong properties. T HEOREM 0.1 [1a]. For a Banach space X the following properties are equivalent: (i) X has the AP. (ii) There exists a reﬂexive Banach space R with basis and with unconditional ﬁnitedimensional decomposition such that for each compact K ⊂ BX and for each ε > 0 there is a compact one-to-one operator T : R → X with T (BR ) ⊃ K and T 1 + ε. (iii) For any compact set K ⊂ X there is an M-basic sequence {xi } in X (with biorthogonal functionals {xi∗ }) such that x = xi∗ (x)xi for each x ∈ K. The key element in Theorem 0.1(ii) is that T is one-to-one. Indeed, every compact set in every Banach space may be covered by an operator range of ( n ⊕n1 )2 . 3. Several surprising results on the stochastic approximation property have just appeared [4a]. Given a Radon probability measure μ on a Banach space X, we say that X has the μ-approximation property (μ-AP, in short) provided there is a sequence {Bn } of ﬁnite-rank operators on X so that x − Bn x → 0 for μ almost every x in X. We say that X has the stochastic AP provided X has the μ-AP for every Radon probability measure μ on X. If X is separable, we say that X has the μ-basis property (μ-BP, in short) if there is an M-basis {xn , xn∗ } for X for which ∗ xn (x)xn = 1. μ x ∈ X: x = n

We say that X has the stochastic BP provided X has the μ-BP for every Radon probability measure μ on X. As we have seen in Chapter 7, there is a whole sequence of distinct properties for a Banach space which lie between the AP and the basis property. However, in [4a] it is shown that for any Radon probability measure μ on a separable Banach space X, X has 1817

1818

Addenda and corrigenda to Chapter 7

the μ-AP if and only if it has the μ-BP. Therefore, the stochastic AP and the stochastic BP are equivalent properties for separable Banach spaces. It is further shown in [4a] that there are Banach spaces failing stochastic AP. Finally, another strong result in [4a] is that stochastic AP implies AP for Banach spaces with non-trivial type. 4. In [3a] connections are made between the bounded approximation property (BAP, in short) and the non-linear theory of Banach spaces: T HEOREM 0.2 [3a]. If X and Y are Lipschitz-isomorphic Banach spaces such that X has the BAP then Y also has the BAP. There are no known examples of separable X and Y which are Lipschitz-isomorphic and not linearly isomorphic. However, Theorem 0.2 also applies to nonseparable Banach spaces where counterexamples are known. D EFINITION 0.3. Let X be a Banach space and λ 1. We say that X has the λ-Lipschitz bounded approximation property (λ-Lip BAP, in short) if for every compact set K ⊂ X and every ε > 0 there exists a Lipschitz map F : X → X with ﬁnite-dimensional range such that F Lip λ and F (x) − x ε for all x ∈ K. T HEOREM 0.4 [3a]. Let X be an arbitrary Banach space. The following are equivalent: (i) X has the λ-BAP. (ii) X has the λ-Lip BAP.

New references [1a] V.P. Fonf, W.B. Johnson, A.M. Plichko and V.V. Shevchyk, Covering a compact set in a Banach space by an operator range of a Banach space with basis, Trans. Amer. Math. Soc., to appear. [2a] V.P. Fonf, W.B. Johnson, G. Pisier and D. Preiss, Stochastic approximation properties in Banach spaces, Studia Math. (Special issue in honor of A. Pełczy´nski on the occasion of his seventieth birthday), submitted. [3a] G. Godefroy and N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. (Special issue in honor of A. Pełczy´nski on the occasion of his seventieth birthday), submitted. [4a] O.I. Reinov, How bad may be a Banach space with the approximation property?, Mat. Zametki 33 (6) (1983), 833–846 (in Russian).

Addenda and Corrigenda to Chapter 8, Local Operator Theory, Random Matrices and Banach Spaces by K.R. Davidson and S.J. Szarek 1. p. 346, the Added in proof section: (i) The paper [1a], which is a revised version of [105], has been circulated in the meantime; it contains additionally some concentration results for not-necessarily-extreme eigenvalues. (ii) More precise (but still presumably far from optimal) results in the same direction as [1a] were obtained in [7a]. 2. p. 346, inequality (4): a factor 1/L in the middle expression is missing. It should read

P(F M + t) 1 − Φ(t/L) < exp −t 2 /2L2 . 3. p. 349, Theorem 2.8: more quantitative results (i.e., estimates valid for any dimension rather than in the limit) were obtained √ in [2a] and [6a]. 4. p. 352, inequality (11): a factor n in the middle expression is missing. It should read

√

P(F 2 + σ t) < 1 − Φ t n < exp −nt 2 /2 . √ 5. p. 353, Theorem 2.13: a factor n in the middle expression in the second displayed formula is missing. It should read # #

max P s1 (Γ ) 1 + β + t , P sm (Γ ) 1 − β − t

√

< 1 − Φ t n < exp −nt 2 /2 . 6. p. 354, Problem 2.14: the existence of the limit was proved in [3a]. 7. p. 357, Problem 2.18: solved in the afﬁrmative in [4a]. 8. The book [5a], and particularly its Section 8.5, overlaps and complements the material presented in Section 2 of the chapter.

New references [1a] N. Alon, M. Krivelevich and V.H. Vu, On the concentration of eigenvalues of random symmetric matrices, Israel J. Math., to appear. [2a] G. Aubrun, A small deviation inequality for the largest eigenvalue of a random matrix, Preprint (2002). 1819

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[3a] F. Guerra and F.L. Toninelli, The thermodynamic limit in mean ﬁeld spin glass models, Comm. Math. Phys. 230 (1) (2002), 71–79. [4a] U. Haagerup and S. Thorbjørnsen, A new application of random matrices: Ext(Cr∗ (F2 )) is not a group, Preprint (2002), available at http://arXiv.org/abs/math.OA/0212265 [5a] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monographs 89, Amer. Math. Soc., Providence, RI (2001). [6a] M. Ledoux, A remark on hypercontractivity and tail estimates for the largest eigenvalues of random matrices, Preprint (2002). [7a] M. Meckes, Concentration of norms and eigenvalues of random matrices, Preprint (2002), available at http://arXiv.org/abs/math.PR/0211192

Addenda and Corrigenda to Chapter 11, Operator Ideals by J. Diestel, H. Jarchow and A. Pietsch The diagram on page 490 should be corrected as follows:

ν1

π2dual

α adj

π2

axis of symmetry

lift

(π1dual )ext

α

π1

dual dual π1 π1 π2 ◦ π2

dual π π2 2

λ1 λ2 λ∞

sur inj λ λ1 ∞

·

ν1 α

α dual ·

axis of symmetry

Since we are in the ﬁnite-dimensional setting the 1-nuclear norm ν 1 coincides with the 1-integral norm ι1 .

1821

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Addenda and Corrigenda to Chapter 15, Inﬁnite Dimensional Convexity by V.P. Fonf, J. Lindenstrauss and R.R. Phelps 1. p. 610, l. 15. In the mean time there appeared a revised (also somewhat expanded) version of [136]. The right reference at this point is [3a, Section 15]. 2. p. 641, Theorem 5.7 and p. 644, Theorem 5.14 ((1) ⇒ (4)). A more streamlined proof of these assertions appears in [2a]. This paper contains some other related results. For example, if X is separable and non-reﬂexive and its unit sphere is covered by a union ∗ of caps {Dn }∞ n=1 of radius a < 1 then for every sequence εn → 0 there is an f ∈ X with f = 1 and such that sup{f (x): x ∈ Dn } 1 − εn for every n. 3. p. 662, Proposition 7.8. In [36] it is only proved that any covering of a reﬂexive space by CCB sets cannot be locally ﬁnite. The stronger statement made in Proposition 7.8 is mentioned in [36] as a remark without proof. We do not know at present whether Proposition 7.8 is really true. 4. p. 663, l. 11. Erase the sentence starting with “Subsequent to this. . . ”, and replace it by the following: A survey of more recent results in this direction is given in [1a]. However, in spite of the many results mentioned in this reference, the problem of convexity of Chebyshev sets in Hilbert space is still open.

New references [1a] V.S. Balaganskii and L.P. Vlasov, The problem of convexity of Chebyshev sets, Russian Math. Surveys 51 (1996), 1127–1190. [2a] V.P. Fonf and J. Lindenstrauss, Boundaries and generation of convex sets, Israel J. Math., to appear. [3a] R.R. Phelps, Lectures on Choquet’s Theorem, 2nd ed., Lecture Notes in Math. 1757, Springer (2001).

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Author Index Roman numbers refer to pages on which the author (or his/her work) is mentioned. Italic numbers refer to reference pages. Numbers between brackets are the reference numbers. No distinction is made between ﬁrst and co-author(s).

Aarts, J. 1802, 1806 [1] Abramovich, Y.A. 87, 88, 90–94, 99–105, 108, 110, 111, 116, 117 [1–9]; 118 [10–21]; 535, 558 [1]; 1665, 1667 [1] Adams, R. 1363, 1365, 1398, 1399, 1403, 1404, 1419 [1] Aharoni, I. 829, 830 [1,2]; 906, 935 [1]; 1757, 1796, 1806 [2] Aizenman, M. 354, 360 [1] Akemann, C.A. 1461, 1510 [1] Akilov, G.P. 106, 107, 120 [74] Al-Husaini, A.L. 905, 935 [2] Albiac, F. 1119, 1127 [1,2] Aldous, D.J. 136, 156 [1]; 237, 265 [1]; 515, 526, 527 [1] Alencar, R. 812, 830 [3] Alesker, S. 714, 731–735, 772 [1–5] Alexander, H. 674, 675, 704 [1] Alexandrov, A.B. 1101, 1120, 1127 [3,4]; 1506, 1510 [2] Alexandrov, A.D. 727, 731, 772 [6,7] Alexandrov, A.G. 1800, 1806 [3] Alexopoulos, J. 515, 528 [2,3] Alfsen, E.M. 310, 313 [1]; 611–614, 620, 621, 627, 665 [1–3] Aliprantis, C.D. 21–24, 83 [1]; 87–90, 92–97, 99–105, 108, 110, 111, 113, 116, 117 [4–9]; 118 [10–14,22–29]; 535, 558 [1] Allekhverdiev, D.E. 453, 490 [1] Allen, G.D. 527, 528 [4] Allexandrov, G. 304, 313 [2] Alon, N. 358, 360 [2]; 770, 772 [8]; 1632, 1632 [1]; 1819, 1819 [1a] Alonso, J. 793, 830 [4] Alspach, D.E. 59, 83 [22]; 133, 135, 147, 151, 154, 156, 156 [2–8]; 279, 304, 313 [3,4]; 581, 595 [1]; 839, 862, 868, 868 [1]; 875, 896 [1]; 1019, 1038,

1051, 1052, 1064, 1065 [1,2]; 1351, 1359 [1]; 1560, 1569, 1580, 1585, 1594, 1598–1600, 1600 [1–7]; 1698, 1700 [1]; 1719, 1739 [1] Alster, K. 1754, 1762, 1806 [4,5] Altshuler, Z. 134, 156 [9]; 511, 527, 528 [5,6] Amann, H. 250, 265 [2] Ambrosio, L. 1523, 1544, 1544 [1,2] Amemiya, I. 89, 118 [30]; 440, 490 [2] Amir, D. 744, 745, 770, 772 [9,10]; 793, 820, 822, 830 [5,6]; 1572, 1577, 1596, 1600 [8]; 1632, 1632 [2,3]; 1714, 1717, 1730, 1739 [2–4]; 1745, 1748, 1755, 1762, 1763, 1806 [6,7] Anantharaman-Delaroche, C. 1431, 1455 [1] Andersen, N.T. 341, 360 [3] Anderson, J. 332–335, 360 [4–7]; 1461, 1510 [1] Andô, T. 110, 118 [31]; 130, 147, 156 [10]; 255, 265 [3]; 904, 935 [3] Androulakis, G. 1065 [3]; 1345, 1352, 1359 [2–4] Angelos, J.R. 320, 360 [8] Ansari, S. 545, 547, 558 [2] Ansel, J.P. 385, 390 [1] Antipa, A. 257, 265 [4] Aoki, T. 1101, 1102, 1127 [6] Apostol, C. 330, 332, 360 [9–11] Arai, H. 704, 704 [2] Araki, H. 1470, 1510 [3]; 1694, 1700 [2] Araujo, A. 1191, 1198 [1–3] Arazy, J. 1151, 1172 [1]; 1461, 1467, 1468, 1477–1479, 1507, 1510 [4–10]; 1511 [11] Archangel’skii, A.V. 1745, 1747, 1748, 1763, 1806 [8–10] Arenson, E.L. 87, 118 [15] Argyros, S. 139, 154, 157 [11,12]; 822, 830 [7]; 1019, 1038–1041, 1050, 1051, 1053, 1056, 1058, 1059, 1062–1065, 1065 [1]; 1066 [4–12]; 1081, 1096 [1]; 1253–1256, 1266, 1272, 1275, 1280, 1281, 1295 [1–3]; 1351, 1352, 1359 [1,5]; 1554, 1560, 1600 [9]; 1601 [10]; 1752, 1754, 1756, 1759, 1761,

1825

1826

Author Index

1763, 1767, 1774, 1775, 1781, 1784, 1800–1802, 1806 [11–14] Arias, A. 1440, 1454, 1455 [2,3] Arias-de-Reyna, J. 716, 772 [11]; 1607, 1632 [4] Arnold, L. 343, 360 [12] Aron, R.M. 555, 558 [3]; 676, 704 [3]; 812, 830 [3] Aronszajn, N. 535, 558 [4]; 1532, 1544 [3] Arvanitakis, A.D. 1554, 1600 [9] Arveson, W.B. 339, 340, 360 [13,14]; 535, 558 [5,6]; 1427, 1428, 1431, 1455 [4]; 1495–1497, 1511 [12,13] Ash, M. 215, 230 [1,2] Asimov, L. 615, 618, 620, 621, 626, 627, 665 [4] Asmar, N.H. 249, 265 [5,6]; 1371, 1419 [2] Asplund, E. 663, 665 [5]; 792, 798, 805, 828, 830 [8,9] Astashkin, S.V. 1155, 1172 [2] Aubin, J.P. 433 [1]; 798, 830 [10] Aubrun, G. 1819, 1819 [2a] Axler, S. 1700, 1701 [3] Azagra, D. 409, 422, 433 [2,3]; 799, 830 [11]; 1748, 1753, 1806 [15,16] Azoff, E. 328, 360 [15] Azuma, K. 1610, 1632 [5] Babenko, K.I. 1638, 1667 [2] Bachelier, L. 369, 390 [2] Bachelis, G.F. 204, 230 [3]; 884, 896 [2] Bachir, M. 411, 416, 418, 433 [4,5] Baernstein, A. 259, 265 [7]; 444, 490 [3] Baernstein II, A. 1608, 1632 [6] Bagaria, J. 1094, 1096 [2] Bai, Z.D. 344, 348, 353, 358, 360 [16–19]; 361 [20]; 366 [187] Baire, R. 1019, 1066 [13] Bakhtin, I.A. 88, 118 [32] Bakry, D. 350, 361 [21] Balaganskii, V.S. 1823, 1823 [1a] Ball, K.M. 163, 165, 171, 177, 183, 185, 187, 192, 193 [1–5]; 716, 718, 722, 724, 725, 728, 772, 772 [11,12]; 773 [13–18]; 901, 918, 935 [4,5]; 1223, 1244 [1]; 1481, 1511 [14]; 1607, 1632 [4] Banach, S. 273, 313 [5]; 444, 490 [4]; 524, 528 [7]; 1249, 1295 [4]; 1705, 1739 [5] Banaszczyk, W. 767, 773 [19] Bang, T. 183, 193 [6] Bañuelos, R. 259, 265 [8–10] Bapat, R.B. 98, 113, 118 [33] Barany, I. 175, 193 [7] Barles, G. 426, 431, 433 [6] Barthe, F. 164, 171, 173, 178, 193 [8,9]; 718, 773 [20]; 921, 935 [6] Bastero, J. 519, 528 [8,9]; 769, 773 [21]; 1103, 1106, 1127, 1127 [7–10]

Bates, S. 1539, 1544 [4] Baturov, D.P. 1747, 1806 [17] Bauer, H. 621, 665 [6] Beauzamy, B. 6, 7, 52, 57, 83 [2]; 444, 474, 479, 480, 490 [5,6]; 491 [7]; 549, 550, 555, 558 [3,7–9]; 792, 804, 830 [12]; 1136, 1137, 1172 [3]; 1340, 1352, 1359 [6] Beck, A. 1181, 1198 [4]; 1303, 1330 [1] Becker, R. 613, 665 [7] Beckner, W. 481, 491 [8] Behrends, E. 1079, 1096 [3]; 1749, 1806 [18] Bekollé, D. 1676, 1701 [4] Bell, M. 1796, 1806 [19] Bellenot, S. 1062, 1066 [14] Bellow, A. 260, 265 [11] Ben Arous, G. 345, 346, 361 [22] Benedek, A. 107, 118 [34] Benitez, C. 793, 830 [4,13] Benjamini, Y. 1392, 1419 [3] Bennett, C. 505, 515, 528 [10]; 1133, 1136, 1147, 1172 [4,5]; 1404, 1407, 1420 [4] Bennett, G. 147, 157 [13]; 230 [4]; 342, 361 [23]; 463, 491 [9,10]; 748, 773 [23]; 866, 867, 868 [2]; 962, 973 [1] Benveniste, E.J. 331, 359, 361 [24] Benyamini, Y. 35, 36, 38, 42, 48, 83 [3]; 279, 313 [6]; 342, 354, 361 [25]; 630, 634, 665 [8]; 766, 773 [22]; 822, 829, 830 [14,15]; 906, 935 [7]; 1087, 1096 [4]; 1158, 1172 [6]; 1251, 1260, 1295 [5]; 1310, 1330 [2]; 1348, 1359 [7]; 1521, 1540, 1544 [5]; 1560, 1569, 1594, 1596–1600, 1600 [5–7]; 1601 [11–14]; 1608, 1632 [7]; 1718, 1739 [6]; 1745, 1751, 1752, 1754–1756, 1780, 1797, 1805, 1806 [20–22] Bercovici, H. 341, 361 [26] Berg, I.D. 323–325, 328, 329, 331, 361 [27–29] Bergh, J. 74, 76–78, 80, 83 [4]; 505, 528 [11]; 577, 595 [2]; 692, 704 [4]; 1133, 1136, 1139, 1172 [7,8]; 1280, 1295 [6]; 1404, 1420 [5]; 1437, 1455 [5,6]; 1466, 1511 [15] Berkson, E. 237, 249, 250, 265 [5,12,13]; 1371, 1411–1413, 1419 [2]; 1420 [6]; 1494, 1511 [16] Berman, A. 98, 113, 118 [35] Berman, K. 336, 361 [30]; 859, 868 [3] Bernstein, A.J. 1657, 1667 [3] Bernstein, A.R. 535, 558 [10] Bernstein, S. 470, 491 [11] Bernués, J. 769, 773 [21]; 1062, 1066 [15]; 1127, 1127 [10] Besbes, M. 1700, 1701 [5] Besov, O.V. 1363, 1398, 1399, 1409, 1420 [7] Bessaga, C. 603, 648, 665 [9,10]; 792, 799, 830 [16,17]; 1029, 1045, 1066 [16]; 1249, 1295 [7];

Author Index 1560, 1571, 1601 [15]; 1745, 1759, 1796, 1800, 1806 [23–25] Bhatia, R. 327, 329, 347, 361 [31–36]; 1461, 1511 [17] Biane, P. 360, 361 [37]; 1482, 1491, 1511 [18,19] Billard, P. 569, 595 [3]; 1594, 1601 [16] Bishop, E. 608, 610, 640, 641, 665 [11–13]; 1733, 1739 [7] Björk, T. 374, 390 [3] Black, F. 369, 371, 390 [4] Blasco, O. 250, 265 [14,15]; 894, 896 [3]; 1682, 1701 [6] Blecher, D. 1427, 1430, 1432, 1435, 1438, 1442, 1444, 1445, 1454, 1455 [7–12]; 1456 [13–16] Blower, G. 264, 265 [16] Boas, R.P. 156, 157 [14]; 1678, 1701 [7] Bobkov, S.G. 350, 358, 361 [38]; 1623, 1632 [8] Bochner, S. 265 [17]; 480, 491 [12] Boˇckariov, S.V. 569, 575, 595 [4–6]; 1693, 1701 [8] Bogachev, V.I. 1528, 1532, 1544 [6–8] Bohnenblust, F. 140, 157 [15]; 465, 491 [13]; 1705, 1739 [8] Bolker, E.D. 525, 528 [12]; 768, 773 [24]; 911, 921, 924, 935 [8] Bollobás, B. 1077, 1078, 1096 [5] Boman, J. 1408, 1409, 1420 [8] Bombieri, E. 207, 230 [5]; 555, 558 [9] Bomze, I.M. 112, 119 [43] Bonami, A. 1481, 1511 [20]; 1676, 1701 [4] Bonic, R. 413, 433 [7]; 799, 830 [18] Bonsall, F.F. 608, 665 [14] Border, K.C. 87, 100, 118 [22] Borell, C. 346, 361 [39]; 717, 773 [25]; 1531, 1544 [9] Borovikov, V. 614, 665 [15] Borsuk, K. 1371, 1420 [9]; 1553, 1601 [17] Borwein, J.M. 396, 399, 418, 433 [8,9]; 664, 665 [16]; 798, 820, 830 [19]; 1521, 1538, 1544 [10,11]; 1749, 1806 [26,27] Bossard, B. 793, 805, 828, 830 [20–22]; 1020, 1042, 1044, 1066 [17–21]; 1787, 1806 [28] Bourbaki, N. 1733, 1739 [9] Bourgain, J. 139, 147, 150, 154, 157 [16–21]; 175, 193 [12]; 201, 204, 206, 209, 213, 214, 219–221, 223, 226, 229, 230, 230 [6–11]; 231 [12–14]; 237, 245, 247, 250, 265 [18–22]; 336, 337, 358, 359, 361 [40–44]; 445, 465, 468, 482, 491 [14–17]; 576, 580, 581, 590, 592, 595 [7–11]; 633, 634, 636, 658, 665 [17–20]; 675, 686, 687, 699, 703, 704, 704 [5–11]; 718, 724, 736, 742, 755, 759, 763, 766–772, 773 [26–40]; 805, 830 [23]; 844, 845, 853, 854, 859, 860, 862–867, 868 [4–13]; 884, 885, 891, 892, 895, 896, 896 [4,5]; 897 [6–8]; 918, 925, 927, 935 [9–12]; 1018–1020, 1039, 1046, 1064, 1066 [22–26]; 1158, 1172 [9]; 1221, 1231, 1235,

1827

1240, 1244 [2–4]; 1254, 1295 [8]; 1305, 1330 [3]; 1351, 1359 [8–10]; 1371, 1396, 1397, 1403, 1405, 1411–1413, 1419 [2]; 1420 [6,10–14]; 1494, 1502, 1511 [21,22]; 1581, 1583, 1587, 1589, 1595, 1599, 1601 [18–21]; 1630, 1633 [9]; 1646, 1647, 1649–1651, 1667, 1667 [4–6]; 1694, 1701 [9,10]; 1716, 1726, 1739 [10–12]; 1748, 1758, 1767, 1776, 1802, 1806 [29,30] Bourgin, R.D. 260, 265 [23]; 633, 634, 665 [21,22]; 793, 830 [24]; 1745, 1789, 1807 [31] Boutet de Monvel, A. 349, 361 [45] Bouziad, A. 1807 [32] Boyd, D.W. 514, 528 [13]; 1151, 1155, 1172 [10]; 1663, 1668 [7] Bo˙zejko, M. 1482, 1511 [23,24] Brascamp, H.J. 164, 193 [10]; 718, 773 [41] Bratteli, O. 620, 665 [23]; 1466, 1511 [25] Braverman, M.Sh. 523, 528 [14,15] Brenier, Y. 173, 193 [11]; 713, 773 [42] Bretagnolle, J. 140, 157 [22]; 524, 527, 528 [16,17]; 855, 868 [14]; 906, 935 [13] Bronk, B.V. 344, 361 [46] Brown, D.J. 113, 116, 118 [23,24] Brown, L.G. 323, 331, 361 [47,48]; 1477, 1511 [26] Brown, S.W. 341, 362 [49]; 543, 558 [11] Brudnyi, Yu.A. 74, 83 [5]; 1108, 1127 [11]; 1133, 1142, 1145, 1149, 1172 [11–13]; 1373, 1383, 1420 [15,16] Brunel, A. 1036, 1066 [27,28]; 1074, 1096 [6]; 1304, 1306, 1307, 1310, 1330 [4,5] Bu, S.Q. 139, 157 [23]; 263, 265 [24] Buchholz, A. 1487, 1511 [27–29] Bukhvalov, A.V. 87, 89, 96, 106, 108, 118 [37,38]; 119 [39–41]; 263, 265 [25–27]; 638, 665 [24]; 1484, 1511 [30] Burago, Y.D. 710, 712, 726, 774 [43] Burger, M. 935 [14] Bürger, R. 112, 119 [42,43] Burkholder, D.L. 10, 83 [23]; 126, 128, 157 [24,25]; 237, 238, 241–246, 248–251, 253, 254, 256–259, 262, 266 [28–44]; 477, 482, 491 [18–22]; 523, 528 [18]; 590, 595 [12]; 895, 897 [9]; 901, 906, 935 [15]; 1484, 1492, 1494, 1495, 1511 [31,32]; 1664, 1668 [8] Burkinshaw, O. 21–24, 83 [1]; 87–90, 92–97, 99–105, 108, 110, 111, 113, 117 [4–9]; 118 [10–13,23,25–29]; 535, 558 [1] Busemann, H. 177, 193 [13]; 918, 935 [16] Cabello Sanchez, F. 805, 830 [25] Caffarelli, L.A. 714, 774 [44] Calderón, A.P. 245, 266 [45,46]; 1139–1141, 1143, 1151, 1165, 1172 [14,15]; 1469, 1511 [33]

1828

Author Index

Calkin, J.W. 439, 491 [23] ˇ Canturija, Z.A. 579, 596 [18] Capon, M. 1048, 1066 [29] Caradus, S.R. 558 [12] Carathéodory, C. 712, 774 [45] Carl, B. 175, 193 [14]; 452, 463, 491 [24,25]; 854, 868 [15]; 958, 959, 962, 973, 973 [2–4]; 1305, 1330 [6] Carlen, E.A. 1461, 1481, 1482, 1491, 1511 [14,34–36] Carleson, L. 223, 231 [15] Carne, T.K. 1443, 1456 [17] Carothers, N.L. 135, 147, 156 [7]; 304, 313 [3]; 504, 505, 511, 523, 525–527, 528 [19–32]; 901, 935 [17] Carro, M.J. 1161, 1172 [16] Cartier, P. 610, 665 [25] Cartwright, D.I. 92, 119 [44]; 1715, 1739 [13] Casazza, P.G. 12, 14, 58, 60, 83 [24]; 133, 136, 140, 157 [26–28]; 276, 279, 285, 286, 291–294, 297–299, 302, 304, 309, 311, 313, 313 [7–18]; 511, 527, 528 [6]; 529 [33,34]; 812, 821, 830 [26,27]; 1063, 1065, 1066 [30,31]; 1094, 1096 [7]; 1157, 1172 [17]; 1255, 1295 [9]; 1418, 1420 [17]; 1646, 1647, 1649–1651, 1657, 1658, 1667 [6]; 1668 [9–12]; 1739, 1739 [14] Cascales, B. 1807 [33] Caselles, V. 110, 119 [45,46] Castillo, J.M.F. 1107, 1127 [12]; 1155, 1172 [18]; 1745, 1789, 1799, 1803, 1807 [34,35] Cauty, R. 1127, 1127 [13,14] ˇ Cech, E. 1753, 1807 [36] Cepedello, M. 813, 828, 830 [28,29]; 831 [30,31]; 1807 [37] Cerda, J. 1161, 1172 [16] Ceretelli, O.D. 1170, 1172 [19] Chaatit, F. 1019, 1036, 1045, 1046, 1066 [32,33]; 1067 [34]; 1348, 1359 [11] Chang, K.-C. 433 [10] Chatterji, S.D. 260, 266 [47,48] Chen, S.T. 515, 529 [35–37] Chen, Z.L. 97, 119 [47] Cheridito, P. 383, 390 [5] Chevet, S. 354, 362 [50]; 461, 491 [26] Chilin, V.I. 1477, 1478, 1484, 1511 [37–40]; 1512 [41] Cho, C. 310, 313 [19] Choi, C. 259, 266 [49,50] Choi, M.D. 101, 119 [48]; 323, 362 [51]; 1431, 1455, 1456 [18,19] Choquet, G. 605, 608, 611, 613, 614, 629, 665 [26–33]; 793, 831 [32]; 1025, 1047, 1048, 1067 [35] Choulli, M. 433 [11] Choulli, T. 389, 390 [6,7] Christensen, E. 340, 362 [52]; 1431, 1440–1442, 1444, 1445, 1456 [20,21]

Christensen, J.P.R. 624, 666 [34]; 1018, 1019, 1048, 1067 [36]; 1531, 1544 [12] Ciesielski, K. 1758, 1807 [38] Ciesielski, Z. 575, 583, 585–587, 595 [13–17]; 973, 973 [5] Clarke, F. 412, 423, 425, 433 [12,13]; 434 [14–16] Clarkson, J.A. 128, 157 [29]; 484, 491 [27]; 1775, 1807 [39] Clément, P. 245, 266 [51] ˇ Coban, M.M. 1752, 1807 [40] Cobos, F. 266 [52]; 1471, 1472, 1512 [42] Coifman, R.R. 1120, 1128 [15]; 1158, 1172 [20]; 1406, 1420 [18]; 1682, 1691–1693, 1696, 1701 [11,12] Cole, B.J. 676, 682, 704 [3]; 705 [12] Colin de Verdiere, Y. 1456 [22] Connes, A. 1452, 1456 [23] Connes, B. 215, 231 [16] Contreras, M.D. 642, 666 [35] Cooke, R. 214, 231 [17] Corson, H.H. 624, 629, 662, 666 [32,36,37]; 1747, 1748, 1750, 1751, 1756, 1757, 1760, 1775, 1800, 1803, 1805, 1807 [41–43] Corvellec, J.-N. 425, 434 [17] Coulhon, T. 250, 266 [53]; 1363, 1423 [95] Cowen, C.C. 320, 360 [8]; 471, 491 [28,29] Crandall, M.G. 420, 421, 426, 431, 434 [18–20] Creekmore, J. 505, 529 [38] Csörnyei, M. 1532, 1544 [13] Cuartero, B. 1116, 1117, 1124, 1128 [16] Cuculescu, I. 1493, 1512 [43,44] Cwikel, M. 505, 529 [39]; 1140–1143, 1145, 1147, 1149, 1151, 1153, 1158, 1159, 1161, 1172 [1,20]; 1173 [21–33] Dacunha-Castelle, D. 140, 157 [22,30]; 455, 491 [30]; 524, 527, 528 [16,17]; 855, 868 [14]; 906, 935 [13]; 1304, 1307, 1330 [7] Dadarlat, M. 332, 362 [53] Dalang, R.C. 374, 390 [8] Dancer, E.N. 112, 119 [49] Danilevich, A.A. 263, 265 [27]; 638, 665 [24]; 1484, 1511 [30] Dar, S. 714, 724, 731, 732, 763, 772 [5]; 774 [46–48] Dashiell, F.K. 1775, 1794, 1807 [44] Daubechies, I. 566, 578, 596 [19] David, G. 250, 266 [54]; 591, 596 [20]; 702, 705 [13] Davidson, K.R. 320, 323–325, 328, 331–333, 339–341, 361 [29]; 362 [54–67]; 859, 868 [16]; 969, 973 [6] Davie, A.M. 283, 313 [20,21]; 673, 705 [14] Davies, R.O. 1529, 1544 [14] Davis, B.J. 258, 266 [43]; 523, 528 [18]; 1170, 1173 [34]

Author Index Davis, C. 327–329, 339, 347, 360 [15]; 361 [34,35]; 362 [68] Davis, W.B. 1020, 1067 [37] Davis, W.J. 96, 119 [50]; 265 [21]; 266 [55]; 443, 491 [31]; 646, 666 [38]; 743, 766, 774 [49]; 792, 793, 821, 831 [33–35]; 850, 868 [17]; 1103, 1128 [17]; 1133, 1136, 1137, 1173 [35]; 1254, 1280, 1295 [10]; 1483, 1512 [45]; 1643, 1668 [13]; 1761, 1766, 1803, 1804, 1807 [45,46] Day, M.M. 792, 798, 821, 831 [36,37]; 1101, 1102, 1128 [18]; 1396, 1420 [19]; 1774, 1775, 1807 [47] de Acosta, A. 1191, 1198 [1] de Branges, L. 105, 118 [36]; 541, 558 [13]; 603, 666 [39] de Figueiredo, D.G. 434 [21] de Leeuw, K. 184, 193 [15]; 608, 610, 665 [12]; 965, 974 [19]; 1700, 1701 [28] de Pagter, B. 110, 111, 121 [106]; 245, 266 [51]; 1465, 1467, 1468, 1477, 1478, 1512 [51–56] de Valk, V. 515, 529 [50] Debs, G. 793, 831 [38]; 1019, 1067 [38] Deddens, J.A. 339, 362 [69] Defant, A. 466, 467, 484, 488, 490, 491 [32]; 1513 [98] Defant, M. 590, 596 [21] Degiovanni, M. 425, 434 [17] Deift, P. 349, 362 [70] Delbaen, F. 374, 377–386, 389, 390 [9–16]; 465, 491 [16]; 658, 665 [18]; 901, 905, 911–915, 935 [18]; 1599, 1601 [20]; 1674, 1701 [13]; 1726, 1739 [11] Deliyanni, I. 1062, 1063, 1065, 1066 [5–7,15]; 1253, 1255, 1256, 1275, 1295 [1]; 1352, 1359 [5] Dellacherie, C. 374–376, 390 [17] Deville, R. 33, 34, 57, 83 [6,25]; 406, 408, 409, 413, 415, 418–422, 433 [2,3,11]; 434 [22–31]; 476, 491 [33]; 644, 659, 666 [40,41]; 792, 793, 795, 798, 799, 805, 812–814, 821, 831 [39–47]; 1745, 1747–1750, 1752–1755, 1757–1767, 1772–1778, 1781–1792, 1794–1799, 1801, 1804, 1806 [15]; 1807 [48–59] Devinatz, A. 1495–1497, 1512 [46] DeVore, R.A. 575, 596 [22]; 1404, 1420 [20,21] Diestel, J. 13, 18, 35, 36, 38–40, 47, 55, 60, 65, 66, 72, 83 [7–9]; 106, 107, 119 [51]; 259, 260, 263, 266 [56]; 445, 455, 458, 459, 464, 466, 471, 474, 479, 480, 483, 484, 491 [17,34–37]; 518, 529 [40]; 558 [14]; 675, 681, 690, 705 [15]; 792, 793, 806, 831 [48–50]; 867, 868 [18]; 879, 881, 882, 887, 897 [10]; 929, 935 [19]; 943, 950, 951, 974 [7]; 1222, 1244 [5]; 1378, 1387–1389, 1396, 1420 [22,23]; 1452, 1456 [24]; 1563, 1601 [22]; 1710, 1711, 1724, 1733, 1736, 1739 [15,16]; 1745, 1748, 1749, 1762, 1770, 1803, 1808 [60,61]

1829

Dilworth, S.J. 504, 511, 518, 519, 523–527, 528 [22–29]; 529 [41–49]; 769, 770, 774 [50,55]; 906, 935 [20]; 1103, 1126, 1128 [19,20]; 1142, 1173 [36]; 1700, 1701 [5] Dineen, S. 676, 705 [16]; 812, 830 [3] Dinger, U. 1530, 1544 [15] Ditor, S. 1552, 1554, 1556, 1593, 1601 [23,24] Dixmier, J. 1388, 1420 [24]; 1461, 1463, 1466, 1480, 1512 [47,48] Dixon, A.C. 451, 491 [38] Dixon, P.G. 286, 310, 313 [22] Dmitrovski˘ı, V.A. 347, 362 [71] Dobrowolski, T. 799, 830 [11]; 1753, 1806 [16] Dodds, P.G. 94, 119 [52]; 1465, 1467, 1468, 1477, 1478, 1480, 1511 [37,38]; 1512 [49–58] Dodds, T.K. 1465, 1467, 1468, 1477, 1478, 1480, 1512 [49–57] Doléans-Dade, C. 389, 390 [18] Domenig, T. 471, 492 [39] Doob, J.L. 260, 267 [57] Dor, L.E. 131, 147, 155, 157 [13,31,32]; 230 [4]; 255, 256, 267 [58]; 565, 596 [23]; 748, 768, 773 [23]; 774 [51]; 862, 866, 867, 868 [2,19]; 906, 935 [21]; 1652, 1668 [14] Dore, G. 250, 267 [59–62] Douglas, R.G. 323, 331, 361 [47,48] Doust, I. 256, 267 [63] Dow, A. 1808 [62] Dowling, P.M. 1477, 1512 [50] Dowling, P.N. 263, 267 [64,65]; 1700, 1701 [5] Downarowicz, T. 617, 666 [43] Dragilev, M.M. 1667, 1668 [15–17] Drewnowski, L. 1105, 1128 [21] Driouich, A. 267 [66] Drnovšek, R. 102, 119 [53] Duan, Y. 515, 529 [36] Dubinsky, E. 473, 492 [40]; 1302, 1303, 1330 [8]; 1667, 1668 [18] Dudley, R.M. 1188, 1190, 1193, 1194, 1198 [5,6]; 1625, 1633 [10] Dufﬁe, D. 374, 390 [19] Dugundji, J. 603, 666 [44]; 1756, 1808 [63] Dunford, N. 106, 107, 119 [54,55]; 439, 444, 492 [41,42]; 1366, 1388, 1395, 1420 [25]; 1579, 1601 [25] Dupire, B. 371, 390 [20] Duren, P.L. 675, 694, 705 [17]; 1102, 1104, 1113, 1120, 1128 [22,23]; 1677, 1681, 1701 [14] Durier, R. 650, 655, 666 [45] Durrett, R. 5, 6, 83 [10] Dvoretzky, A. 136, 157 [33]; 458, 475, 492 [43,44]; 720, 735, 774 [52–54]; 1303, 1315, 1330 [9–11]; 1606, 1625, 1626, 1633 [11,12]

1830

Author Index

Dykema, K.J. 345, 357, 358, 362 [72]; 366 [178]; 1171, 1173 [37,38]; 1445, 1458 [103]; 1466, 1487, 1516 [196] Dynkin, E.B. 616, 666 [42] Eaton, M. 910, 935 [22] Ebenstein, S.E. 204, 230 [3]; 884, 896 [2] Èdelšte˘ın, I.S. 149, 157 [34]; 1644, 1645, 1668 [19] Edelstein, M. 664, 666 [46] Edgar, G.A. 260, 263, 267 [65,67–69]; 631, 633, 636, 639, 665 [22]; 666 [47–49]; 1484, 1512 [59]; 1745, 1758, 1764, 1766, 1789–1791, 1802, 1803, 1808 [64–67] Edmunds, D.E. 969, 974 [8] Edwards, D.A. 607, 615, 620, 666 [50–52] Edwards, R.E. 538, 558 [15] Effros, E.G. 310, 313 [1]; 621, 622, 626, 666 [53,54]; 1427, 1431, 1432, 1434, 1435, 1438–1447, 1452, 1454, 1455, 1456 [18–20,25–43]; 1510, 1512 [60] Egghe, L. 260, 267 [70] Einstein, A. 369, 390 [21–23] Ekeland, I. 395, 409, 434 [32,33]; 798, 830 [10]; 831 [51] El Haddad, E.M. 418, 420, 421, 434 [27,28] El Karoui, N. 382, 390 [24] El-Gebeily, M.A. 156, 157 [35] El-Mennaoui, O. 267 [66] Ellentuck, E. 1049, 1067 [39]; 1077, 1096 [8] Ellis, A.J. 615, 618, 620, 621, 626, 627, 665 [4] Elton, J. 1046, 1049, 1050, 1056, 1064, 1067 [40] Emery, M. 378, 379, 390 [25] Enﬂo, P. 98, 119 [56]; 129, 133, 134, 156 [2]; 157 [36,37]; 235, 237, 267 [71]; 273, 279, 280, 283, 285, 313 [4,23]; 446, 451, 479, 489, 492 [45,46]; 543, 545, 547, 549, 550, 555, 558 [2,3,9,16–19]; 766, 772, 774 [56,57]; 804, 805, 829, 831 [52–54]; 1115, 1128 [24]; 1157, 1173 [39]; 1304, 1330 [12]; 1348, 1359 [12]; 1805, 1808 [68] Engelking, R. 1756, 1793, 1802, 1808 [69] Engle, P. 1397, 1420 [26] Erdös, P. 359, 362 [73]; 444, 492 [47]; 1058, 1067 [41] Evans, L. 1367, 1420 [27] Evans, W.D. 969, 974 [8] Exel, R. 326, 362 [74]; 1495, 1512 [61] Fabes, E.B. 1411, 1420 [28] Fabian, M. 413, 418, 434 [34–37]; 792, 793, 798, 812, 814, 820–822, 828, 831 [41,55–59]; 832 [60–65]; 1537, 1544 [16]; 1745, 1747–1759, 1761–1769, 1771, 1772, 1774–1782, 1785–1789, 1792, 1795, 1797–1799, 1801–1805, 1806 [26]; 1808 [70–91]; 1809 [92]

Fack, T. 1464, 1469, 1480, 1481, 1512 [62–64] Fakhoury, H. 445, 492 [48] Falconer, K. 220, 231 [18] Fang, G. 425, 434 [38–41]; 435 [42] Farahat, J. 1079, 1096 [9] Faris, W.G. 1403, 1420 [29] Farmaki, V. 822, 830 [7]; 1019, 1035, 1038, 1041, 1042, 1045, 1046, 1067 [42–48]; 1754, 1766, 1780, 1781, 1806 [12]; 1809 [93] Feder, M. 304, 313 [24] Federer, H. 1363, 1401, 1420 [30]; 1522, 1523, 1544 [17] Fefferman, C. 220, 223, 231 [19,20]; 267 [72] Feige, U. 1610, 1633 [13] Feldman, J. 535, 558 [5]; 615, 666 [55] Fell, J.M. 610, 665 [25] Feller, W. 606, 617, 666 [56]; 804, 832 [66] Felouzis, V. 1064, 1066 [8]; 1254, 1266, 1272, 1280, 1281, 1295 [2] Fenchel, W. 727, 774 [58] Ferenczi, V. 1252–1254, 1265, 1266, 1295 [11–15]; 1729, 1739 [17] Ferguson, T.S. 906, 935 [23] Ferleger, S.V. 1492, 1512 [65,66] Fernandez, D.L. 250, 267 [73] Fernique, X. 347, 350, 362 [75,76] Fetter, H. 1758, 1809 [94] Fichtenholz, G.M. 1373, 1420 [31] Figá-Talamanca, A. 1466, 1512 [67] Figiel, T. 96, 119 [50,57]; 208, 231 [21]; 249, 267 [74,75]; 276, 283, 289, 309, 313 [25–28]; 443, 475, 491 [31]; 492 [49]; 511, 529 [51]; 583, 585–587, 591–594, 595 [15–17]; 596 [24,25]; 715, 716, 726, 735, 737, 747, 748, 751, 768, 770, 772, 773 [36]; 774 [59–61]; 792, 832 [67]; 853, 854, 858, 863, 868 [20]; 869 [21–23]; 916, 925, 935 [24]; 973, 973 [5]; 1060, 1062, 1067 [49]; 1133, 1136, 1137, 1171, 1173 [35,37]; 1222, 1233, 1234, 1244 [6,7]; 1254–1256, 1280, 1295 [10,16,17]; 1305, 1306, 1330 [13,14]; 1342, 1359 [13]; 1387, 1420 [32]; 1589, 1601 [26]; 1608, 1633 [14]; 1649, 1664, 1668 [20,21]; 1691, 1692, 1697, 1701 [15]; 1761, 1766, 1803, 1807 [46] Fillmore, P.A. 323, 331, 339, 361 [47,48]; 362 [69] Finet, C. 434 [26]; 805, 832 [68]; 1020, 1067 [50]; 1757, 1758, 1767, 1809 [95–97] Fleming, R.J. 901, 935 [25] Flinn, P. 279, 313 [6]; 314 [29]; 511, 528 [30] Floret, K. 466, 467, 484, 488, 490, 491 [32] Foia¸s, C. 330, 332, 360 [9,10] Föllmer, H. 387, 388, 390 [26,27] Fonf, V.P. 34, 35, 83 [26]; 611, 637, 641, 644, 646–650, 653–659, 661–665, 666 [40,57–62]; 667 [63–71]; 668 [96]; 793, 813, 831 [42,43];

Author Index 832 [69]; 1019, 1046, 1067 [51,52]; 1747, 1749, 1795, 1798, 1799, 1807 [51,52]; 1809 [98,99]; 1817, 1818 [1a,2a]; 1823, 1823 [2a] Force, G. 147, 151, 154, 157 [38] Forrester, P.J. 345, 362 [77] Fosgerau, M. 1752, 1804, 1809 [100] Fourie, J. 484, 491 [34] Fradelizi, M. 921, 935 [6] Frampton, J. 413, 433 [7]; 799, 830 [18] Frankiewicz, R. 1805, 1809 [101] Frankl, P. 1613, 1625, 1633 [15] Franklin, Ph. 575, 596 [26] Frankowska, H. 433 [1] Frazier, M. 1141, 1173 [40] Fréchet, M. 906, 935 [26] Fremlin, D.H. 89, 92, 94, 119 [52,58,59]; 1019, 1020, 1066 [25]; 1117, 1128 [25]; 1748, 1758, 1767, 1802, 1806 [30] Friedman, Y. 1509, 1510 [9]; 1512 [68]; 1600, 1601 [27] Friis, P. 325, 362 [78]; 363 [79] Frobenius, G. 98, 119 [60] Frolík, Z. 1802, 1809 [102] Frontisi, J. 828, 832 [70]; 1798, 1809 [103,104] Fry, R. 813, 828, 832 [71] Fuhr, R. 626, 627, 667 [72] Füredi, Z. 175, 193 [7]; 1613, 1625, 1633 [15] Gagliardo, E. 1381, 1398, 1420 [33,34] Gamboa de Buen, B. 1758, 1809 [94] Gamelin, T.W. 60, 84 [27]; 675–677, 682, 704 [3]; 705 [12,18–20]; 879, 895, 897 [11]; 1675, 1701 [16] Gamlen, J.L.B. 129, 157 [39] Gantmacher, V.R. 443, 492 [50] García, C.L. 285, 313 [10] García del Amo, A. 527, 529 [52] Garcia-Cuerva, J. 505, 529 [53] Gardner, R.J. 177, 193 [16–18]; 918, 919, 935 [27–29] Garling, D.J.H. 137, 139, 157 [40]; 246, 263, 264, 266 [55]; 267 [76–78]; 463, 492 [51]; 516, 519, 527, 529 [54,55]; 681, 705 [21]; 793, 831 [33]; 905, 936 [30]; 952, 974 [9]; 1103, 1128 [17]; 1142, 1173 [41]; 1483, 1484, 1494, 1495, 1512 [45,69,70] Garnett, J.B. 674, 675, 705 [22]; 1638, 1668 [22]; 1693, 1701 [17] Garsia, A.M. 209, 231 [22]; 1490, 1512 [71] Gasparis, I. 1051, 1056, 1059, 1064, 1065, 1066 [9]; 1067 [53,54]; 1599, 1601 [28] Gaudet, R.J. 129, 157 [39] Gaudin, M. 345, 364 [122] Geiss, S. 267 [79] Gelfand, I.M. 902, 908, 919, 920, 936 [31,32]

1831

Geman, S. 344, 353, 363 [80] Georgiev, P. 828, 832 [72] Gevorkian, G.G. 565, 573, 596 [27,28] Geyler, V.A. 87, 89, 90, 118 [16]; 119 [40] Ghoussoub, N. 34, 83 [25]; 97, 119 [61–63]; 263, 267 [80–82]; 402–405, 425, 434 [41]; 435 [42–50]; 445, 483, 492 [53]; 635, 637–639, 667 [73–75]; 795, 798, 799, 821, 831 [34,44]; 832 [73]; 1019, 1067 [55]; 1589, 1598, 1601 [26,29]; 1752, 1753, 1775, 1783, 1807 [53] Giannopoulos, A.A. 47, 84 [28]; 164, 169, 177, 180, 193 [19–22]; 342, 358, 363 [81]; 719, 722, 725, 726, 729, 737, 754, 755, 766, 774 [62–70]; 844, 859, 867, 869 [24]; 918, 936 [33]; 1203, 1221, 1224, 1225, 1244 [8]; 1626, 1627, 1633 [16] Giesy, D.P. 1303, 1330 [15,16] Giga, M. 250, 267 [84] Giga, Y. 250, 267 [83,84] Giles, J.R. 1745, 1809 [105] Gillespie, T.A. 237, 249, 250, 265 [5,12,13]; 1133, 1162, 1173 [42]; 1347, 1359 [14]; 1494, 1511 [16] Gillman, L. 1712, 1739 [18] Giné, E. 1181, 1186, 1190, 1191, 1194, 1195, 1198 [1–3,7,8]; 1199 [9,10] Ginibre, J. 219, 231 [23] Girardi, M. 443, 492 [52]; 518, 529 [44] Girko, V.L. 344, 363 [82] Glasner, E. 619, 620, 667 [76] Gleit, A. 658, 667 [77] Glimm, J. 542, 558 [20] Gluskin, E.D. 175, 193 [23]; 454, 492 [54]; 743, 765, 766, 774 [71]; 775 [72–74]; 854, 869 [25]; 946, 968, 974 [10]; 1208, 1212, 1222, 1224, 1244 [9,10]; 1245 [11]; 1254, 1296 [18,19] Godefroy, G. 33, 34, 57, 83 [6]; 84 [29]; 97, 119 [64]; 154, 156, 157 [11,41]; 235, 268 [85]; 285, 295, 310, 314 [30,31]; 406, 408, 409, 415, 434 [24,25]; 435 [51]; 476, 491 [33]; 644, 645, 666 [41]; 667 [78]; 792, 793, 798, 799, 805, 812–814, 821, 822, 828, 829, 830 [22]; 831 [38,45,46]; 832 [60,73–84]; 1016, 1018–1020, 1046, 1067 [50,56–61]; 1081, 1096 [1]; 1112, 1128 [26]; 1560, 1601 [10]; 1745, 1747–1750, 1752–1755, 1757–1767, 1772–1778, 1780–1792, 1795–1802, 1804, 1805, 1806 [28]; 1807 [54–58]; 1808 [74–77]; 1809 [96,106–122]; 1818, 1818 [3a] Godement, R. 616, 667 [79] Godun, B.V. 1767, 1804, 1809 [123]; 1810 [124] Goethals, J.P. 916, 936 [34] Gohberg, I.C. 439, 492 [55]; 1464, 1465, 1496, 1498, 1512 [72]; 1513 [73] Gol’dstein, V.M. 1372, 1420 [35] Gonzalez, M. 812, 832 [85]; 1107, 1127 [12]; 1155, 1172 [18]; 1745, 1789, 1799, 1803, 1807 [34,35]

1832

Author Index

Gonzalo, R. 413, 434 [30]; 812, 813, 831 [47]; 832 [85] Goodey, P. 911, 936 [35] Goodman, V. 147, 157 [13]; 230 [4]; 342, 361 [23]; 463, 491 [10]; 748, 773 [23]; 866, 867, 868 [2]; 1185, 1190, 1199 [11] Goodner, D.A. 1712, 1739 [19] Gordon, Y. 278, 314 [32]; 342, 352, 354, 361 [25]; 363 [83–85]; 454, 463, 466, 492 [51,54,56]; 735, 740, 749, 750, 759, 766, 769, 773 [22]; 775 [75–79]; 858, 869 [26]; 896, 897 [12]; 925, 936 [36]; 946, 952, 968, 974 [9,10]; 1127, 1128 [27]; 1222, 1245 [12]; 1387–1389, 1420 [36]; 1478, 1507, 1513 [74] Gorelik, E. 829, 832 [86] Gorin, E.A. 905, 936 [37] Götze, F. 350, 358, 361 [38] Goullet de Rugy, M. 622, 667 [80] Gowers, W.T. 18, 84 [30]; 136, 157 [42]; 222, 231 [24]; 277, 304, 310, 314 [33]; 770, 775 [80,81]; 812, 813, 832 [87,88]; 1016, 1036, 1051, 1064, 1067 [62,63]; 1087, 1089, 1094–1096, 1096 [10,11]; 1097 [12]; 1101, 1106, 1110, 1128 [28,29]; 1158, 1173 [43]; 1250–1255, 1260, 1261, 1263, 1265–1268, 1271, 1273, 1274, 1276, 1283, 1288, 1296 [20–27]; 1310, 1330 [17]; 1344, 1349, 1359 [15,16]; 1632, 1633 [17,18]; 1656, 1657, 1668 [23] Graham, C.C. 874, 897 [13] Graham, R.L. 1074, 1097 [13] Granas, A. 603, 666 [44] Grandits, P. 389, 390 [28] Granville, A. 207, 230 [5] Greenleaf, F.P. 876, 897 [14] Gripenberg, G. 573, 596 [29] Gripey, R. 1367, 1420 [27] Grobler, J.J. 110, 111, 119 [65,66] Gromov, M. 347, 363 [86]; 714, 717, 744, 745, 775 [82–86]; 1606, 1607, 1633 [19–21] Gronbaek, N. 286, 314 [34] Gross, L. 1301, 1330 [18,19]; 1481, 1482, 1513 [75,76] Grothendieck, A. 273, 281, 282, 288, 289, 309, 314 [35]; 440, 444, 449–452, 457, 459, 466, 467, 483, 486, 489, 490, 492 [57–62]; 911, 936 [38]; 964, 974 [11]; 1021, 1044, 1067 [64]; 1301, 1302, 1330 [20]; 1378, 1392, 1420 [37]; 1421 [38]; 1432, 1438, 1439, 1443, 1446, 1457 [44]; 1464, 1513 [77]; 1581, 1601 [30]; 1723, 1724, 1740 [20,21] Gruenhage, G. 1810 [125] Grünbaum, B. 930, 936 [39] Grz¸as´lewicz, R. 515, 529 [56] Grzech, M. 1805, 1809 [101]

Guédon, O. 718, 749, 775 [78,87,88] Guerra, F. 1819, 1820 [3] Guerre-Delabrière, S. 134–136, 139, 158 [43,44]; 519, 529 [57]; 813, 833 [89]; 1745, 1810 [126] Guionnet, A. 345, 346, 361 [22]; 363 [87] Gul’ko, S.P. 1755, 1762, 1810 [127] Gundy, R.F. 128, 157 [25]; 242, 257, 258, 266 [42–44]; 268 [86]; 523, 528 [18]; 880, 897 [15] Gurarii, P.I. 581, 596 [30] Gurarii, V.I. 581, 596 [30]; 626, 667 [81]; 1771, 1810 [128] Gutiérrez, J.A. 237, 268 [87]; 676, 705 [23] Gutman, A.E. 87, 106, 108, 119 [41] Haagerup, U. 358, 363 [88,89]; 1305, 1330 [21]; 1428, 1431, 1440, 1449, 1452, 1456 [25]; 1457 [45–47]; 1470, 1474, 1476, 1480, 1483, 1484, 1487, 1495, 1506–1509, 1513 [78–82]; 1819, 1820 [4a] Habala, P. 6, 7, 20, 34, 36, 41, 42, 63, 83 [11,12]; 792, 832 [62]; 833 [90]; 1253, 1269, 1295 [15]; 1296 [28]; 1537, 1544 [18]; 1745, 1747–1758, 1761–1766, 1768, 1769, 1771, 1772, 1774, 1775, 1777–1780, 1782, 1785, 1787–1789, 1792, 1797, 1802, 1804, 1805, 1808 [80]; 1810 [129] Hadwiger, H. 733, 775 [89] Hadwin, D. 329, 363 [90] Hagler, J. 1569, 1595, 1601 [31,32]; 1752, 1810 [130] Hahn, H. 1705, 1740 [22] Hájek, P. 6, 7, 20, 34, 36, 41, 42, 63, 83 [11,12]; 434 [36]; 659, 666 [40]; 667 [85]; 792, 799, 813, 821, 828, 831 [31,42,43]; 832 [61,62]; 833 [90–92]; 1537, 1544 [18]; 1745, 1747–1758, 1761–1766, 1768, 1769, 1771, 1772, 1774–1782, 1785, 1787–1789, 1792, 1793, 1795–1799, 1802, 1804, 1805, 1807 [37,51,52]; 1808 [77–80]; 1810 [129,131–140] Hajłasz, P. 1399, 1403, 1421 [39] Halbeisen, L. 1358, 1359 [17] Halberstam, H. 200, 231 [25] Halmos, P.R. 106, 119 [67]; 535, 558 [21] Halperin, I. 505, 526, 529 [58] Halpern, H. 336, 361 [30]; 859, 868 [3] Hamana, M. 1453, 1457 [48,49] Hammand, P. 1046, 1067 [65] Hansell, R.W. 1745, 1791, 1792, 1810 [141] Harcharras, A. 1501–1505, 1513 [83,84] Hardin, C.D., Jr. 901, 902, 905, 936 [40–42] Hardy, G.H. 529 [59]; 956, 974 [12]; 1173 [44] Hare, D. 799, 831 [45] Harmand, P. 310, 314 [36]; 829, 833 [93]; 1745, 1810 [142] Harper, L.H. 745, 775 [90]; 1613, 1625, 1633 [22]; 1657, 1668 [24]

Author Index Harrison, J.M. 374, 376, 377, 391 [29,30] Hart, S. 1657, 1668 [25] Hasumi, M. 1714, 1740 [23] Hayakawa, K. 1142, 1173 [45] Haydon, R.G. 140, 158 [45]; 408, 409, 435 [52–54]; 526, 528 [32]; 611, 615, 644, 667 [82–84]; 799, 813, 833 [94,95]; 901, 935 [17]; 1019, 1030, 1034, 1036, 1045, 1046, 1068 [66]; 1339, 1359 [18]; 1715, 1740 [24]; 1776, 1777, 1784–1787, 1789, 1791, 1794–1796, 1798, 1799, 1810 [139,143–147] Heath-Brown, D.R. 229, 231 [26] Heinrich, S. 305–308, 314 [37]; 443, 444, 455, 492 [63–65]; 793, 829, 833 [96]; 1543, 1544 [19] Helson, H. 1495–1497, 1513 [85,86] Henkin, G.M. 1392, 1396, 1421 [40] Hensgen, W. 250, 263, 268 [88,89] Hensley, D. 175, 176, 193 [24,25]; 724, 775 [91] Henson, C.W. 1107, 1128 [30] Hernández, F.L. 518, 527, 529 [52,60,61]; 530 [62–66]; 1661, 1668 [26] Herrero, D.A. 329–333, 359, 362 [62]; 363 [91–95] Hervé, M. 608, 667 [86] Herz, C. 906, 936 [43] Hestenes, M.R. 1373, 1421 [41] Hewitt, E. 616, 667 [87] Hilbert, D. 439, 441, 492 [66] Hille, E. 442, 492 [67]; 968, 974 [13] Hilsum, M. 1470, 1513 [87] Hindman, N. 1082, 1097 [14] Hirsberg, B. 626, 667 [88] Hitczenko, P. 257, 268 [90]; 1154, 1173 [46]; 1610, 1633 [23] Hoeffding, W. 519, 530 [67] Hoffman, A.J. 328, 363 [96] Hoffman, K. 675, 678, 705 [24]; 875, 878, 887, 897 [16]; 1677, 1701 [18] Hoffman-Jørgensen, J. 473, 492 [68]; 769, 775 [92]; 1180, 1181, 1183–1185, 1188, 1191, 1199 [12–15]; 1302, 1303, 1330 [22,23] Holbrook, J. 327, 361 [36] Holický, P. 798, 833 [97]; 1752, 1758, 1792, 1810 [148,149] Holmstedt, T. 1154, 1173 [47] Holub, J.R. 960, 974 [14] Hörmander, L. 223, 231 [27]; 731, 775 [93]; 1369, 1370, 1373, 1407, 1421 [42] Horn, A. 447, 492 [69] Hsu, Y.-P. 526, 529 [45] Huang, C.-F. 374, 390 [19] Hudzik, H. 515, 527, 529 [37,56]; 530 [68] Huff, R.E. 633, 667 [89,90] Huijsmans, C.B. 96, 119 [68] Hunt, R.A. 505, 530 [69]; 1173 [48] Hurewicz, W. 1019, 1068 [67]

1833

Hustad, O. 626, 627, 667 [91,92] Hutton, C.V. 441, 493 [70] Hyers, D.H. 1108, 1128 [31] Il’in, V.P. 1363, 1398, 1399, 1408, 1409, 1420 [7]; 1421 [43] Ioffe, A.D. 418, 435 [55]; 798, 833 [98] Ionescu Tulcea, A. 260, 268 [91] Ionescu Tulcea, C. 260, 268 [91] Isac, G. 1108, 1128 [31] Isbell, J.R. 1714, 1740 [25] Ishii, H. 420, 421, 431, 434 [18] Ivanov, M. 419, 434 [29] Iwaniec, T. 259, 268 [92,93] Izumi, H. 1470, 1513 [88,89] Jacka, S.D. 385, 391 [31] Jackson, S. 1530, 1544 [20] Jacobson, C. 214, 231 [28] Jacod, J. 376, 391 [32] Jahandideh, M.T. 102, 119 [69] Jain, N.C. 1188, 1190, 1199 [16,17] Jajte, R. 1493, 1513 [90,91] James, R.C. 235, 268 [94]; 275, 314 [38]; 477, 479, 493 [71,72]; 581, 596 [31]; 643, 667 [93]; 792, 804, 833 [99,100]; 1052, 1068 [68]; 1250, 1252, 1256, 1257, 1263, 1296 [29–31]; 1303, 1304, 1306, 1307, 1330 [24–27]; 1331 [28]; 1335, 1336, 1359 [19]; 1758, 1810 [150] Jameson, G.J.O. 87, 119 [70]; 458, 459, 493 [73] Jamison, J.E. 901, 935 [25] Janovsky, L.P. 92, 118 [17] Janssen, G. 1307, 1331 [29] Jaramillo, J.A. 413, 434 [30]; 676, 705 [23]; 812, 813, 831 [47]; 832 [85] Jarchow, H. 47, 55, 60, 65, 66, 72, 83 [9]; 286, 309, 313 [11]; 455, 458, 459, 464, 466, 471, 473, 474, 480, 491 [35]; 493 [74–76]; 518, 529 [40]; 675, 681, 690, 705 [15]; 867, 868 [18]; 879, 881, 882, 887, 897 [10]; 901, 905, 911–915, 929, 935 [18,19]; 943, 950, 951, 974 [7]; 1222, 1244 [5]; 1378, 1387–1389, 1420 [22]; 1452, 1456 [24]; 1490, 1500, 1513 [92]; 1563, 1601 [22]; 1674, 1684, 1701 [13,19]; 1710, 1711, 1724, 1736, 1739 [16] Jarosz, K. 702, 703, 705 [25,26] Jawerth, B. 1141, 1145, 1147, 1149, 1161, 1173 [26,40,49] Jayne, J.E. 631, 668 [94]; 1745, 1761, 1788, 1791, 1794, 1803, 1804, 1810 [147,151–155]; 1811 [156,157]; 1814 [269] Jensen, R. 420, 435 [56] Jerison, M. 1712, 1739 [18] Jevti´c, M. 1682, 1701 [20]

1834

Author Index

Ji, G. 1496, 1513 [93] Jiménez Sevilla, M. 664, 665 [16]; 799, 833 [101]; 1758, 1784, 1804, 1811 [158] Johansson, K. 346, 363 [97] John, F. 463, 493 [77]; 718, 775 [94] John, K. 489, 493 [78]; 805, 833 [102]; 1763, 1772, 1787–1789, 1798, 1799, 1811 [159–164] Johnson, W.B. 88–91, 96, 97, 105, 108, 119 [50,62,63,71,72]; 125, 126, 129, 131, 134–136, 140–143, 145–149, 154, 156, 156 [8]; 157 [13]; 158 [46–57]; 164, 190, 193 [26]; 230 [4]; 257, 262, 268 [95,96]; 273–280, 285, 288–296, 298–300, 305, 306, 309, 310, 312, 313 [10,12,19,27]; 314 [28,39–51]; 336, 338, 363 [98]; 443, 445, 459, 483, 491 [31]; 492 [52,53]; 493 [79,80]; 511, 521–523, 529 [51]; 530 [70–72]; 563, 573, 581, 588, 596 [32,33]; 630, 634, 636, 640, 641, 655, 668 [95]; 744, 748, 755, 769, 770, 773 [23]; 775 [95–97]; 784, 792, 793, 805, 821, 829, 831 [35]; 832 [67]; 833 [103–108]; 839–845, 850, 851, 853–855, 858–860, 863, 864, 866, 867, 868 [2,20]; 869 [21,22,27–34]; 906, 925, 929, 936 [44–46]; 950, 951, 953, 957, 961, 964, 965, 974 [15,16]; 1020, 1060, 1062, 1067 [37,49]; 1068 [69]; 1103, 1116, 1119, 1128 [32]; 1133, 1136, 1137, 1157, 1173 [35,50]; 1181, 1192, 1199 [18,19]; 1207, 1208, 1212, 1222, 1224, 1233, 1234, 1242, 1244 [6]; 1245 [13–17]; 1254–1256, 1280, 1295 [10]; 1296 [17]; 1304, 1306, 1331 [30,31]; 1342, 1359 [13]; 1374, 1375, 1377, 1378, 1380, 1387–1389, 1393, 1396, 1404, 1407, 1420 [32]; 1421 [44]; 1454, 1457 [50]; 1478, 1513 [94]; 1535, 1537, 1539, 1543, 1544 [4,21,22]; 1552, 1572, 1579, 1589, 1598–1600, 1601 [26,29,33–36]; 1607, 1610, 1614, 1626, 1628–1630, 1633 [24–29]; 1649, 1652, 1654–1656, 1660–1664, 1668 [20,27]; 1673, 1682, 1701 [21]; 1710, 1716, 1717, 1726, 1731–1733, 1735–1737, 1740 [26–33]; 1745, 1747, 1750, 1752, 1756–1758, 1761, 1765, 1766, 1769, 1771–1773, 1796, 1797, 1800, 1803–1805, 1807 [45,46]; 1811 [165–170]; 1817, 1818 [1a,2a] Jones, L. 148, 158 [47]; 842, 843, 869 [27] Jones, P. 305, 314 [52]; 1373, 1397, 1421 [45,46]; 1699, 1701 [22] Jordan, P. 906, 936 [47] Josefson, B. 1749, 1773, 1811 [171,172] Journé, J.L. 250, 266 [54]; 591, 596 [20] Juhász, I. 1811 [173] Junge, M. 590, 596 [21]; 724, 775 [98,99]; 1447–1452, 1456 [26]; 1457 [51–57]; 1487, 1492, 1493, 1509, 1510, 1513 [95–102] Junilla, H. 1808 [62]

Kadets, M.I. 142, 147, 158 [58]; 463, 493 [81]; 510, 530 [73]; 580, 596 [35]; 611, 647, 667 [68]; 668 [96]; 792, 793, 798, 833 [109–112]; 883, 897 [18]; 929, 936 [48]; 1304, 1331 [32]; 1509, 1514 [103]; 1752, 1771, 1810 [128]; 1811 [174] Kadison, R.V. 333, 334, 363 [99]; 859, 869 [35]; 1463, 1514 [104] Kaftal, V. 336, 361 [30]; 859, 868 [3] Kahan, W.M. 339, 362 [68] Kahane, J.P. 184, 193 [15]; 363 [100]; 644, 645, 668 [97]; 873, 897 [17]; 965, 974 [19]; 1181, 1199 [20]; 1307, 1331 [33] Kakosyan, A.V. 905, 939 [134] Kakutani, S. 443, 465, 493 [82–84] Kalenda, O. 822, 833 [113]; 1537, 1544 [23]; 1545 [24]; 1745, 1756, 1766, 1801, 1811 [175–178] Kalton, N.J. 74, 81, 84 [31]; 95, 120 [73]; 143, 156, 157 [41]; 158 [59,60]; 278, 279, 291–295, 297, 301, 302, 310, 311, 313 [13–16]; 314 [30,53–55]; 403, 435 [57]; 505, 511, 518, 527, 530 [74–77]; 569, 596 [34]; 697, 705 [27]; 769, 775 [77]; 792, 793, 821, 829, 830 [26]; 832 [77–81]; 833 [114]; 843, 854, 868 [7]; 869 [36]; 886, 897 [19]; 901, 936 [49]; 1101, 1103–1127, 1127 [2,11]; 1128 [26,27,30,33–50]; 1129 [51–60]; 1133, 1143, 1144, 1151–1159, 1161, 1162, 1164–1168, 1170, 1171, 1172 [17]; 1173 [27,28,38,51–53]; 1174 [54–63]; 1253, 1255, 1296 [32,33]; 1357, 1359 [20,21]; 1382, 1421 [47]; 1647, 1649–1651, 1657, 1658, 1661–1667, 1668 [9–11,26,28–31]; 1696, 1701 [23]; 1711, 1727, 1734, 1735, 1740 [34,35]; 1796, 1805, 1809 [113–116]; 1818, 1818 [3a] Kami´nska, A. 515, 527, 529 [37]; 530 [68,78–81] Kamont, A. 565, 596 [27] Kanellopoulos, V. 1039–1041, 1053, 1066 [10] Kannan, R. 1625, 1633 [30] Kanter, M. 177, 193 [27]; 524, 530 [82] Kantorovich, L.V. 87, 91, 92, 106, 107, 120 [74–76] Karadzov, G.E. 1140, 1174 [64] Karatzas, I. 374, 391 [33] Kashin, B.S. 336, 338, 342, 358, 363 [101–103]; 575, 596 [36]; 749, 762, 775 [100]; 1222, 1245 [18]; 1305, 1331 [34] Katavolos, A. 1432, 1445, 1457 [58] Kato, T. 445, 493 [85] Katz, N. 222, 231 [29,30] Katznelson, Y. 184, 193 [15]; 569, 596 [37]; 965, 974 [19]; 1410, 1421 [48] Kaufman, R. 793, 828, 830 [22]; 833 [115]; 1020, 1068 [70,71]; 1601 [37]; 1716, 1740 [36]; 1787, 1806 [28]

Author Index Kazarian, K.S. 505, 529 [53] Kazhdan, J.L. 621, 622, 666 [54] Kechris, A.S. 1013, 1018–1020, 1032, 1045, 1068 [72–74]; 1811 [179] Keleti, T. 1530, 1545 [25] Keller, O.-H. 603, 668 [98] Kelley, J.L. 1712, 1740 [37] Kelly, B.P. 249, 265 [6] Kemperman, J.H.B. 914, 936 [50] Kendall, D.G. 613, 614, 668 [99] Kenderov, P. 435 [58]; 1537, 1545 [26]; 1752, 1804, 1807 [40]; 1811 [180,181] Kesten, H. 358, 363 [104] Ketonen, T. 859, 869 [37] Kheifets, A. 1700, 1701 [24] Khovanskii, A.G. 732, 776 [102] Kirchberg, E. 1431, 1440, 1448, 1450, 1452, 1454, 1457 [59–62]; 1551, 1601 [38] Kirchheim, B. 1523, 1543, 1544, 1544 [1,2]; 1545 [27] Kiriakouli, P. 1019, 1046, 1068 [75–77] Kishimoto, A. 1440, 1456 [27] Kislyakov, S.V. 60, 64, 84 [27,32]; 468, 493 [86]; 675, 679, 690, 691, 695, 697, 699–701, 705 [28–32]; 879, 882, 895, 897 [11,20,21]; 1133, 1174 [65–68]; 1392, 1393, 1405, 1406, 1412, 1415, 1416, 1418, 1421 [49–52]; 1505, 1514 [105]; 1675, 1701 [16] Kitover, A.K. 87, 100, 118 [15]; 120 [77] Klain, D. 733, 775 [101] Klebanov, L.B. 905, 939 [134] Klee, V.L. 602, 603, 629, 650, 652, 661–663, 666 [32]; 668 [100–107]; 793, 798, 833 [116]; 1111, 1129 [61] Klemes, I. 875, 897 [22]; 1698, 1701 [25] Knaust, H. 821, 833 [117]; 1354, 1357, 1359 [22] Knöthe, H. 713, 776 [103] Koëthe, G. 1454, 1457 [63] Koldobsky, A. 71, 84 [33]; 177, 193 [18,28]; 524, 529 [46]; 853, 869 [38]; 905, 906, 909–911, 918, 919, 921–923, 935 [20,29]; 936 [37,51–59]; 937 [60–62] Kolmogoroff, A.N. 370, 391 [34] Kolyada, V.I. 1399, 1403, 1404, 1421 [53–55] Komisarski, A. 1020, 1068 [78] Komorowski, R. 279, 314 [56]; 315 [57]; 1252, 1296 [34,35]; 1805, 1809 [101] König, H. 71, 84 [33]; 268 [97,98]; 452, 463, 469, 470, 493 [88–90]; 724, 770, 776 [104,105]; 853, 869 [38]; 915–917, 930–934, 937 [63–69]; 944, 953, 954, 957, 959, 961, 962, 964–966, 968, 970, 971, 973, 974 [15,17,18]; 1234, 1245 [19]; 1392, 1403, 1421 [56]

1835

Koosis, P. 327, 361 [34]; 1677, 1678, 1688, 1699, 1701 [26] Korotkov, V.B. 87, 106, 108, 119 [41]; 120 [78] Kosaki, H. 1464, 1466, 1469–1471, 1474, 1476, 1477, 1480, 1512 [64]; 1514 [106–108] Koskela, P. 1399, 1403, 1421 [39] Köthe, G. 457, 493 [91]; 570, 596 [38]; 1667, 1668 [32]; 1728, 1740 [38] Krasnoselsky, M.A. 87, 88, 118 [32]; 120 [79,80] Kraus, J. 1440, 1457 [46] Krawczyk, A. 1811 [182] Krawczyk, L. 389, 390 [6,7,28] Krée, P. 1491, 1511 [34] Krein, M.G. 88, 99, 120 [81,82]; 439, 492 [55]; 1464, 1465, 1496, 1498, 1512 [72]; 1513 [73] Krein, S. 74, 78, 81, 83 [13]; 87, 120 [83]; 1133, 1174 [69] Krengel, U. 87, 120 [84] Kreps, D.M. 374, 376, 391 [29,35] Kriecherbauer, T. 349, 362 [70] Krieger, H.J. 110, 120 [85] Krishnaiah, P.R. 353, 366 [187] Krivelevich, M. 346, 363 [105]; 1819, 1819 [1a] Krivine, J.L. 136, 138, 139, 158 [61,62]; 284, 315 [58]; 455, 491 [30]; 515, 524, 528 [16]; 530 [83]; 812, 833 [118]; 906, 935 [13]; 937 [70]; 1303–1305, 1307, 1314, 1330 [7]; 1331 [35,36]; 1339, 1359 [23]; 1479, 1514 [109] Kruglyak, N.Ya. 74, 83 [5]; 1133, 1142, 1143, 1145, 1147, 1149, 1172 [11,12]; 1173 [29]; 1174 [70] Krygin, A.W. 1478, 1484, 1511 [39,40] Kuelbs, J. 1179, 1182, 1185, 1190, 1199 [11,21–23] Kunen, K. 268 [99] Kunze, R. 1461, 1466, 1514 [110] Kuratowski, K. 1019, 1068 [79] Kurzweil, J. 798, 813, 833 [119,120]; 1811 [183] Kusraev, A.G. 87, 106, 108, 119 [41] Kutateladze, S.S. 87, 106, 108, 119 [41] Kutzarova, D. 304, 313 [2]; 821, 834 [122]; 1062, 1063, 1065, 1066 [6]; 1068 [80]; 1781, 1782, 1811 [184] Kwapie´n, S. 140, 158 [63,64]; 254, 257, 268 [100]; 275, 315 [59]; 338, 347, 363 [106,107]; 465, 466, 473, 481, 493 [92–94]; 527, 530 [84,85]; 770, 776 [106]; 820, 834 [121]; 855, 856, 858, 869 [23,39]; 875, 879, 880, 886, 887, 889, 897 [23]; 934, 937 [71]; 1115, 1129 [62]; 1181, 1191, 1199 [24,25]; 1222, 1244 [7]; 1304, 1331 [37]; 1392, 1393, 1421 [57,58]; 1446, 1457 [64]; 1478, 1495, 1498, 1514 [111]; 1617, 1633 [31]; 1697, 1701 [27] Kye, S.-H. 1447, 1455, 1457 [65] Kyriazis, G. 579, 596 [39]

1836

Author Index

Laba, I. 222, 231 [29] Lacey, H.E. 465, 493 [95]; 862, 869 [40]; 1716, 1740 [39]; 1745, 1772, 1811 [185] Lamb, C.W. 260, 268 [101] Lamberton, D. 250, 266 [53]; 374, 391 [36] Lammers, M.C. 1650, 1668 [12] Lamperti, J. 905, 937 [72] Lance, E.C. 339, 364 [108] Lancien, G. 792, 793, 804, 805, 821, 829, 832 [78,79]; 834 [123,124]; 1018, 1020, 1068 [81,82]; 1791, 1796, 1805, 1809 [115,116]; 1811 [186,187] Landes, T. 515, 530 [86] Lapeyre, B. 374, 391 [36] Lapresté, J.-T. 1340, 1352, 1359 [6] Larman, D.G. 177, 193 [29]; 735, 776 [107]; 918, 937 [73] Larman, R.R. 1812 [188] Larson, D.R. 341, 364 [109] Latała, R. 460, 493 [96]; 718, 776 [108] Latter, R.H. 1693, 1701 [17] Lazar, A.J. 614, 622, 624–626, 646, 658, 667 [88]; 668 [108–113] Le Hoang Tri 1112, 1127 [5] Le Merdy, C. 1436, 1444, 1452, 1455 [11]; 1457 [53,67,68] Leach, E.B. 798, 834 [125]; 1812 [189] Lebourg, G. 798, 831 [51] Lebowitz, J.L. 354, 360 [1] Ledoux, M. 52, 84 [34]; 338, 350, 361 [21]; 364 [110–112]; 472, 493 [97]; 523, 530 [87]; 740, 756, 776 [109]; 848, 849, 869 [41]; 1179, 1180, 1183, 1185, 1186, 1188, 1190–1196, 1199 [26–32]; 1606, 1616, 1620, 1623–1625, 1632 [8]; 1633 [32–34]; 1634 [35]; 1819, 1820 [5,6] Leduc, M. 407, 409, 435 [59]; 812, 834 [126] Ledyaev, Y. 412, 423, 433 [13]; 434 [14–16] Lee, J.M. 238, 268 [102,103] Lee, P.Y. 1019, 1068 [83] Leeb, K. 1074, 1097 [13] Lehto, O. 259, 268 [104] Leindler, L. 1607, 1634 [36] Leinert, M. 1465, 1470, 1514 [113,114] Lemberg, H. 1314, 1331 [38]; 1339, 1359 [24] Lennard, C.J. 526, 528 [27,28]; 529 [47]; 1477, 1512 [50]; 1700, 1701 [5] LePage, R. 1192, 1199 [33]; 1629, 1634 [37] Leranoz, C. 569, 596 [34]; 1119, 1127 [1,2]; 1129 [53,63]; 1666, 1667, 1668 [30,33] Leung, D.H. 504, 505, 530 [88–92]; 658, 668 [114]; 1019, 1068 [84] Levental, S. 384, 391 [37] Levy, M. 511, 530 [93]; 813, 833 [89]; 1137, 1174 [71]

Lévy, P. 739, 744, 776 [110]; 906, 907, 937 [74]; 1605, 1634 [38] Lewandowski, M. 905, 937 [75] Lewis, D.R. 129, 158 [65]; 278, 279, 313 [6]; 314 [32]; 466, 492 [56]; 493 [98]; 726, 752, 776 [111]; 840, 858, 869 [26,42,43]; 930, 937 [66]; 1222, 1245 [12,20]; 1387–1389, 1420 [36]; 1478, 1507, 1513 [74]; 1600, 1601 [39]; 1717, 1727, 1740 [40] Li, D. 156, 157 [41]; 829, 832 [80,82]; 1046, 1067 [59] Lidski˘ı, V.B. 451, 493 [99]; 968, 974 [20] Lieb, E.H. 164, 173, 193 [10,30]; 718, 773 [41]; 1461, 1481, 1482, 1511 [14,35,36]; 1514 [112] Lifshits, E.A. 87, 120 [80] Lima, Å. 310, 314 [36]; 626, 668 [115,116] Lin, B.L. 133, 157 [27]; 511, 527, 528 [6]; 529 [33,34] Lin, H. 325, 364 [113] Lin, P.K. 471, 493 [100]; 526, 528 [32]; 901, 930, 935 [17]; 937 [66]; 1065, 1068 [80]; 1461, 1510 [10] Linde, W. 475, 484, 493 [101,102]; 905, 937 [78–82] Lindeman, A. 259, 265 [8] Lindenstrauss, J. 7, 10–14, 18, 21–27, 30, 33–36, 38, 42, 48, 50, 51, 78–80, 83 [3,14,15,26]; 84 [35]; 87–91, 105, 108, 119 [72]; 120 [86,87]; 125, 126, 129, 132–136, 140–143, 145, 146, 154, 156, 158 [48,49,66–70]; 159 [71,72]; 164, 175, 190, 193 [12,26]; 208, 231 [21]; 251, 257, 258, 262, 263, 267 [81]; 268 [95,105–107]; 273, 275, 278, 281, 282, 284, 289, 295, 299–301, 305–307, 309, 311, 312, 314 [46,47]; 315 [60–67]; 403, 435 [45]; 440, 443, 459, 465, 466, 475, 492 [49]; 493 [80,103]; 494 [104,105]; 505, 511, 514, 515, 518, 530 [94]; 531 [95–99]; 563, 573, 574, 577, 580, 581, 588, 596 [32,40–43]; 602, 618, 624–626, 630, 634–636, 638–641, 645, 646, 651, 653–655, 658, 661–663, 665 [8]; 666 [37,38]; 667 [69,73]; 668 [95,113,117–123]; 681, 705 [33]; 715, 716, 735–737, 742, 744, 747, 748, 766, 768–770, 773 [30–32,37–39]; 774 [61]; 775 [95]; 776 [112,114,115]; 784, 792, 793, 798, 805, 821, 822, 829, 830 [1,2,6,14]; 831 [34,54]; 832 [69]; 833 [104,106,107]; 834 [127–130]; 839–842, 844, 845, 850, 855, 858, 860, 865, 866, 868 [8]; 869 [28]; 906, 916, 925, 927–929, 935 [7,10–12,24]; 936 [44]; 937 [76,77]; 950, 951, 953, 963–965, 974 [16,21–23]; 1019, 1067 [52]; 1087, 1096 [4]; 1103, 1116, 1119, 1128 [32]; 1129 [64]; 1151, 1155, 1157, 1158, 1167, 1172 [6]; 1173 [39,50]; 1174 [72]; 1181, 1199 [18]; 1207, 1208, 1212, 1222, 1233, 1245 [14,21]; 1249, 1251–1253, 1256, 1260, 1263, 1264, 1285, 1295 [5]; 1296 [36–39]; 1301, 1304–1306, 1310, 1330 [2,13]; 1331 [31,39,40]; 1336, 1348,

Author Index 1359 [7,25]; 1374, 1375, 1377, 1378, 1380–1383, 1387–1389, 1392, 1393, 1396, 1404, 1407, 1419 [3]; 1421 [44,59,60]; 1454, 1457 [66]; 1478, 1480, 1481, 1507, 1511 [11]; 1513 [94]; 1514 [115]; 1521, 1533–1535, 1537–1541, 1543, 1544 [4,5,21,22]; 1545 [28–31]; 1552, 1572, 1579, 1590, 1594, 1597–1600, 1601 [14,33,40,41]; 1602 [42,43]; 1607, 1608, 1626, 1630, 1633 [9,14,24,25]; 1639–1643, 1646, 1647, 1649–1651, 1658, 1664, 1667 [6]; 1668 [34–39]; 1673, 1675, 1682, 1701 [21,29]; 1707–1711, 1714–1716, 1723–1729, 1732, 1733, 1736, 1737, 1740 [27,41–50]; 1745, 1747–1752, 1754, 1756–1760, 1762–1768, 1771–1773, 1775, 1790, 1794–1798, 1800, 1801, 1803–1805, 1806 [2,7,20]; 1807 [42–44]; 1809 [98,99]; 1811 [167–169]; 1812 [190–198]; 1823, 1823 [2a] Lindsey, J.H. 1657, 1668 [40] Linhart, J. 769, 776 [113] Lions, J.L. 1136, 1139, 1140, 1174 [73–76]; 1471, 1472, 1514 [116] Lions, P.L. 420, 421, 426, 431, 434 [18–20] Lisitsky, A. 909, 937 [83] Littlewood, J.E. 956, 974 [12]; 1173 [44] Litvak, A. 736, 737, 746, 767, 769, 773 [19]; 776 [116–118]; 1224, 1245 [11] Litvinov, G.L. 536, 558 [22] Llavona, J.L. 676, 705 [23] Löfström, J. 74, 76–78, 80, 83 [4]; 505, 528 [11]; 577, 595 [2]; 692, 704 [4]; 1133, 1136, 1172 [8]; 1280, 1295 [6]; 1404, 1420 [5]; 1437, 1455 [6]; 1466, 1511 [15] Lomonosov, V.I. 102, 105, 120 [88,89]; 536, 538, 543, 545, 558 [22–26]; 641, 668 [124]; 1397, 1420 [26] Lonke, Y. 910, 911, 937 [62,84] Loomis, L.H. 668 [125] Lopachev, V.A. 905, 910, 937 [85–87] López, G. 1020, 1046, 1066 [21]; 1068 [85,86] Lopez, J.M. 886, 897 [24] Lopez Abad, J. 1094, 1096 [2] Lorentz, G.G. 505, 524, 526, 531 [100–102]; 575, 596 [22] Lorentz, R.A. 578, 596 [44] Loring, T.A. 325, 326, 362 [74]; 364 [114,115] Lotz, H.P. 92, 119 [44]; 455, 456, 484, 494 [106]; 531 [103]; 1589, 1602 [44]; 1715, 1740 [51] Louveau, A. 1019, 1020, 1032, 1045, 1067 [60]; 1068 [73,74]; 1809 [117] Lovaglia, A.R. 792, 834 [131] Lovász, L. 1625, 1633 [30]; 1634 [39] Lowdenslager, D. 1495, 1497, 1513 [86] Lozanovsky, G.Ya. 87, 89, 106, 107, 119 [39]; 120 [90,91]; 1162, 1174 [77]; 1347, 1359 [26] Lubotzky, A. 338, 359, 364 [116]; 1451, 1457 [69]

1837

Lucchetti, R. 435 [58] Luecking, D. 471, 494 [107]; 1684, 1701 [30] Lukacs, E. 937 [88] Lusin, N.N. 1019, 1068 [87,88] Lusky, W. 273, 302, 303, 315 [68–70]; 581, 596 [45–47]; 626, 668 [126,127]; 905, 937 [89] Lust-Piquard, F. 192, 193 [31]; 1486, 1488, 1489, 1514 [117–121] Lutwak, E. 918, 937 [90] Lutzer, D. 1802, 1806 [1] Luxemburg, W.A.J. 87, 96, 119 [68]; 120 [92–94] Lyubich, Y. 914, 916, 917, 937 [91,92]; 938 [93] Maaden, A. 435 [60] MacCluer, B.D. 471, 491 [29] MacGregor, T. 1682, 1701 [31] Mackey, G. 1771, 1812 [199] Magajna, B. 1444, 1457 [70] Magidor, M. 444, 492 [47]; 1058, 1067 [41] Magill, M. 115, 120 [95] Maiorov, V.E. 973, 974 [24] Makai, E. 959, 974 [25] Makarov, B.M. 87, 106, 108, 119 [41] Maleev, R. 814, 834 [132] Maligranda, L. 515, 531 [104]; 1151, 1174 [78] Mandelbrot, B.B. 371, 391 [38] Mandrekar, V. 1191, 1198 [3] Mangheni, P.J. 1716, 1740 [52] Mani, P. 735, 768, 776 [107,119] Mankiewicz, P. 47, 84 [36]; 277, 304, 315 [71–73]; 358, 364 [117]; 633, 668 [128]; 766, 776 [120,121]; 793, 829, 833 [96]; 1216, 1219, 1220, 1222–1226, 1230–1233, 1235–1237, 1239, 1240, 1242–1244, 1245 [22–35]; 1254, 1255, 1296 [40]; 1532, 1543, 1544 [19]; 1545 [32] Manoussakis, A. 1062, 1063, 1065, 1066 [6,7]; 1068 [89,90] Marˇcenko, V.A. 343, 344, 353, 357, 364 [118] Marcinkiewicz, J. 250, 268 [108]; 522, 531 [105] Marciszewski, W. 1019, 1068 [91]; 1796, 1806 [19]; 1812 [200–202] Marcolino, J. 1476, 1479, 1514 [122] Marcus, M.B. 523, 531 [106]; 1190, 1192, 1199 [17,34] Margulis, G.A. 359, 364 [119,120] Marsalli, M. 1496, 1497, 1499, 1514 [123–126] Martin, D.A. 144, 159 [73]; 1358, 1359 [27] Martin, G. 259, 268 [93] Martín, M. 1046, 1068 [86] Marton, K. 1625, 1634 [40] Masani, P. 1495, 1497, 1516 [198] Mascioni, V. 277, 300, 306, 315 [74,75]; 1019, 1036, 1046, 1066 [33]

1838

Author Index

Mastyło, M. 527, 530 [68]; 1143, 1154, 1173 [29]; 1174 [79,80] Masuda, T. 1470, 1510 [3] Mateljevi´c, M. 1677, 1701 [32] Matheron, E. 1797, 1807 [59] Matheson, A. 875, 896 [1] Matoušek, J. 798, 834 [133]; 925, 928, 938 [94] Matoušková, E. 798, 834 [133,134]; 1532, 1533, 1538, 1545 [28,33–35] Mattner, L. 905, 938 [95] Mauldin, R.D. 458, 494 [108]; 1530, 1544 [20] Maurey, B. 11, 51, 53, 84 [37,38]; 125, 133, 134, 136, 139, 140, 148, 158 [45,50,62]; 159 [74–77]; 237, 242, 251, 263, 267 [80–82]; 268 [109]; 277, 284, 304, 310, 314 [33]; 315 [76]; 402–405, 435 [45–49]; 459, 473, 474, 477, 494 [109–112]; 511, 515, 523, 530 [70,83]; 635, 637–639, 667 [73–75]; 745, 770, 771, 776 [122–125]; 799, 812, 821, 829, 832 [73]; 833 [118]; 834 [135,136]; 841, 842, 845, 855, 857, 858, 867, 869 [29,44–46]; 883, 884, 897 [25]; 906, 921, 935 [1,6]; 936 [46]; 953, 957, 961, 964, 965, 974 [15,26]; 1019, 1051, 1055, 1056, 1065, 1067 [55,63]; 1068 [92,93]; 1076, 1081, 1082, 1089, 1090, 1094, 1096, 1097 [12,15,16]; 1101, 1106, 1110, 1128 [29]; 1181, 1191, 1199 [35,37]; 1250–1255, 1260, 1261, 1263, 1265, 1267, 1268, 1271, 1273, 1274, 1276, 1283, 1288, 1296 [26,27,41]; 1301–1306, 1320, 1331 [41–45]; 1344, 1349–1358, 1359 [16,28–31]; 1389, 1421 [61]; 1479, 1514 [109]; 1607, 1612, 1632, 1634 [41,42]; 1652, 1654–1656, 1660–1664, 1668 [27,41]; 1710, 1735, 1740 [53] Maynard, H.B. 634, 668 [129] Mazurkiewicz, S. 1560, 1602 [45]; 1722, 1741 [54]; 1787, 1812 [203] Mazya, V.G. 1363, 1365, 1367, 1373, 1398, 1399, 1401, 1403, 1404, 1421 [62] McAsey, M. 1496, 1514 [127] McCann, R.J. 173, 193 [32]; 713, 776 [126] McCarthy, Ch.A. 568, 596 [48]; 1465, 1480, 1507, 1514 [128]; 1639, 1669 [42] McConnell, T.R. 250, 264, 268 [110–112] McGehee, O.C. 576, 577, 596 [49]; 874, 897 [13] McGuigan, R. 658, 667 [77] McIntosh, A. 327, 329, 347, 361 [35] McLaughlin, D. 1797, 1812 [204] McLaughlin, K.T.-R. 349, 362 [70] McMullen, P. 732, 733, 776 [127,128] McWilliams, R.D. 1045, 1068 [94] Meckes, M. 1819, 1820 [7] Medzhitov, A. 515, 531 [107] Mehta, M.L. 342, 344, 345, 364 [121,122] Mejlbro, L. 1529, 1545 [36] Memin, J. 382, 391 [39]

Mendelson, S. 663, 669 [130] Mercourakis, S. 1019, 1046, 1050, 1058, 1064, 1065, 1066 [11]; 1068 [95,96]; 1587, 1602 [46]; 1745, 1752, 1756, 1761, 1763, 1766, 1767, 1770, 1773–1775, 1784, 1800–1802, 1806 [13,14]; 1812 [205,206] Merton, R.C. 369, 371, 391 [40] Merucci, C. 1151, 1174 [81] Meyer, M. 179, 194 [33]; 724, 728, 749, 759, 775 [78,79]; 776 [105,129]; 777 [130,131] Meyer, P.A. 374–376, 389, 390 [17,18]; 610, 614, 616, 665 [25]; 666 [33]; 669 [131] Meyer, Y. 566, 577, 578, 596 [50]; 1696, 1701 [11] Meyer-Nieberg, P. 87, 89, 92, 96, 120 [96] Mézard, M. 354, 364 [123] Miao, B. 348, 360 [17] Michael, E. 596 [51]; 1731, 1741 [55]; 1755, 1812 [207] Michaels, A.J. 103, 120 [97] Milman, D.P. 792, 834 [137] Milman, M. 1141, 1145, 1147, 1149, 1159, 1161, 1173 [26,28,30]; 1174 [82]; 1407, 1421 [63] Milman, V.D. 47, 48, 51, 53, 83 [16]; 84 [28]; 169, 175, 193 [12,21]; 208, 231 [21,31]; 342, 347, 358, 363 [81,86]; 364 [124]; 471, 475, 492 [49]; 494 [113]; 523, 531 [108]; 710, 713–719, 722–724, 726, 729, 731, 732, 735–750, 752, 754, 755, 758, 759, 762, 766–772, 772 [5,8–10]; 773 [33,34,36–40]; 774 [49,61,64–68]; 775 [85,86]; 776 [114,117,118]; 777 [132–154]; 792, 821, 834 [138]; 844, 845, 848, 850, 852, 859, 865–867, 868 [8,17]; 869 [24,47]; 884, 892, 897 [8,26]; 914–916, 924, 925, 927, 935 [12,24]; 938 [96,97]; 1063, 1068 [97]; 1127, 1129 [65]; 1203, 1221, 1224, 1225, 1230, 1231, 1234, 1243, 1244 [8]; 1245 [36,37]; 1246 [38,39]; 1250, 1296 [42,43]; 1305, 1306, 1310, 1315, 1317, 1325, 1326, 1330 [3,13]; 1331 [46–49]; 1335, 1338, 1350, 1353–1358, 1359 [30,32]; 1360 [33–35]; 1606–1608, 1612, 1625–1627, 1630, 1632, 1632 [1–3]; 1633 [9,14,16,20,21]; 1634 [43–46]; 1745, 1812 [208] Milne, H. 676, 705 [34] Milutin, A.A. 1551, 1572, 1602 [47] Minc, H. 98, 120 [98] Mirsky, L. 329, 364 [125] Misiewicz, J. 906, 909, 911, 938 [98–101] Mityagin, B.S. 459, 494 [114]; 906, 935 [1]; 1174 [83]; 1373, 1409, 1421 [64,65]; 1667, 1669 [43,44] Molto, A. 792, 834 [139]; 1778, 1790–1792, 1812 [209–214] Monat, P. 389, 390 [10] Monniaux, S. 250, 268 [113]

Author Index Montesinos, V. 832 [62]; 1745, 1747–1758, 1761–1769, 1771, 1772, 1774–1780, 1782, 1785, 1787–1790, 1792, 1797, 1802, 1804, 1805, 1808 [80–85]; 1812 [209] Montgomery, H.L. 198, 224–226, 231 [32]; 345, 364 [126]; 555, 558 [9] Montgomery-Smith, S.J. 74, 81, 84 [31]; 249, 259, 265 [6,7]; 523, 527, 529 [48]; 531 [109]; 1118, 1129 [54]; 1142, 1153–1155, 1173 [41,46]; 1174 [61]; 1175 [84–86] Moors, W.B. 1521, 1534, 1537, 1538, 1544 [10,11]; 1545 [26,37]; 1804, 1811 [180,181] Moreno, J.P. 664, 665 [16]; 799, 833 [101]; 1758, 1784, 1804, 1811 [158] Morris, P. 633, 667 [89,90] Morton, A. 374, 390 [8] Morzocchi, M. 425, 434 [17] Moschovakis, Y.N. 1019, 1068 [98] Muckenhoupt, B. 579, 596 [52] Muhly, P.S. 237, 249, 250, 265 [12,13]; 1445, 1455 [12]; 1494, 1496, 1511 [16]; 1514 [127] Mujica, J. 677, 705 [35]; 1800, 1812 [215] Müller, C. 926, 938 [102] Müller, P.F.X. 130, 131, 159 [78–80]; 1693, 1694, 1696, 1698, 1699, 1702 [33–37] Murray, F.J. 129, 159 [81]; 484, 494 [115]; 1770, 1812 [216] Muscalu, C. 1484, 1514 [129] Musiela, M. 374, 391 [41] Musielak, J. 515, 531 [110] Nachbin, L. 88, 120 [99]; 1712, 1741 [56] Nagasawa, M. 702, 705 [36] Naimark, M.A. 607, 669 [132] Nakano, H. 87, 89, 120 [100] Namioka, I. 87, 88, 120 [101]; 798, 834 [140]; 1745, 1751, 1752, 1789, 1791, 1792, 1794, 1802–1804, 1807 [33]; 1810 [147,151–155]; 1812 [217–220]; 1814 [257] Naor, A. 845, 869 [48] Narayan, S.K. 320, 360 [8] Nash-Williams, C.St.J.A. 1076, 1097 [17] Natanson, I.P. 507, 531 [111]; 1749, 1801, 1812 [221] Nathanson, M. 207, 231 [33] Nawrocki, M. 1666, 1669 [45] Nazarov, F. 184, 192, 194 [34] Negrepontis, S. 139, 157 [12]; 659, 669 [133]; 1019, 1046, 1068 [77,96]; 1600, 1602 [48]; 1745, 1755, 1756, 1758, 1763, 1767, 1774, 1775, 1784, 1801, 1802, 1806 [14]; 1812 [206,222,223] Neidinger, R.D. 1280, 1296 [44,45] Nelson, E. 1464, 1481, 1514 [130,131] Nemirovski, A.M. 812, 813, 834 [141]

1839

Neufang, M. 1813 [224] Neuwirth, S. 1505, 1513 [84] Newman, C.M. 147, 157 [13]; 230 [4]; 342, 361 [23]; 463, 491 [10]; 748, 773 [23]; 866, 867, 868 [2] Newman, D.J. 1700, 1702 [38] Neyman, A. 906, 938 [103] Ng, K.F. 87, 122 [143] Ng, P.W. 1457 [71] Nguyen Nhu 1112, 1127 [5] Nica, A. 345, 357, 358, 366 [178]; 1445, 1458 [103]; 1466, 1487, 1516 [196] Niculescu, C. 480, 494 [117] Nielsen, N.J. 146, 159 [82]; 268 [98]; 277, 279, 286, 304, 315 [71,77–79]; 518, 531 [112]; 1240, 1245 [28]; 1447, 1457 [54]; 1510, 1513 [99]; 1600, 1602 [49]; 1739, 1739 [14] Nikishin, E.M. 516, 531 [113]; 1302, 1316, 1331 [50,51] Nikol’ski˘ı, S.M. 575, 597 [53]; 1363, 1398, 1399, 1409, 1420 [7] Nilsson, P. 1153, 1173 [31] Nirenberg, L. 1398, 1421 [66] Nissenzweig, A. 1749, 1813 [225] Nordgren, E.A. 101, 119 [48] Nördlander, G. 793, 834 [142] Norin, N.V. 1528, 1545 [38] Novikov, I. 523, 531 [114]; 595, 597 [54,55] Novikov, S.Ya. 527, 531 [115,116] Nussbaum, R.D. 99, 112, 120 [102,103]; 121 [104] Oberlin, D. 685, 705 [37] Odell, E. 59, 83 [22]; 133, 139, 141, 143–145, 147, 149, 150, 154, 156 [2]; 158 [51,52]; 159 [83–85]; 255, 256, 267 [58]; 276, 279, 313 [4,17]; 565, 581, 595 [1]; 596 [23]; 612, 645, 669 [134]; 813, 820, 821, 833 [117]; 834 [143,144]; 839, 862, 868, 868 [1]; 1016, 1019, 1022, 1025, 1030, 1034, 1036, 1044–1046, 1048, 1050, 1052, 1056, 1060, 1061, 1063–1065, 1065 [2,3]; 1066 [30]; 1068 [66]; 1069 [99–105]; 1081, 1082, 1090, 1097 [18,19]; 1110, 1129 [66]; 1133, 1162, 1175 [87]; 1251, 1260, 1296 [46]; 1336, 1339, 1343, 1347–1349, 1351, 1352, 1354, 1355, 1357, 1358, 1359 [2,17,18,22]; 1360 [36–46]; 1748, 1759, 1775, 1813 [226–230] Odlyzko, A.M. 345, 362 [77]; 364 [127] Ogrodzka, Z. 1414, 1422 [67] Oikhberg, T. 275, 285, 314 [48]; 1454, 1457 [50,72–75]; 1551, 1602 [50] Oja, E. 1748, 1813 [231] Oleszkiewicz, K. 460, 493 [96]; 1505, 1513 [84] Olevskiˇı, A.M. 213, 231 [34]; 251, 269 [114,115] Olin, R.F. 341, 364 [128] Olsen, G.H. 618, 668 [122]

1840

Author Index

Oncina, L. 1791, 1792, 1813 [232] O’Neil, R. 505, 511, 531 [117,118] Oprea, A.G. 1493, 1512 [44] Ordower, M. 340, 362 [63] Orihuela, J. 792, 834 [139]; 1756, 1762, 1764, 1766, 1767, 1778, 1789–1792, 1801, 1803, 1812 [209–213]; 1813 [233–235] Orlicz, W. 458, 473, 494 [118] Ørno, P. 92, 121 [105]; 1749, 1813 [236] Ornstein, D. 1409, 1422 [68] Ortynski, A. 1666, 1669 [45] Ostrovskii, M.M. 1143, 1144, 1174 [63] Otto, F. 350, 364 [129] Ovchinnikov, V.I. 1151, 1154, 1174 [78,80]; 1175 [88]; 1464–1467, 1514 [132–134] Oxtoby, J.C. 1813 [237] Ozawa, N. 1447, 1452, 1454, 1455, 1456 [28]; 1457 [55,71]; 1458 [76,77] Pajor, A. 175, 179, 193 [14]; 194 [33]; 454, 492 [54]; 718, 722–724, 728, 750, 756, 759, 762, 767, 769, 773 [19]; 776 [105,117]; 777 [130,131,146–148,155,156]; 854, 868 [15]; 924, 938 [97]; 946, 968, 974 [10]; 1225, 1226, 1246 [40] Paley, R.E.A.C. 250, 269 [116]; 1664, 1669 [46]; 1688, 1702 [39] Pallaschke, D. 1113, 1129 [67] Palmon, O. 767, 778 [157] Panzone, R. 107, 118 [34] Paouris, G. 765, 778 [158] Papadimitrakis, M. 164, 193 [22]; 725, 774 [69]; 918, 938 [104] Papadopoulou, S. 660, 669 [135] Papini, L. 650, 655, 666 [45] Parisi, G. 354, 364 [123] Parrott, S.K. 339, 364 [130] Partington, J.R. 1775, 1813 [238] Pastur, L.A. 343, 344, 349, 353, 357, 364 [118,131–133] Paulsen, V.I. 320, 362 [64]; 364 [134]; 1427–1432, 1435, 1438, 1440, 1442, 1445, 1455 [12]; 1456 [13,14]; 1458 [78–81] Pavlovi´c, M. 1677, 1682, 1701 [20,32] Payá, R. 642, 666 [35]; 1046, 1068 [86] Pearcy, C.M. 322, 364 [135]; 536, 539, 558 [27]; 559 [28]; 1170, 1175 [89] Pechanec, J. 1795, 1813 [239] Peck, N.T. 278, 301, 302, 314 [54]; 403, 435 [57]; 1101, 1104, 1107, 1111, 1113–1115, 1125, 1128 [30]; 1129 [55,56,68]; 1157, 1158, 1174 [62]; 1666, 1668 [31] Pedersen, G.K. 1461, 1510 [1]

Peetre, J. 481, 494 [119]; 1136, 1140, 1145, 1173 [32]; 1174 [76]; 1175 [90,91]; 1363, 1382, 1400, 1407, 1422 [69–71]; 1466, 1468, 1471, 1472, 1514 [116,135] Peirats, V. 518, 529 [60] Pelant, J. 832 [62]; 1745, 1747–1758, 1761–1766, 1768, 1769, 1771, 1772, 1774, 1775, 1777–1780, 1782, 1785, 1787–1789, 1792, 1795–1797, 1802, 1804, 1805, 1808 [62,80]; 1809 [118]; 1813 [240] Pełczy´nski, A. 96, 119 [50]; 125, 129, 132, 135, 142, 146, 147, 156, 158 [58,66,67]; 159 [86]; 251, 255, 256, 268 [105]; 269 [117–119]; 274, 277, 290, 300, 305, 315 [80–82]; 440, 443, 445, 458, 459, 462–466, 473, 479, 491 [31]; 492 [40]; 494 [104,114,120–125]; 510, 530 [73]; 577, 580–582, 596 [35,41,51]; 597 [56,57]; 603, 648, 665 [9,10]; 675, 688, 703, 704, 705 [39]; 792, 830 [17]; 858, 869 [23]; 875, 876, 878–880, 883, 886, 887, 889, 894, 896 [3]; 897 [18,19,23,27,28]; 901, 905, 911–915, 935 [18]; 951, 963, 974 [21,27]; 1018, 1028, 1029, 1045, 1066 [16]; 1069 [106,107]; 1133, 1136, 1137, 1173 [35]; 1219, 1222, 1223, 1240, 1244 [7]; 1246 [41,42]; 1249, 1253, 1254, 1280, 1295 [7,10]; 1296 [47]; 1301–1304, 1330 [8]; 1331 [32,39]; 1366, 1367, 1372, 1377, 1380–1383, 1387, 1393, 1396, 1400–1403, 1410–1413, 1417–1419, 1420 [6]; 1421 [47,58]; 1422 [72–74,76–81]; 1478, 1495, 1498, 1509, 1514 [103,111]; 1525, 1545 [39]; 1552, 1560, 1569, 1571, 1580, 1590, 1593, 1594, 1601 [15]; 1602 [42,51–54]; 1639, 1642, 1652, 1668 [35]; 1669 [47–49]; 1674, 1675, 1697, 1701 [13,27,29]; 1709, 1711, 1717, 1723, 1727, 1728, 1732–1735, 1740 [35,44,45]; 1741 [57–59]; 1745, 1761, 1765–1767, 1796, 1800, 1803, 1806 [24,25]; 1807 [46]; 1813 [241,242] Peller, V.V. 1500, 1501, 1505, 1506, 1510 [2]; 1514 [136,137]; 1515 [138,139] Pena, A. 769, 773 [21]; 1127, 1127 [10] Peressini, A.L. 87, 121 [107] Perissinaki, I. 722, 774 [70] Perron, O. 98, 121 [108] Persson, A. 462, 494 [126] Petrushev, P. 579, 596 [39]; 597 [58] Pettis, B.J. 444, 492 [41]; 792, 834 [145]; 1579, 1601 [25] Petty, C.M. 164, 177, 193 [13]; 194 [35]; 725, 778 [159]; 918, 935 [16] Petunin, Yu.I. 74, 78, 81, 83 [13]; 87, 120 [83]; 1133, 1174 [69] Pezzotta, A. 662, 667 [70] Pfaffenberger, W.E. 558 [12] Pﬁtzner, H. 1477, 1515 [140] Phelps, R.R. 34, 35, 83 [26]; 435 [61]; 610, 611, 614, 626, 627, 634, 640, 641, 645, 646, 659, 664,

Author Index 665 [13]; 667 [72]; 668 [121]; 669 [136–140]; 793, 798, 832 [69]; 834 [140]; 1019, 1025, 1067 [52]; 1069 [108]; 1532, 1535, 1537, 1538, 1545 [40,41]; 1733, 1739 [7]; 1745, 1747, 1749, 1752, 1783, 1789, 1795, 1798, 1809 [99]; 1812 [188]; 1813 [243]; 1814 [257]; 1823, 1823 [3a] Phillips, N.C. 1431, 1457 [62] Phillips, R. 338, 359, 364 [116]; 1451, 1457 [69] Piasecki, M. 264, 269 [120,121] Picardello, M. 1466, 1512 [67] Pichorides, S.K. 482, 494 [127] Pietsch, A. 250, 269 [122]; 439, 440, 444, 448, 451–455, 458, 459, 462–464, 466, 469–471, 475, 476, 480, 484, 493 [102]; 494 [126,128–136]; 495 [137–139]; 590, 597 [59]; 929, 935 [19]; 943–945, 947, 950, 951, 953, 954, 957, 961, 966, 969, 973, 974 [7,28–32]; 1222, 1246 [43]; 1301, 1331 [52]; 1466, 1468, 1515 [141,142]; 1563, 1601 [22] Pigno, L. 576, 577, 596 [49] Pinsker, A.G. 87, 106, 107, 120 [75] Pintz, J. 207, 230 [5] Pisier, G. 47, 51, 53, 83 [17]; 206, 231 [35,36]; 235, 237, 269 [123,124]; 276, 277, 284–286, 305, 314 [49]; 315 [76,83–85]; 320, 338, 346, 365 [136–141]; 466, 468, 471, 473–475, 477, 479, 483, 489, 494 [112]; 495 [140–150]; 518, 523, 531 [106,119]; 695, 705 [38]; 710, 726, 752, 754, 760, 767, 770–772, 776 [124,125]; 777 [149]; 778 [160–164]; 793, 804, 805, 812, 831 [54]; 834 [136,146–149]; 843, 845, 849, 870 [49–51]; 879, 882, 886, 888, 891, 893, 894, 897 [29–32]; 964, 974 [33]; 1129 [69]; 1133, 1144, 1157, 1164, 1173 [39]; 1175 [92–98]; 1181, 1188, 1190–1192, 1194, 1199 [15,34,37]; 1200 [38–42]; 1222, 1224–1226, 1230–1232, 1234, 1241, 1242, 1245 [16]; 1246 [38,44]; 1303–1306, 1315, 1320, 1322, 1331 [44,45,48,53–55]; 1332 [56–61]; 1406, 1422 [82]; 1429, 1431, 1432, 1438, 1441, 1442, 1444, 1446–1451, 1457 [56,73]; 1458 [82–92]; 1471, 1478, 1483–1493, 1495, 1499, 1500, 1502, 1503, 1508, 1513 [80,81]; 1514 [121]; 1515 [143–157]; 1599, 1601 [21]; 1607, 1610, 1625, 1628, 1634 [47–49]; 1726, 1739 [12]; 1741 [60]; 1818 [2a] Pitt, L.D. 94, 121 [109]; 905, 936 [42] Pittenger, A.O. 257, 269 [125] Pitts, D.R. 341, 362 [65,66]; 365 [142] Plemmons, R.J. 98, 113, 118 [35] Plichko, A. 304, 313 [2]; 1759, 1764, 1765, 1767, 1769, 1789, 1800, 1803, 1804, 1806 [3]; 1807 [35]; 1813 [244–248]; 1817, 1818 [1a] Pliska, S.R. 374, 377, 391 [30]

1841

Plotkin, A.I. 901–903, 905, 910, 937 [86,87]; 938 [105–109] Plymen, R.J. 1466, 1515 [158] Pol, R. 1019, 1069 [109]; 1745, 1750, 1751, 1754, 1757, 1758, 1762, 1792, 1794, 1802, 1806 [5]; 1807 [38]; 1812 [220]; 1813 [249–254] Polya, G. 907, 938 [110,111]; 956, 974 [12]; 1173 [44] Polyrakis, I.A. 116, 118 [14] Pommerenke, Ch. 701, 705 [40] Pompe, W. 573, 597 [60] Poornima, S. 1403, 1422 [83] Popescu, G. 341, 365 [143] Popovici, I.M. 97, 98, 121 [110,111] Pospíšil, B. 1753, 1807 [36] Poulsen, E.T. 617, 669 [141] Power, S.C. 339, 341, 362 [67]; 365 [144]; 1496, 1515 [159] Preiss, D. 37, 42, 84 [39]; 396, 399, 413, 418, 425, 433 [8]; 434 [37]; 435 [50,62]; 793, 798, 820, 821, 829, 830 [19]; 833 [107]; 834 [140,150,151]; 1529–1534, 1538, 1539, 1541, 1543, 1544 [4,22]; 1545 [25,28–31,36,42–47]; 1747, 1752, 1777, 1789, 1813 [255,256]; 1814 [257,258]; 1818 [2a] Prékopa, A. 1607, 1634 [50] Privalov, A.A. 578, 597 [61] Privalov, I.I. 694, 705 [41] Protter, P. 376, 391 [42] Prüss, J. 250, 268 [113]; 269 [126,127] Przelawski, K. 793, 830 [13] Przeworska-Rolewicz, D. 445, 495 [151] Ptak, V. 1052, 1069 [110] Pukhlikov, A.V. 732, 776 [102] Pustylnik, E.I. 87, 120 [79] Quenez, M.C. 382, 390 [24] Quinzii, M. 115, 120 [95] Rabinovich, L. 917, 938 [112] Radjavi, H. 99, 101, 119 [48]; 121 [112]; 542, 549, 559 [29,30] Raghavan, T.E.S. 98, 113, 118 [33] Rainwater, J. 609, 669 [142]; 1783, 1814 [259] Raja, M. 805, 828, 834 [152]; 835 [153]; 1787, 1790–1794, 1798, 1814 [260–263] Randrianantoanina, B. 515, 531 [120]; 901, 905, 936 [49]; 938 [113,114] Randrianantoanina, N. 1490, 1492, 1493, 1498, 1509, 1515 [160–165] Range, R.M. 675, 683, 705 [42] Ransford, T.J. 1126, 1128 [20] Rao, M. 612, 669 [143] Rassias, T.M. 1108, 1128 [31]

1842

Author Index

Raynaud, Y. 137, 139, 140, 158 [43]; 159 [87–89]; 519, 523, 527, 528 [8,9]; 531 [121–126]; 532 [127,128]; 1479, 1509, 1515 [166–169] Read, C.J. 98, 102, 121 [115–117]; 274, 295, 297, 299, 301, 315 [86]; 549, 550, 555, 556, 559 [31–36]; 1216, 1240, 1246 [45]; 1254, 1296 [48]; 1644, 1669 [50] Reese, M.L. 1103, 1129 [70] Reif, J. 1814 [264] Reinov, O. 462, 484, 495 [152,153]; 1817, 1818, 1818 [4a] Reisner, S. 527, 532 [129]; 759, 775 [79]; 778 [165]; 896, 897 [12] Retherford, J.R. 953, 957, 961, 964, 965, 974 [15] Revalski, J. 434 [31] Revuz, D. 372, 373, 378, 379, 391 [43] Rezniˇcenko, E.A. 1747, 1814 [265] Reznick, B. 916, 917, 938 [115] Rhandi, A. 112, 121 [113,114]; 433 [11] Ribarska, N.K. 1789, 1803, 1814 [266,267] Ribe, M. 829, 835 [154,155]; 1104, 1107, 1108, 1110, 1129 [71,72]; 1157, 1175 [99]; 1347, 1360 [47] Ricard, E. 1457 [74] Rieffel, M.A. 484, 495 [154]; 634, 669 [144] Riesz, F. 442, 495 [155,156] Riesz, M. 244, 269 [128] Ringrose, J.R. 339, 365 [145]; 439, 495 [157]; 1463, 1514 [104] Riss, E.A. 1530, 1545 [48] Rivière, N.M. 1411, 1420 [28]; 1422 [84] Roberts, J.W. 403, 435 [57]; 603, 669 [145]; 1101, 1107–1109, 1111–1114, 1117, 1118, 1125, 1129 [56–58,73–76]; 1255, 1296 [33]; 1666, 1668 [31] Robinson, A. 535, 558 [10] Robinson, D.W. 620, 665 [23]; 1466, 1511 [25] Robinson, P.L. 1466, 1515 [158] Rochberg, R. 703, 705 [43]; 706 [44,45]; 1120, 1128 [15]; 1133, 1158, 1159, 1161, 1172 [20]; 1173 [28,49]; 1175 [100,101] Rodé, G. 641, 669 [146,147]; 1814 [268] Rodin, V.A. 519, 532 [130]; 1155, 1175 [102] Rodriguez-Salinas, B. 518, 527, 529 [61]; 530 [62–64] Rogalski, M. 611, 669 [148]; 1048, 1069 [111] Rogers, C.A. 177, 193 [29]; 458, 492 [44]; 631, 668 [94]; 720, 735, 774 [54]; 918, 937 [73]; 1303, 1330 [11]; 1626, 1633 [12]; 1745, 1761, 1788, 1789, 1791, 1794, 1803, 1804, 1810 [146,147,151–155]; 1811 [156,157]; 1814 [269] Rogers, L.C.G. 376, 383, 391 [44,45] Rohlin, V.A. 616, 669 [149] Rolewicz, S. 445, 495 [151]; 769, 778 [166]; 1101, 1102, 1129 [77,78]

Romberg, B.W. 1104, 1113, 1120, 1128 [23] Ropela, S. 575, 597 [62] Rørdam, M. 325, 362 [78]; 363 [79] Rosenoer, S. 340, 365 [146] Rosenthal, H.P. 20, 59, 84 [40]; 125, 128, 129, 131, 134, 140, 145–148, 150, 151, 153, 154, 157 [11,21,37]; 158 [54,68]; 159 [90–93]; 204, 231 [37]; 255, 265 [22]; 268 [99]; 269 [119]; 274, 277, 279, 290, 291, 293, 294, 296, 300, 305, 314 [50,51]; 315 [81,87]; 445, 466, 473, 480, 492 [40]; 494 [105]; 495 [158,159]; 521, 532 [131]; 576, 581, 595 [11]; 596 [33]; 612, 614, 636, 644, 645, 665 [19]; 669 [134,150,151]; 813, 821, 833 [105]; 835 [156,157]; 868, 870 [52]; 875, 876, 897 [33]; 1015, 1016, 1018–1020, 1022–1025, 1028, 1030, 1031, 1034, 1036, 1044–1048, 1051, 1055, 1056, 1058, 1060, 1064, 1065, 1066 [26,33]; 1067 [34]; 1068 [66,93]; 1069 [102,112–120]; 1076, 1081, 1096 [1]; 1097 [20,21]; 1242, 1245 [17]; 1260, 1261, 1296 [41]; 1301–1303, 1330 [8]; 1332 [62]; 1339, 1340, 1349, 1352, 1359 [18,31]; 1360 [38,48,49]; 1454, 1455 [3]; 1457 [66,75]; 1458 [93]; 1506–1509, 1513 [82]; 1515 [170]; 1551, 1560, 1561, 1580, 1581, 1587, 1589, 1592, 1594, 1595, 1598–1600, 1601 [10,34,40,41]; 1602 [44,50,55–62]; 1652, 1654, 1669 [48,51]; 1697, 1702 [40]; 1708, 1714, 1717, 1722, 1723, 1725, 1727–1729, 1732, 1740 [28,46,47]; 1741 [61–64]; 1748, 1755, 1758, 1762, 1767, 1771, 1772, 1800, 1802, 1804, 1805, 1808 [68]; 1811 [170]; 1813 [228]; 1814 [270–274] Rosenthal, P. 101, 119 [48]; 542, 549, 559 [29,30] Rosinski, J. 905, 938 [116] Ross, K.A. 886, 897 [24] Rosset, S. 730, 778 [167] Roth, K. 200, 231 [25] Rothschild, B.L. 1074, 1097 [13] Royden, H.L. 6, 13, 15, 20, 62, 83 [18]; 1751, 1814 [275] Ruan, Z.J. 1427, 1432–1436, 1438–1440, 1442–1447, 1452–1455, 1456 [15,26,28–41]; 1457 [54,55,57,65]; 1458 [94–97]; 1510, 1512 [60]; 1513 [99] Rubio de Francia, J.L. 250, 269 [129–131] Rudelson, M. 737, 767, 778 [168–170] Rudin, M.E. 1751, 1752, 1754–1756, 1780, 1806 [21]; 1812 [207] Rudin, W. 19, 37, 83 [19]; 147, 159 [94]; 197, 205, 206, 231 [38]; 359, 365 [147]; 603, 669 [152]; 883, 897 [34]; 901, 902, 938 [117]; 1419, 1422 [85]; 1502, 1516 [171]; 1674, 1675, 1677, 1679, 1694, 1700, 1701 [28]; 1702 [41,42] Rudnick, Z. 345, 365 [148] Ruelle, D. 354, 360 [1]; 619, 620, 669 [153]

Author Index Ruiz, C. 527, 530 [65,66] Russo, B. 1600, 1601 [27] Russu, G.I. 1466, 1516 [172] Ruston, A.F. 449, 450, 484, 495 [160–162] Rutkowski, M. 374, 391 [41] Rutman, M.A. 99, 120 [82] Ruzsa, I. 205, 231 [39] Ryan, R. 263, 269 [132] Rychtáˇr, J. 1782, 1814 [276,277] Ryff, J.V. 1151, 1175 [103] Ryll-Nardzewski, Cz. 906, 938 [101] Saab, P. 95, 120 [73] Saakyan, A.A. 575, 596 [36] Saccone, S.F. 682, 685, 686, 706 [46,47] Sagher, Y. 505, 529 [39]; 1141, 1158, 1172 [20]; 1173 [30,33] Sahakian, A. 578, 596 [44] Saint Raymond, J. 633, 669 [154]; 759, 778 [171]; 793, 831 [38] Saito, K.S. 1465, 1496, 1497, 1513 [93]; 1514 [127]; 1516 [173–177] Sakai, S. 1463, 1516 [178] Saks, S. 444, 490 [4] Salem, R. 873, 897 [17] Salinas, N. 324, 332, 333, 360 [11]; 362 [62]; 365 [149]; 536, 539, 558 [27] Saloff-Coste, L. 1363, 1423 [95] Samet, D. 1730, 1741 [65] Samuel, C. 286, 305, 315 [88]; 316 [89]; 1560, 1569, 1602 [63] Samuelson, P.A. 370, 391 [46] Saphar, P.D. 285, 310, 314 [31]; 461, 495 [163,164]; 792, 829, 832 [81] Sarason, D. 1495, 1497, 1516 [179]; 1700, 1702 [43] Sarnak, P. 338, 345, 359, 364 [116]; 365 [148]; 1451, 1457 [69] Savage, L.J. 616, 667 [87] Scalora, F.S. 260, 269 [133] Schachermayer, W. 263, 267 [82]; 374, 377–386, 389, 390 [10–16]; 391 [47,48]; 404, 435 [49]; 633, 636, 669 [155,156]; 798, 799, 832 [73]; 835 [158]; 1767, 1800, 1803, 1809 [97]; 1813 [234]; 1814 [278] Schaefer, H.H. 87, 88, 92, 99, 110, 111, 121 [120–124]; 821, 835 [159] Schäffer, J.J. 805, 835 [160] Schatten, R. 439, 484, 486, 495 [165–169] Schauder, J. 273, 316 [90]; 442, 495 [170] Schechter, M. 1159, 1175 [104] Schechtman, G. 47, 48, 51, 53, 83 [16]; 129–132, 134, 140, 148, 150, 153, 154, 156, 157 [21]; 158 [50,53,55]; 159 [78,79,95–97]; 176, 194 [36]; 208, 231 [31]; 257, 268 [96]; 278, 314 [46]; 336,

1843

338, 346, 347, 363 [98]; 364 [124]; 365 [150]; 471, 475, 494 [113]; 511, 521–523, 530 [70–72]; 531 [108]; 576, 581, 595 [11]; 710, 713, 718, 735, 737, 739–742, 745, 746, 748, 749, 755, 769, 770, 775 [96,97]; 776 [118]; 777 [150–152]; 778 [172–175]; 792, 793, 821, 829, 833 [106,107]; 840, 844, 845, 847, 848, 851–853, 855, 857–859, 862–864, 866, 867, 869 [21,22,29–34,46,47]; 870 [53–56]; 884, 897 [26]; 906, 925, 936 [45,46]; 1018, 1019, 1066 [26]; 1157, 1173 [50]; 1186, 1192, 1199 [19]; 1200 [43]; 1234, 1242, 1245 [15]; 1246 [39]; 1310, 1315, 1317, 1325, 1326, 1331 [49]; 1338, 1360 [33]; 1403, 1422 [86]; 1539, 1543, 1544 [4,22]; 1606, 1608, 1610, 1612, 1614, 1624–1626, 1628–1630, 1633 [13,26–29]; 1634 [45,46,51–55]; 1642, 1652, 1654–1656, 1660–1664, 1668 [27]; 1669 [52]; 1698, 1699, 1702 [37]; 1716, 1740 [29]; 1796, 1805, 1811 [169] Schep, A.R. 108, 121 [125] Scherer, K. 1404, 1420 [20] Schipp, F. 576, 597 [63] Schlüchtermann, G. 1478, 1512 [58] Schlumprecht, Th. 139, 143, 144, 159 [84,85]; 176, 177, 193 [18]; 194 [36]; 813, 820, 821, 833 [117]; 834 [143,144]; 919, 935 [29]; 1061–1063, 1069 [103,104,121]; 1081, 1082, 1090, 1097 [18,19]; 1110, 1129 [66]; 1133, 1162, 1175 [87]; 1251, 1256, 1260, 1296 [46,49]; 1335, 1339, 1343–1349, 1352, 1354, 1355, 1357, 1359 [3,4,18,22]; 1360 [38–44,50,51]; 1775, 1813 [229,230] Schmidt, E. 439, 441, 496 [171]; 716, 778 [176] Schmuckenschläger, M. 1625, 1634 [53] Schnaubelt, R. 112, 121 [114] Schneider, R. 710, 720, 726, 734, 778 [177,178]; 911, 921, 938 [118–120]; 1607, 1634 [56] Schoenberg, I.J. 906, 938 [121] Scholes, M. 369, 371, 390 [4] Schonbek, T. 1471, 1472, 1512 [42] Schreier, J. 1051, 1064, 1069 [122]; 1581, 1602 [64] Schur, I. 439, 442, 496 [173,174] Schütt, C. 140, 158 [63,64]; 159 [88]; 471, 496 [172]; 525–527, 530 [84,85]; 532 [128,132,133]; 855, 856, 869 [39]; 933, 934, 937 [67,71]; 1234, 1245 [19]; 1478, 1516 [180]; 1655, 1669 [53] Schwartz, J.T. 106, 107, 119 [54]; 439, 492 [42]; 1366, 1388, 1395, 1420 [25]; 1639, 1669 [42] Schwartz, L. 471, 496 [175] Schwarz, H.U. 87, 121 [126]; 488, 496 [176] Schweizer, M. 387–389, 390 [10,26] Sciffer, S. 1537, 1545 [26]; 1804, 1811 [180,181] Sedaev, A.A. 527, 532 [134]; 1477, 1511 [37] Segal, I. 1461, 1516 [181] Seidel, J.J. 916, 936 [34]

1844

Author Index

Seifert, C.J. 479, 491 [36] Semadeni, Z. 625, 669 [157]; 1714, 1740 [25]; 1745, 1803, 1814 [279,280] Semenov, E.M. 74, 78, 81, 83 [13]; 87, 120 [83]; 519, 523, 527, 531 [114,116]; 532 [130]; 597 [55]; 812, 813, 834 [141]; 1133, 1155, 1174 [69]; 1175 [102] Sersouri, A. 798, 835 [158]; 1019, 1069 [123] Shapiro, J.H. 470, 471, 496 [177–180]; 1104, 1113, 1120, 1129 [59,60,79,80]; 1683, 1684, 1702 [44–46] Sharpley, C. 1404, 1420 [21] Sharpley, R. 505, 515, 528 [10]; 1133, 1136, 1147, 1172 [4,5]; 1404, 1407, 1420 [4] Shashkin, Yu.A. 640, 669 [158] Shatalova, O.A. 917, 937 [92] Shcherbina, M. 349, 361 [45]; 364 [133] Shelah, S. 1255, 1297 [50,51]; 1759, 1814 [281] Shevchyk, V.V. 1817, 1818 [1a] Shields, A.L. 322, 364 [135]; 559 [28]; 1104, 1113, 1120, 1128 [23]; 1700, 1701 [3] Shiga, K. 440, 490 [2] Shilov, G.E. 902, 908, 920, 936 [31] Shimogaki, T. 1175 [105] Shiryaev, A.N. 374, 391 [49] Shlyakhtenko, D. 360, 365 [151]; 1447, 1448, 1458 [92] Shreve, S. 374, 391 [33] Shteinberg, A. 1172 [13] Shultz, F. 627, 665 [3] Shura, T.J. 136, 157 [28]; 276, 313, 313 [18]; 812, 830 [27]; 1065, 1066 [31]; 1094, 1096 [7]; 1255, 1295 [9] Shvartsman, P. 1373, 1383, 1420 [15,16] Sidorenko, N.G. 1412, 1415, 1416, 1418, 1421 [51]; 1422 [87] Sierpinski, W. 1019, 1068 [88]; 1560, 1602 [45]; 1722, 1741 [54] Silver, J. 1049, 1069 [124] Silverstein, J.W. 349, 353, 365 [152] Silverstein, M.L. 266 [44] Simon, B. 439, 496 [181]; 1465, 1516 [182] Simon, P. 576, 597 [63]; 1747, 1755, 1814 [258,282] Simoniˇc, A. 105, 121 [127,128]; 535, 559 [37] Simonovits, M. 1625, 1633 [30]; 1634 [39] Simons, S. 644, 669 [159] Sina˘ı, Ya.G. 349, 365 [153] Sinclair, A. 1431, 1440–1442, 1444, 1445, 1456 [15,20,21] Singer, I. 1020, 1069 [125]; 1639, 1669 [49]; 1745, 1777, 1814 [283,284] Singer, I.M. 333, 334, 363 [99]; 859, 869 [35] Sisson, P. 1114, 1129 [81] Sjölin, P. 595, 597 [64] Skorochod, A.V. 1526, 1545 [52]

Skorohod, A.S. 384, 391 [37] Sledd, W.T. 1681, 1702 [47] Smickih, S.V. 99, 122 [146] Šmídek, M. 798, 833 [97]; 1752, 1758, 1810 [149] Smith, B. 576, 577, 596 [49] Smith, K.T. 535, 558 [4] Smith, M. 1777, 1814 [285] Smith, R. 1431, 1440, 1442, 1456 [16]; 1458 [81] Smoluchowski, M. 369, 391 [50] Smolyanov, O.G. 1527, 1528, 1545 [38,49–51] Šmulyan, V.L. 793, 835 [161] Snobar, M.G. 463, 493 [81]; 929, 936 [48] Sobczyk, A. 1454, 1458 [98]; 1553, 1602 [65]; 1705, 1707, 1716, 1739 [8]; 1741 [66] Sobecki, D. 1142, 1173 [36] Sobolev, A.V. 87, 120 [80] Sobolev, S.L. 1365, 1373, 1398, 1422 [88] Sobolevsky, P.E. 87, 120 [79] Sohr, H. 250, 267 [83,84]; 269 [127] Sokolov, G.A. 1767, 1814 [286] Solecki, S. 1532, 1545 [53] Sölin, P. 223, 231 [15] Solonnikov, V.A. 1415, 1422 [89] Somasundaram, S. 1534, 1537, 1545 [37] Sondermann, D. 387, 390 [27] Soria, J. 1161, 1172 [16] Soshnikov, A.B. 346, 349, 365 [153,154] Souslin, M.M. 1019, 1069 [126] Spalsbury, A. 545, 559 [38] Sparr, G. 1151, 1175 [106,107]; 1466, 1468, 1514 [135] Speicher, R. 1491, 1511 [19] Srinivasan, T.P. 702, 706 [48] Srivatsa, V.V. 1792, 1814 [287] Starbird, T. 129, 131, 147, 157 [31,36]; 822, 830 [15]; 1115, 1128 [24]; 1652, 1668 [14]; 1754, 1806 [22] Stegall, C. 129, 158 [65]; 483, 496 [182]; 798, 835 [162–164]; 1020, 1069 [127]; 1252, 1296 [38]; 1532, 1545 [35]; 1591, 1600, 1601 [39]; 1602 [66]; 1717, 1727, 1740 [40]; 1752, 1758, 1766, 1803, 1812 [196]; 1814 [288–290] Stegenga, D.A. 1681, 1702 [47] Stein, E.M. 215, 231 [40]; 244, 267 [72]; 269 [134–136]; 505, 532 [135]; 894, 897 [35]; 1363, 1371, 1373, 1398, 1399, 1403, 1405, 1410, 1422 [90–92]; 1531, 1546 [54]; 1688, 1696, 1701 [11]; 1702 [48] Stephani, I. 452, 491 [25]; 958, 959, 973, 973 [3] Stephenson, K. 905, 938 [122] Stepr¯ans, J. 1255, 1297 [51]; 1759, 1814 [281] Stern, R. 412, 434 [15,16] Sternfeld, Y. 618, 625, 668 [122]; 669 [160] Stezenko, V.Ya. 88, 118 [32] Stiles, W.J. 1116, 1129 [82]

Author Index Stinespring, W. 1461, 1516 [183] Størmer, E. 620, 669 [161,162] Stout, E.L. 675, 706 [49] Stout, W.F. 1610, 1634 [57] Strassen, V. 1188, 1190, 1198 [6] Straszewicz, S. 628, 669 [163] Stratila, S. 1463, 1516 [184,185] Strichartz, R. 217, 231 [41] Stricker, C. 374, 385, 389, 390 [1,6,7,10]; 391 [51] Strömberg, J.O. 575, 597 [65]; 1531, 1546 [54] Study, E. 712, 774 [45] Sucheston, L. 260, 267 [69]; 1036, 1066 [27,28]; 1074, 1096 [6]; 1304, 1306, 1307, 1310, 1330 [4,5] Sudakov, V.N. 756, 778 [179] Sukochev, F. 515, 531 [107]; 1477, 1478, 1484, 1492, 1506–1509, 1511 [37–40]; 1512 [41,50,55–58,65,66]; 1513 [82]; 1516 [186–188] Sullivan, F.E. 1752, 1810 [130] Sunder, V.S. 106, 119 [67]; 327, 365 [155] Swart, J. 484, 491 [34] Synnatzschke, J. 92, 106, 121 [129] Szankowski, A. 275, 283–285, 305, 316 [91–94]; 489, 496 [183]; 735, 766, 776 [115]; 778 [180]; 1440, 1458 [99]; 1478, 1516 [189] Szarek, S.J. 146, 159 [98]; 273, 295, 299, 301, 302, 316 [95]; 322, 331, 345, 358, 359, 361 [24]; 363 [95]; 365 [156–164]; 460, 496 [184]; 580, 597 [66]; 736, 737, 749, 765–767, 770, 773 [19,35]; 774 [55]; 778 [181–186]; 779 [187]; 859, 868 [16]; 870 [57]; 969, 973 [6]; 1158, 1175 [108]; 1212, 1213, 1218, 1220–1222, 1224, 1225, 1235, 1236, 1239, 1243, 1244, 1244 [4]; 1245 [29]; 1246 [46–55]; 1254, 1297 [52,53]; 1305, 1332 [64,65]; 1525, 1545 [39]; 1681, 1688, 1702 [49] Szego, G. 907, 938 [111] Szlenk, W. 805, 835 [165]; 1020, 1069 [128]; 1560, 1561, 1602 [67]; 1719, 1741 [67]; 1765, 1813 [242] Szulga, J. 1617, 1633 [31] Tacon, D.G. 1789, 1815 [291] Takesaki, M. 1463, 1477, 1516 [190] Talagrand, M. 96, 121 [130]; 181, 194 [37]; 201, 231 [42]; 338, 344, 346, 350, 354, 359, 364 [112]; 365 [165,166]; 366 [167–169]; 472, 473, 493 [97]; 496 [185]; 523, 530 [87]; 633, 665 [20]; 737, 740, 755, 756, 766, 769, 776 [109]; 778 [186]; 779 [188,189]; 828, 835 [166]; 844, 845, 848, 849, 869 [41]; 870 [58,59]; 885, 898 [36]; 925, 939 [123]; 1019, 1020, 1047, 1066 [25]; 1067 [61]; 1069 [129]; 1117, 1129 [83]; 1179, 1180, 1183, 1185, 1186, 1188, 1190–1196, 1199 [30–32];

1845

1200 [44–47]; 1221, 1246 [53]; 1614–1617, 1619, 1623–1625, 1630, 1632, 1634 [35,58–63]; 1745, 1748, 1758, 1763, 1767, 1775–1777, 1790, 1791, 1794, 1795, 1802, 1806 [30]; 1809 [119]; 1815 [292–297] Tam, S.C. 1106, 1129 [84] Tamarkin, J. 968, 974 [13] Tang, W.K. 793, 835 [167]; 1019, 1068 [83,84]; 1782, 1801, 1815 [298,299] Tao, T. 222, 223, 231 [29,30,43,44] Taylor, A.E. 265 [17] Taylor, B.A. 1608, 1632 [6] Taylor, P.D. 470, 496 [180]; 1684, 1702 [46] Temme, D. 1700, 1702 [50] Terenzi, P. 1784, 1815 [300–302] Terevcak, I. 1107, 1128 [30] Terp, M. 1464, 1466, 1470, 1473–1476, 1480, 1516 [191,192] Thompson, A.C. 664, 666 [46] Thomson, J.E. 341, 364 [128] Thorbjørnsen, S. 358, 363 [88,89]; 1449, 1457 [47]; 1819, 1820 [4a] Tišer, J. 1529, 1531, 1533, 1541, 1543, 1545 [36,45,46]; 1546 [55] Todorˇcevi´c, S. 1019, 1069 [130]; 1745, 1748, 1802, 1815 [303–305] Tokarev, E.V. 527, 531 [116] Tolias, A. 1065, 1066 [12]; 1255, 1295 [3] Tomas, P. 215, 231 [45] Tomczak-Jaegermann, N. 43, 45–47, 51, 53, 60, 64, 66, 83 [20]; 84 [36]; 266 [55]; 277, 279, 287, 315 [57,72,73,78]; 316 [96]; 358, 364 [117]; 365 [164]; 459, 464, 471, 473, 496 [186–188]; 710, 722, 726, 736, 743, 749–751, 756, 757, 765–767, 770, 774 [49,60]; 776 [116,121]; 777 [155,156]; 779 [187,190–195]; 793, 831 [33]; 835 [168]; 841, 850, 853, 854, 858, 864, 868 [17]; 869 [25]; 870 [60]; 930–934, 937 [67–69]; 1060, 1063, 1065, 1068 [97]; 1069 [105,131]; 1103, 1128 [17]; 1216, 1222–1226, 1230–1234, 1236, 1240, 1242–1244, 1245 [11,19,30–35]; 1246 [40,54–56]; 1252, 1296 [34,35]; 1305, 1306, 1330 [14]; 1332 [65,66]; 1350–1358, 1359 [30]; 1360 [34,45,46,52,53]; 1392, 1423 [93]; 1477, 1480, 1481, 1483, 1484, 1512 [45,70]; 1516 [193–195]; 1735–1737, 1741 [68]; 1745, 1815 [306] Tonge, A. 47, 55, 60, 65, 66, 72, 83 [9]; 455, 458, 459, 464, 466, 471, 474, 491 [35]; 518, 529 [40]; 675, 681, 690, 705 [15]; 867, 868 [18]; 879, 881, 882, 887, 897 [10]; 1222, 1244 [5]; 1378, 1387–1389, 1420 [22]; 1452, 1456 [24]; 1710, 1711, 1724, 1736, 1739 [16] Toninelli, F.L. 1819, 1820 [3] Töplitz, O. 1667, 1668 [32]

1846

Author Index

Topping, D. 1170, 1175 [89] Torrea, J.L. 250, 269 [131]; 505, 529 [53] Torunczyk, H. 799, 835 [169]; 1797, 1799, 1811 [159]; 1815 [307] Tracy, C.A. 349, 358, 366 [170,171] Tran Van An 1112, 1127 [5] Trautman, D.A. 519, 526, 527, 528 [28,29]; 529 [49] Triana, M.A. 1116, 1117, 1124, 1128 [16] Triebel, H. 463, 495 [138]; 958, 973 [4]; 1363, 1400, 1404, 1423 [94]; 1515 [142] Troitsky, V.G. 102, 121 [131,132]; 556, 558, 559 [39,40] Troyanski, S.L. 635, 669 [164]; 792, 805, 814, 821, 832 [83,84]; 834 [122,132,139]; 835 [170]; 1759, 1767, 1778, 1781–1785, 1787, 1788, 1790–1792, 1798, 1805, 1808 [86]; 1809 [120–122]; 1810 [124]; 1811 [184]; 1812 [209–214]; 1815 [308–314] Tsarpalias, A. 1019, 1050, 1058, 1064, 1065, 1066 [11]; 1755, 1812 [223] Tsay, J. 348, 360 [17] Tsirelson, B.S. 136, 140, 159 [99]; 276, 316 [97]; 799, 812, 835 [171]; 1060, 1069 [132]; 1250, 1255, 1297 [54]; 1335, 1342, 1343, 1360 [54]; 1649, 1669 [54] Tsolomitis, A. 722, 774 [70] Turett, B. 505, 528 [31] Turpin, P. 1101, 1113, 1119, 1120, 1122–1124, 1129 [85,86]; 1130 [87] Tzafriri, L. 7, 10–14, 18, 21–27, 30, 33, 50, 51, 78–80, 83 [14,15]; 84 [41]; 87, 92, 120 [86,87]; 121 [133]; 125, 133–135, 140, 143, 146, 147, 158 [50,69,70]; 159 [71,72,100]; 257, 258, 268 [106,107]; 276, 279, 281, 282, 284, 289, 300, 301, 305–307, 309, 313 [12,15]; 315 [64–67]; 336, 337, 361 [42–44]; 505, 511, 514, 515, 518, 523, 529 [51]; 530 [70,94]; 531 [95–99]; 574, 577, 580, 596 [42,43]; 681, 705 [33]; 784, 834 [130]; 853–855, 859, 860, 862–864, 866, 867, 868 [7,9–13]; 869 [25,29]; 870 [61]; 906, 928, 936 [46]; 937 [77]; 965, 974 [23]; 1116, 1119, 1129 [64]; 1130 [88]; 1151, 1155, 1167, 1174 [72]; 1245 [21]; 1252, 1256, 1260, 1263, 1264, 1285, 1296 [39]; 1304, 1305, 1331 [40]; 1336, 1359 [25]; 1380, 1383, 1387, 1420 [32]; 1421 [60]; 1480, 1481, 1514 [115]; 1639, 1641–1643, 1646, 1647, 1649–1652, 1654–1658, 1660–1664, 1667 [6]; 1668 [11,27,36–38]; 1708, 1711, 1714, 1715, 1724, 1727, 1740 [48–50]; 1745, 1748, 1757, 1759, 1764–1768, 1771, 1775, 1800, 1801, 1804, 1805, 1812 [197,198] Uglanov, A.V. 1528, 1546 [56]

Uhl, J.J. 35, 36, 38–40, 83 [8]; 106, 107, 119 [51]; 259, 260, 263, 266 [56]; 479, 480, 483, 484, 491 [37]; 558 [14]; 793, 831 [50]; 1396, 1420 [23] Uriz, Z. 1103, 1127 [9]

Vaaler, J.D. 175, 194 [38] Valdivia, M. 1755, 1763, 1764, 1766, 1767, 1771, 1784, 1788, 1790, 1792, 1800, 1801, 1803, 1812 [211–213]; 1813 [234,235]; 1815 [315–321] Valette, A. 1450, 1458 [100] van Dulst, D. 515, 529 [50] van Mill, J. 603, 669 [165] van Rooij, A.C.M. 91, 121 [118,119] Vanderwerff, J. 434 [36]; 828, 835 [172]; 1749, 1766, 1767, 1771, 1797, 1800, 1806 [27]; 1815 [322–324] Varadhan, S.R.S. 1188, 1200 [48] Varga, R. 555, 559 [41] Vargas, A. 222, 231 [44] Varopoulos, N.Th. 1363, 1423 [95] Vašák, L. 1764, 1770, 1815 [325] Vaserstein, L. 916, 917, 938 [93] Vasin, A.V. 905, 939 [124] Veech, W.A. 1454, 1458 [101]; 1716, 1741 [69] Vega, L. 222, 231 [44] Veksler, A.I. 87, 89, 119 [39,40] Venakides, S. 349, 362 [70] Venni, A. 250, 267 [60–62] Vera, G. 1807 [33] Vershynin, R. 1784, 1815 [326] Vesely, L. 650, 655–657, 667 [71]; 669 [166] Vilenkin, N.Ya. 919, 936 [32] Villa, R. 716, 772 [11]; 1607, 1632 [4] Villadsen, J. 627, 669 [167] Villani, C. 350, 364 [129] Vincent-Smith, G.F. 612, 669 [168] Virasoro, M.A. 354, 364 [123] Vlasov, L.P. 663, 670 [169]; 1823, 1823 [1a] Vodop’yanov, S.K. 1372, 1420 [35] Voiculescu, D. 323, 326, 330–332, 345, 347, 356–360, 360 [9,10]; 366 [172–178]; 1445, 1451, 1458 [102,103]; 1466, 1487, 1516 [196] von Koch, H. 451, 493 [87] von Neumann, J. 439, 447, 484, 494 [115,116]; 495 [169]; 670 [170]; 906, 936 [47] von Weizsäcker, H. 1527, 1545 [50,51] Vu, V.H. 346, 363 [105]; 1819, 1819 [1a] Vukoti´c, D. 1682, 1702 [51] Vulikh, B.Z. 87, 88, 91, 92, 106, 107, 120 [75,76]; 121 [134–136] Vuza, D.T. 97, 98, 121 [110,111]

Author Index Wachter, K.W. 343, 353, 366 [179] Wade, W.R. 576, 597 [63] Waelbroeck, L. 1113, 1123, 1130 [89,90] Wage, M. 1751, 1752, 1754–1756, 1780, 1806 [21] Wagner, G. 769, 779 [196]; 925, 939 [125] Wagner, R. 1060, 1065, 1069 [105]; 1352, 1356, 1360 [35,46,55] Walsh, B. 95, 121 [137] Wang, G. 215, 230 [1,2]; 259, 265 [9,10]; 269 [137] Wang, J.-K. 702, 706 [48] Wang, X. 1521, 1544 [11] Wassermann, S. 332, 359, 366 [180,181]; 1431, 1449, 1452, 1458 [104,105] Wattbled, F. 1471, 1516 [197] Webster, C. 1438, 1456 [42] Wegmann, R. 358, 366 [182] Weil, W. 906, 911, 936 [35]; 938 [120]; 939 [126] Weinberger, W.F. 339, 362 [68] Weis, L. 445, 446, 496 [189]; 860, 870 [62]; 1589, 1602 [68] Weiss, B. 619, 620, 667 [76] Weiss, G. 336, 361 [30]; 505, 532 [135]; 859, 868 [3]; 1133, 1158, 1159, 1161, 1170, 1171, 1172 [20]; 1173 [37,49]; 1175 [101,109,110]; 1363, 1371, 1399, 1403, 1406, 1420 [18]; 1422 [92]; 1682, 1691–1693, 1701 [12] Wells, J.H. 906, 939 [127] Wenzel, J. 250, 269 [122,138,139]; 471, 476, 495 [139]; 590, 597 [59] Wermer, J. 673–675, 704 [1]; 706 [50] Werner, D. 143, 158 [60]; 821, 829, 833 [93,114]; 1046, 1067 [65]; 1357, 1359 [21]; 1745, 1810 [142] Werner, E. 798, 835 [158] Werner, J. 116, 118 [24] Werner, W. 829, 833 [93]; 1046, 1067 [65]; 1745, 1810 [142] West, G. 1496, 1497, 1499, 1514 [124–126] Weston, A. 1127, 1130 [91] Weyl, H. 326, 366 [183]; 439, 496 [190]; 948, 974 [34] Wheeler, R.F. 636, 666 [49]; 1745, 1789, 1802, 1808 [67] White, M.C. 968, 974 [35] Whitﬁeld, J.H.M. 413, 434 [37]; 792, 798, 812, 832 [63,83,84]; 834 [125]; 835 [173]; 1766, 1767, 1784, 1785, 1791, 1795, 1797, 1798, 1800, 1803, 1808 [87,88]; 1809 [118,120–122]; 1812 [189]; 1813 [239]; 1815 [324] Whitney, H. 484, 496 [191]; 1373, 1423 [96] Wickstead, A.W. 91, 93, 95, 96, 105, 118 [13,18–21]; 121 [138]; 122 [139–142] Widder, D.V. 605, 670 [171] Widom, H. 349, 358, 366 [170,171] Wiegerinck, J. 1700, 1702 [50]

1847

Wielandt, H.W. 328, 363 [96] Wiener, N. 1495, 1497, 1516 [198] Wigner, E.P. 342, 358, 366 [184,185]; 1221, 1246 [57] Williams, D. 376, 391 [45] Williams, L.R. 906, 939 [127] Williamson, J.H. 1113, 1130 [92] Willinger, W. 374, 390 [8] Willis, G.A. 286, 309, 314 [34]; 316 [98] Wils, I.M. 334, 366 [186] Winkler, S. 1438, 1456 [43] Wittstock, G. 1428, 1458 [106] Wo-Sang Young 576, 597 [67] Wodzicki, M. 1171, 1173 [37] Wojciechowski, M. 1366, 1367, 1377, 1380, 1381, 1383, 1387, 1392, 1400–1403, 1409, 1411–1413, 1417–1419, 1420 [6]; 1422 [76–81]; 1423 [97–100] Wojtaszczyk, P. 7, 13, 16, 58, 66, 83 [21]; 84 [42]; 146, 149, 157 [34]; 159 [82]; 274, 279, 311, 313 [16]; 315 [79,82]; 459, 471, 477, 496 [192]; 518, 532 [136]; 566, 567, 569, 570, 573, 575, 577, 578, 580, 596 [34]; 597 [68–74]; 674, 675, 686, 687, 689, 690, 702, 703, 706 [51]; 841, 870 [63]; 905, 936 [30]; 1119, 1129 [53]; 1130 [93]; 1389, 1396, 1423 [101]; 1600, 1602 [49]; 1644, 1645, 1664–1667, 1667 [1]; 1668 [19,21,30]; 1669 [55,56]; 1677, 1682, 1691, 1692, 1696, 1697, 1701 [15,23]; 1702 [52–54]; 1745, 1800, 1816 [327] Wolenski, P. 412, 434 [16] Wolfe, J.E. 1599, 1602 [69]; 1714, 1741 [70–72] Wolff, T. 221, 223, 231 [46]; 232 [47]; 1407, 1423 [102] Wolfson, H. 767, 771, 772, 773 [40]; 777 [153,154] Wolniewicz, T. 704, 706 [52]; 1681, 1688, 1694, 1702 [49,55] Wolnik, B. 573, 596 [28] Wong, Y.C. 87, 122 [143] Wood, G.V. 278, 314 [55] Woodroofe, M. 1192, 1199 [33]; 1629, 1634 [37] Wo´zniakowski, K. 577, 578, 597 [69]; 1371, 1423 [103] Wulbert, D. 626, 668 [123]; 1600, 1602 [43] Xu, Q. 477, 495 [150]; 700, 705 [32]; 879, 898 [37]; 1133, 1174 [67,68]; 1175 [98]; 1405, 1406, 1421 [52]; 1438, 1447, 1457 [54]; 1458 [107]; 1465, 1468, 1472, 1483–1485, 1487, 1491–1493, 1497, 1503, 1507–1510, 1513 [99–102]; 1515 [156,157,169]; 1516 [188,199–203]; 1517 [204–206] Yahdi, M. 828, 835 [174] Yan, J.A. 380, 391 [52]

1848

Author Index

Yeadon, F.J. 1517 [207] Yin, Y.Q. 344, 353, 358, 360 [18,19]; 361 [20]; 366 [187] Yood, B. 558 [12]; 559 [42] Yor, M. 372, 373, 378, 379, 391 [43] Yosida, K. 443, 496 [193] Yost, D. 793, 830 [13]; 835 [175]; 1745, 1757, 1759, 1765, 1789, 1799, 1803, 1807 [35]; 1813 [248]; 1816 [328,329] Yurinskii, V.V. 1182, 1200 [49,50]; 1612, 1634 [64] Zaanen, A.C. 87, 89, 99, 106, 110, 120 [93,94]; 122 [144,145] Zabre˘ıko, P.P. 87, 99, 120 [79]; 122 [146] Zachariades, Th. 139, 157 [12] Zahariuta, V.P. 1667, 1669 [57] Zahorski, Z. 1539, 1546 [57] Zaidenberg, M.G. 901, 939 [130] Zajíˇcek, L. 435 [63]; 798, 828, 832 [64]; 833 [97]; 834 [151]; 1532, 1533, 1536, 1545 [47]; 1546 [58,59]; 1752, 1758, 1786, 1808 [89]; 1810 [149] Zalgaller, V.A. 710, 712, 726, 774 [43] Zanco, C. 662, 667 [70] Zastavnyi, V.P. 909, 939 [128,129] Zeitouni, O. 346, 363 [87] Zelazko, W. 1125, 1130 [94,95] Zelený, M. 1530, 1546 [60] Zemanek, J. 959, 974 [25] Zhang, G. 177, 194 [39]; 918, 919, 939 [131–133] Zhao, D. 1019, 1068 [83] Zhong, Y. 101, 119 [48] Zhou, X. 349, 362 [70] Zhu, K. 471, 494 [107]; 1677, 1682, 1683, 1701 [31]; 1702 [56]

Zhu, Q. 418, 433 [9] Ziemer, W.P. 1367, 1423 [104]; 1526, 1546 [61] Zimmermann, F. 250, 269 [140] Zinger, A.A. 905, 939 [134] Zinn, J. 52, 84 [34]; 129, 158 [55]; 176, 194 [36]; 1181, 1182, 1185, 1186, 1190–1192, 1194, 1195, 1198 [3,8]; 1199 [9–11,23,33]; 1200 [42,51]; 1610, 1624, 1629, 1633 [29]; 1634 [37,54,55] Zinnmeister, M. 1019, 1069 [133] Zippin, M. 19, 84 [43]; 131, 134, 140, 141, 146, 149, 158 [54,56,57]; 159 [101]; 274, 277, 279, 290, 293, 294, 296, 300, 314 [50]; 316 [99]; 581, 596 [33]; 829, 833 [108]; 1242, 1245 [17]; 1454, 1458 [108]; 1596, 1598–1600, 1601 [34–36]; 1602 [70,71]; 1640, 1641, 1650, 1668 [39]; 1669 [58]; 1707, 1714, 1717, 1721, 1726, 1731–1733, 1740 [28,30–33]; 1741 [73–77] Zizler, V. 6, 7, 13, 20, 33, 34, 36, 41, 42, 57, 63, 83 [6,11,12]; 84 [44]; 406, 408, 409, 413, 415, 434 [24,25,37]; 476, 491 [33]; 644, 666 [41]; 783, 789, 792, 793, 798, 799, 805, 812–814, 820–822, 828, 831 [45,46]; 832 [60–65,83,84]; 833 [90,92,102]; 835 [173,176]; 1537, 1544 [18]; 1600, 1602 [72]; 1745, 1747–1769, 1771–1792, 1795–1802, 1804, 1805, 1807 [55–58]; 1808 [75–85,88–91]; 1809 [92,118,120–122]; 1810 [129,140]; 1811 [159–164]; 1813 [239]; 1815 [324]; 1816 [330–332] Zlatov, P. 1107, 1128 [30] Zobin, N. 1373, 1423 [105,106] Zolotarev, V.M. 907, 939 [135] Zsidó, L. 1463, 1498, 1516 [185]; 1517 [208] Zvavitch, A. 840, 845, 869 [48]; 870 [56,64] Zygmund, A. 214, 232 [48]; 245, 266 [45,46]; 269 [141]; 522, 531 [105]; 678, 706 [53]; 875, 898 [38]; 1175 [111]; 1370, 1412, 1423 [107]

Subject Index approximation property (AP), 12, 97, 275, 488, 1478 – bounded (BAP), 12, 274 – bounded compact, 308 – commuting bounded (CBAP), 12, 291 – commuting compact, 310 – commuting metric, 291 – commuting unconditional metric, 295 – compact, 308 – metric (MAP), 12, 287, 488 – metric uniform, 307 – positive, 286 – stochastic, 1811 – unconditional, 291, 295 – uniform (UAP), 60, 305 – uniform projection (UPAP), 305 arbitrarily distortable, 1063 arithmetic diameter, 891, 893 Asplund space, 410, 411, 795, 1140, 1537, 1754 associate space, 512 asymptotic p space, 1060, 1063 asymptotic c0 space, 1060, 1063 asymptotic freeness of matrices, 356, 359 asymptotic models, 1358 asymptotic order, 320 asymptotic set, 1250 asymptotic structure, 1352 atomic, 1106 atomic space, 1103 Auerbach lemma, 45 automorphism, 3 Azuma inequality, 1317, 1610

α-th oscillation of f , 1031 A-convex, 404 A-martingale, 402 Aδ -set, 404 w∗

A , 1746 Ap -condition, 1638 absolutely continuous vector measure, 39 abstract Lp space, 22 abstract M space, 22 adjoint – Banach ideal, 457 – ideal norm, 457 admissible class of perturbations, 395, 398, 399 admissible cones of perturbations, 401 Aldous theorem, 515 Alexandrov theorem, 420 Alexandrov–Fenchel inequalities, 727 algebra – C ∗ , 627 – Jordan–Banach, 627 allocation, 113 almost isometric embedding, 925 almost sure convergence, 343, 352, 353, 357, 360 Amir–Cambern theorem, 702 analytic continuation, 908, 921 analytic decomposition of unity, 699 analytic distribution, 921 analytic family, 1158 analytic function on Banach space, 675, 806 – m-homogeneous polynomials, 676 analytic map, 806 analytic operator valued functions, 1496 analytic Radon–Nikodým property (ARNP), 236, 262, 638, 1484 analytic subset, 1010 analytic UMD, 1495 Androulakis–Odell lemma, 1052 angelic compact, 1748 anisotropic Sobolev space, 1408 approximate identity, 882 approximate ultraﬁlters, 1086, 1087 approximating sequence, 288, 292, 296, 297 approximation numbers, 452, 945

β-differentiable, 406 þ Bp,q (Rn ), 1400 (B(X, Y ), τ ), 281, 282 BV(k) (Ω), 1366 B2 -sequences, 917 Baire space, 1786 Baire’s space Σ := NN , 1760 balayage, 401, 610 Banach couple, 74, 1133, 1141, 1142 1849

1850 Banach function spaces, 1141 Banach ideal, 448 – adjoint, 457 Banach lattice, 21, 60, 89, 125, 1696 – λ-lattice injective ((λ-L)-injective), 1715 – p-concave, 27, 504, 855, 964 – p-concavity constant, 27 – p-convex, 27, 504, 841, 855, 964 – p-convexity constant, 27 – absolute value, 21 – discrete, 1715 – functional calculus, 26 – lattice injective (L-injective), 1715 – order complete, 23 – order continuous, 23, 1716 – order continuous, functional representation, 24 – symmetric, 21, 81 Banach space, 1062, 1063 – B-convex, 52, 474, 894 – C(K), 19, 1547 – Cσ (K), 1600 – c1 (Σ × Γ ), 1770 – D, 1756 – JL2 of Johnson and Lindenstrauss, 1757 – K convexity constant, 53 – K-convex, 53, 1320 – L∞ (μ), 14 – Lp (μ), 1 p < ∞, 13 – Lp (μ, X), 1 p < ∞, 37 – cotype q, 48, 1307, 1315 – genus, 1650 – Lorentz space, 21 – Orlicz space, 21 – predual of L1 (μ), 1599 – predual of C(K) and L1 (μ), 20, 625, 657 – quasi, 1665 – reﬂexive, 10, 31, 443 – Schlumprecht’s space S, 1335, 1344 – separable conjugate, 10, 38, 635, 639 – smooth, 30 – Sobolev space, 585 – stable, 515, 519 – strictly convex, 30 – superreﬂexive, 33, 56, 235, 237, 479 – Tsirelson’s space, 1335 – Tsirelson’s space T , 1342 – type p, 48, 845, 1306, 1315 – uniformly convex, 31 – uniformly smooth, 31 – weakly sequentially complete, 4 – with enough symmetries, 74 Banach–Dieudonné theorem, 381, 1751 Banach–Mazur compactum, 765

Subject Index Banach–Mazur distance, 3, 43, 858, 862, 924, 1626, 1627 Banach–Saks property, 444, 1581 Bang lemma, 185, 188, 192 barrier, 638 – PSH, 638 – strong, 638 barycenter, 604 barycentric calculus, 1048 basic sequence, 7, 1104 basis, 7, 274, 287, 585 – K-equivalence, 8 – asymptotically non-equivalent, 1656 – bimonotone, 7 – boundedly complete, 10, 275, 285 – conditional, 1639 – constant, 7 – equivalence, 8 – Faber–Schauder, 9, 564 – Haar, 9, 125, 250 – Markuschevich, 13 – monotone, 7, 255, 256, 277 – perfectly homogeneous, 1650 – problem, 273 – quasi-equivalent, 1667 – shrinking, 10, 278, 294, 299 – subsymmetric, 11, 1650 – symmetric, 11, 837, 854–858, 1637 – unconditional, 9, 126, 251, 274, 301, 855, 858, 1637 – unconditionally monotone, 9 – universal, 11 basis constant, 1675 basis constant of n-dimensional space, 1212 – asymptotically sharp estimate, 1212 Beck’s convexity, 1181 Berg technique, 331 Bernoulli selectors, 338 Bernstein theorem, 606 Berry–Esséen theorem, 934 Besicovitch set, 198, 220 Besov space, 575, 585, 971, 1400 Bessaga–Pełczy´nski theorem, 648 Bessel inequality, 950, 951 Bessel process, 378, 384 Beurling–Ahlfors transform, 236, 259 biorthogonal functionals, 7 biorthogonal system, 1765 biquasitriangular operators, 332 Bishop–Phelps theorem, 34, 385, 395, 396, 641, 1750 Black–Scholes model, 371 Blaschke–Santaló inequality, 728 block basis, 7, 131, 134

Subject Index BMO, 389, 591, 695 Bochner integrable functions, 36, 969 Bochner integral, 36, 402 Bochner theorem, 906, 911 Bochner–Riesz multiplier, 222 Borell lemma, 717 Borwein–Preiss theorem, 396 boundary, 639 – Choquet, 640 – minimal, 640, 656 – of the spectrum, 1308 boundedly complete ﬁnite dimensional decompositions, 299 Bourgain algebra, 687 Bourgain projections, 699 Bourgain theorem, 687, 689 Bourgain–Stegall minimization principle, 414 Boyd index, 257, 258, 514, 527, 1151, 1155, 1663 Brascamp–Lieb inequality, 164 Brenier map, 713 Bröndsted and Rockafellar theorem, 395 Brown–Douglas–Fillmore theorem, 320, 323, 331 Brownian motion, 369 Brunn–Minkowski inequality, 178, 711, 1606, 1608 Brunn–Minkowski theorem, 921 bump function, 400, 794, 1746 Busemann–Petty problem, 901, 918, 919 C (k) (Ω), 1364 (k) C0 (Ω), 1364 C0S (Ω), 1408 C1 , 285 C2 , 295, 301 Cp , 279, 280 conv K, 1746 C[0, ω1 ], 1756 C S (Ω), 1408 C 1 function, 37 C n function, 37 C ∞ function, 37 C(Tn )-module, 1393 (c)-sequence, 1023 c0 -index theorem, 1039 c0 -theorem, 1028 c.b. multilinear maps, 1442 C.C.C. (countable chain condition), 1774 Calderón couple, 80, 1133, 1144, 1150–1153 Calderón–Mitjagin theorem, 1150 Calderón–Zygmund singular integral operator, 245, 894, 1405 Calkin algebra, 323, 334 Calkin theorem, 447

1851

Cameron–Martin–Girsanov theorem, 373 canonical embedding, 4, 21, 1367, 1409 Cantor–Bendixson index, 1062 Carathéodory theorem, 602, 913–915, 927 CBAP, 1440 ˇ Cech complete ball, 1802 central limit theorem, 850, 1188, 1195 change of density, 1630 characteristic function, 6 Chebyshev’s inequality, 1611 Choquet, Bishop–de Leeuw theorem, 610 Choquet, integral representation, 607 Christensen–Sinclair factorization, 1442 Ciesielski–Pol CP space, 1758 Clarkson inequality, 479, 1479 closed – hull of an ideal, 441 – ideal, 441 coanalytic set, 1011 codomain, 440 coefﬁcient converging, 1030 coefﬁcient problem, 163, 183, 189 Cohen idempotent theorem, 874 column vectors, 1431 commodity space, 113 commutant, 539, 540 commutator, 557 commutator estimate, 1159, 1167 commuting projections, 1323 compact family of ﬁnite subsets of N, 1049 compact operator, see operator, compact compactiﬁcation – Bohr, 645, 876 – Stone–Czech, 645 compatible couple, 1404 complementary function, 512 complemented subspaces, 4, 129, 143, 155, 837, 839, 863–868 complemented subspaces of H1 (D), 1698 complemented subspaces theorem, 928, 965, 967 complete contraction, 1429 complete isometry, 1429 completely bounded, 1427, 1428 completely isometric, 1429 completely isomorphic, 1429 complex convexity, 637 complex interpolation, 1158, 1160, 1407 complex interpolation method, 1138 complex interpolation spaces, 964, 1142 component of a vector in a lattice, 23 component of an operator ideal, 441 composition inequality, for p-summing operators, 65 composition operator, 470, 1683

1852

Subject Index

concentration, 744, 1177, 1603, 1605, 1606, 1621, 1622, 1629, 1630, 1632 concentration inequality, 737, 1186, 1606, 1614, 1617, 1623–1625, 1627, 1628, 1630 conditional expectation, 1610 cone generating, 87 constructivity, 330, 332, 358–360 continuum hypothesis, 1746 contraction principle, 848, 850 convergence in distribution, 343, 355 convergence in probability, 343, 352 convex block basis, 1024 convex body, 164, 169, 174, 646, 710 convex function, 1535, 1614, 1615 convex unconditionality, 1064 convolution inequalities, 163 corona problem, 674 Corson compact, 1754 coset ring, 874, 875 Cotlar’s trick, 1499 cotype, 48, 126, 286, 505, 526, 882, 892, 1177, 1188, 1307, 1315 – Gaussian, 472, 1710, 1738 – Haar, 477 countable tightness, 1748 coupon collector’s problem, 847 covering number, 756 critical point theory, 423 crossnorm, 485 – general, 486 – reasonable, 485 – uniform, 486 CSL algebra, 340 cubature formulas, 916 current, 1524 cylindrical measure, 475 Δ2 condition, 513 Δ02 condition, 513 D(Ω), 1364 Davie theorem, 673 Davis interpolation method, 1643 Day’s norm, 1783 decomposition method, 14, 125, 129, 135, 149, 151, 866, 1565, 1641 decomposition, p , 12 decomposition, monotone, 12 decoupling inequalities, 338 decreasing rearrangement, 1611, 1659 DENS space, 1760 density, 840 – change of, 837 – Lewis change of, 840, 849

– Maurey change of, 842 – Pisier change of, 843 density character or density (dens T ), 1746 density hypothesis, 225 dentable, 35, 397, 634 derivation, 340, 1012 descriptive, 1793 deviation inequality, 1618 diagonal argument, 1645 differentiability – almost Fréchet, 1543 – Fréchet (F), 37, 405, 788, 1537, 1541 – Gâteaux (G), 37, 408, 788, 1536, 1539 – metric, 1543 – of a convex function, 41, 409, 1535 – of a vector measure, 39 – of Lipschitz functions, 42, 1534 – weak∗ , 1543 differential games, 426 differential subordination, 253, 258 dilation, 610 dimension conjecture, 703 direct sum, 4 direct sum, inﬁnite, 5 Dirichlet problem, 622 Dirichlet series, 198 discrepancy theory, 927 disk algebra, 673, 874, 879, 1667 distance maximal, 837, 859 distortion – λ-bounded, 1349 – λ-distortion, 1337 – arbitrary, 1337 – biorthogonal, 1344, 1348 – of p , 1335, 1348 – of Hilbert space, 1335 – problem, 1335 distribution function, 5 distributional partial derivative, 1364 domain – (ε, δ), 1372 – Lipschitz, 1373 – quasi-Euclidean, 1372 – with the segment property, 1365 domination problem, 94 Doob–Meyer theorem, 383 Doubling strategy, 376, 377 DP (= the Dunford–Pettis property), 1396 dual ideal, 441 duality of operator spaces, 1435 Duhamel integral formula, 218 Dunford theorem, 107 Dunford–Pettis property (DP), 61, 444, 687, 1056, 1579

Subject Index Dvoretzky theorem, 47, 475, 735, 740, 844, 915, 1338, 1606, 1625 Dvoretzky–Rogers factorization, 737, 1221 Dvoretzky–Rogers lemma, 47, 720 dyadic – martingale, 476 – representation, 453 dyadic tree, 0-separated, 56 ε-(c.c.) sequence, 1032 ε-entropy, 891 Eberlein compact, 1753 Effros–Borel structure, 1018 eigenfunctions, 213 eigenvalues, 213, 943 Ekeland theorem, 395, 396 ellipsoid of maximal volume, 46, 164, 719, 1626 energy function, 1619 entropy, 1194, 1620 entropy function, 1110, 1161, 1162 entropy numbers, 202, 946, 958 epigraph of f , 1794 equiangular lines, 917, 931 equiangular vectors, 931 equimeasurability theorem, 902, 903, 912 evolution case, 426 exactness, 1447 exchange economy, 113 expander graphs, 1632 expectation, 5 expectation, conditional, 6 exposed point, 35, 628, 790 extension, 1155, 1161 extension of isometries, 902 extension property (EP), 1705 – C(K) (C(K) EP), 1709, 1730, 1732 – λ (λ-EP), 1705, 1707 – λ-C(K) (λ-C(K) EP), 1709, 1729–1734 – λ-separable (λ-SEP), 1722, 1723 extension theorem, 902, 904 extrapolation principle, 15 extrapolation theorem, 1143 extremal vectors, 545, 546, 602 F (X) = F (X, X), 280 F (X, Y ), 280 ϕ-function, 514, 527 F. and M. Riesz theorem, 878 Fabes–Rivière criterion, 1412 face, 602 – closed, 614 factorization of operators, 96 factorization property, 443

1853

far-out convex combinations, 1025 Fatou norm, 89 ﬁlter, 55 ﬁltration, 236, 253, 254 ﬁne embeddings, 1627 ﬁnite decomposition, 837, 865–868 ﬁnite dimensional decomposition (FDD), 11, 140, 296, 1353, 1354 ﬁnite dimensional expansion of the identity, 300 ﬁnite geometries, 359 ﬁnite nuclear norm, 966 ﬁnite rank operator, 106, 441, 880 ﬁnite variation, measures of, 39 ﬁnitely representable, 136, 138, 1306 ﬁrst Baire class, 1013 ﬁrst order Hamilton–Jacobi equations, 426 ﬁrst order smooth minimization principle, 406, 408 ﬁrst order sub- and super-differentials, 415 ﬁxed point property, 526 Fock space, 356 Föllmer–Schweizer decomposition, 389 Fourier transform, 481, 901 – restriction to surfaces, 198, 216 Fourier type, 1407 fractional Brownian motion, 383 fractional derivative, 920 fragmented, 1803 Fréchet-differentiable norm, 408, 418 Fredholm operator, 1264 Fredholm resolvent, 441 free central limit theorem, 355 free Poisson distribution, 353 free probability, 354–359 freeness, 355 function – afﬁne, 607 – afﬁne continuous, 612 – almost periodic, 644 – completely monotonic, 605 – concave, 251, 608 – convex, 608 – ﬁrst Baire class, 611 – Haar, 476 – inﬁnitely divisible, 606 – Lipschitz, 1338 – plurisubharmonic, 397, 637 – positive deﬁnite, 606 – Rademacher, 460 – stabilizes, 1337, 1338 – upper envelope of, 608 – upper semicontinuous, 608 – upper semicontinuous envelope of, 612 function spaces on compact smooth manifolds, 583–588

1854

Subject Index

– decomposition of function spaces, 587 – decomposition of the manifold, 586 – spaces on subsets, 585 – spaces with boundary conditions, 586 function, biconcave, 236, 251 function, biconvex, 235, 237 function, maximal, 237 functional – w ∗ -support, 628 – support, 628, 641 functions – stabilization principles, 1338 fundamental biorthogonal system, 1804 fundamental lemma, 1145 fundamental polynomial, 1409 fundamental theorem of asset pricing, 374, 379 γ1 , 858 γ2 , 841 G-viscosity superdifferential, 419 Gagliardo complete, 1134, 1149, 1159 games in Banach spaces, 1090–1093 Garsia conjecture, 209 Gâteaux differentiability space, 1537 Gâteaux differentiable norm, 408, 788 Gauss measure, 475, 905, 1623, 1624 Gaussian correlation problem, 179 Gaussian isoperimetric inequality, 1185 Gaussian processes, 338, 351, 848, 1624, 1625, 1630, 1631 Gaussian variables, 5, 16, 68, 839, 850, 853 Gelfand numbers, 454, 945, 954 general equilibrium, 113 general perturbed minimization principle, 397, 398 generalized Hankel operator, 681 generating cone, 87 GL (= Gordon–Lewis property), 1222, 1388 gl constant, 837, 858 Glicksberg problem, 680 Gordon–Lewis local unconditional structure (G-L l.u.st.), 59, 278–280, 302, 466, 680, 834, 872 Gorelik principle, 826 Gromov–Hausdorff distance, 1144 Grothendieck constant KG , 67, 467, 860, 1639 Grothendieck inequality, 67, 190, 842, 860, 1639 Grothendieck space, 505 Grothendieck theorem, 338, 467, 688, 872, 1748 Grothendieck–Maurey theorem, 1389 Grothendieck-type factorization theorem, 1488 H p -convex, 1483 h-elliptic polynomial, 1411

Haagerup tensor product, 1440 Haagerup’s approximation theorem, 1462, 1476 Haar – function, 476 – polynomial, 476 Haar measure, 1605, 1608, 1609, 1624–1626 Haar null set, 42, 1531 Haar system, 1664, 1666 Hadamard lacunary sequence, 875, 883, 886 Hahn–Banach extension property, 1104 Hahn–Banach theorem, 1616 Hamilton–Jacobi equation, 419 Hamming cube, 1612, 1625 Hamming distance, 1616 Hamming metric, 1613, 1617, 1618 Hankel operators, 1500 Hardy inequality, 500, 887, 894, 956 Hardy operators, 1154 Hardy space, 874, 1666 – dyadic, 1666 harmonic measures, 1625 Hausdorff dimension, 220 Hausdorff metric, 924, 1609 Hausdorff–Young inequality, 77, 504, 970 Hedging problem, 116 Henkin measure, 1394 hereditarily indecomposable Banach spaces, 1062, 1089, 1094, 1263, 1759 hereditary family, 1049 Hilbert space, 1189 – characterization of, 965 Hilbert transform, 235, 244, 481, 1169, 1638 Hilbert–Schmidt norm, 328, 1608 Hille–Tamarkin kernel, 469 Hindman’s theorem, 1082 Hoeffding inequality, 519 Hölder continuous, 425 holomorphic semi-group, 1324 homogeneous function, 910 homogeneous polynomials, 915 Hörmander–Mikhlin criterion, 1370 hyper-reﬂexive, 339 hypercontractivity, 50, 350 hyperﬁnite II1 factor, 1465, 1476, 1481, 1506, 1510 hyperplane C0 [0, ω1 ], 1756 hyperplane problem, 722, 924 ideal, 21, 90, 440 – p-Banach, 448 – Banach, 448 – closed, 441 – dual, 441

Subject Index – idempotent, 444 – injective, 446 – quasi-Banach, 448 – regular, 446 – Schatten–von Neumann, 447 – sequence, 446 – surjective, 446 – symmetric, 441 – ultrapower-stable, 455 ideal p-norm, 448 ideal norm, 448 – adjoint, 457 – non-normalized, 448 ideal property, 61 ideal quasi-norm, 448 ideals of operators on Hilbert space, 66 idempotent ideal, 444 idempotent measure, 875 indecomposable space, 1263 independent, 5 indicator function, 5 induction, 1614, 1615, 1617 inevitable set, 1250, 1260 information theory, 350 injection, 444 injective, 18, 58, 285, 1452 – hull of an ideal, 445 – ideal, 446 – tensor norm, 486 injective tensor product, 882 injective, separably, 18 insurance, 115 integral operator, see operator, integral intermediate space, 75 interpolation method, 1134 interpolation of non-commutative Lp -spaces, 1466, 1471 interpolation pair, 75 interpolation space, 502, 511, 526, 527, 1280 interpolation, K-method, 78 interpolation, complex method, 76 intersection body, 918, 919 invariant mean, 872, 876, 877 invariant subspace, 98, 533 invariant under spreading, 1311 inverse Blaschke–Santaló inequality, 759 inverse limit, 624 isometric embedding, 906 – into Lp , 524, 901 – into lp , 911 isometry, 3, 515, 526 isomorphic classiﬁcation of non-commutative Lp -spaces, 1506–1510 isomorphic symmetrization, 759

1855

isomorphism, 3 isoperimetric inequality, 163, 173, 346, 715, 1605, 1607–1609 – approximate, 1605–1608, 1625 – Levy’s, 1608 isoperimetric problem, 1613, 1614 isotropic – constant, 723 – measure, 722 – position, 722 – vectors, 911 J -functional, 1135 James theorem, 34, 385, 643 James tree space JT , 1758 James’ weak compactness theorem, 1747 Jayne–Rogers selector J , 1788 Jensen theorems, 421 JET = Jones Extension Theorem, 1372 John position, 718 John representation of the identity, 721 John theorem, 46, 169, 170, 718 Johnson–Lindenstrauss subspace JL0 , 1757 joint densities of eigenvalues, 344, 349 Josefson–Nissenzweig theorem, 1749 K-closed couple, 1404 K-convexity, 53, 483, 845, 894, 1320 k-cube, 1657 K-divisibility theorem, 80 K-functional, 78, 502, 523, 1135, 1404 k-intersection body, 923 Kσ δ subset, 1778 K-divisibility, 1149 K-divisibility principle, 1145 K-monotone interpolation space, 1145, 1149 Kadets distance, 1143 Kadets–Klee norm, 1782 Kadets–Klee property, 515, 527, 1782 Kadison–Singer problem, 333, 859 Kahane–Khintchine inequality, 50, 472, 1307 Kakeya maximal function, 221 Kakutani representation theorem, 22 Kashin decomposition, 359, 360 Kato theorem, 1324 KB-space, 89 Khintchine inequality, 16, 26, 460, 472, 519, 717, 850, 934, 1486, 1654 Kislyakov theorem, 679, 691 Knöthe map, 712 Kolmogorov number, 454 Kolmogorov rearrangement problem, 208 Krein–Milman property (KMP), 633

1856

Subject Index

Krein–Milman theorem, 602, 928, 1111, 1713 Krein–Rutman theorem, 99 Krein–Šmulian theorem, 87, 381 Krivine theorem, 48, 1310, 1339, 1353 Kunen’s C(K) space, 1758 Kwapie´n–Schütt inequality, 855 Ky Fan norms, 328 Λϕ,w (I ), 527 λ-equivalent, 1037 Λp -set, 197, 854, 872 Λ(p)cb -set, 1504 L2 (p ), 523 Lϕ , 511, 527 Lϕ (0, 1), 518 p L(k) (Ω), 1364 Lp (Lq ), 523, 527 Lp,∞ , 500, 518 Lp,q , 500, 505, 523 Lp,q (Rn ), 1399 p LS (Ω), 1408 Lw,q , 524, 526, 527 p,∞ , 500, 505, 519 p,q , 500 ϕ , 518 w,q , 525 L-functions, 198 -position, 751 ξ ξ p , (resp. c0 ) spreading model, 1057 p -spaces – ﬁnite direct sums of, 1644, 1646 – inﬁnite direct sums of, 1646 L∞ -space, 1598 L1 -space, 302 Lp,λ -space, 57, 129 Lp -space, 57, 126, 146, 279, 287 Laplacian on the torus, 213 large deviations for random matrices, 344, 346, 348 lattice norm, 89, 820 lattice of measurable functions, 694 – BMO-regular lattice, 696 lattice order, 613 lattice-convexity, 1118 left approximate identity, 286 Legendre polynomials, 926 length of a ﬁnite metric space, 1612 Leontief model, 113 Levi norm, 89 Lévy families, 744 Lévy processes, 370 Lewis lemma, 45 Liapunov theorem, 602 Lidski˘ı trace formula, 451, 463

lifting property, 17, 1454, 1708, 1726 limit order of an ideal, 469 Lindenstrauss Lifting Principle = LLP, 1382 linear extension operator, 1372 linear perturbation principle, 403 Liouville theorem, 905 Lipschitz constant, 1605, 1608, 1618, 1621–1623, 1626 Lipschitz function, 1539, 1605, 1614, 1620–1622, 1624 Lipschitz isomorphic Banach spaces, 826, 1539 Littlewood–Paley decomposition, 879 local basis property, 302, 303 local martingale, 378, 380 local reﬂexivity in operator spaces, 1451 local reﬂexivity principle, 53 local theory, 0, 43, 321, 455, 710 local unconditional structure, 1387, 1478 localization lemma, 1625 locally bounded, 375 locally depends on ﬁnitely many coordinates, 1794 locally uniformly convex norm, 784 locally uniformly lower semicontinuous, 418 locally uniformly rotund (LUR), 1782 logarithmic Sobolev inequality, 350, 1620, 1622, 1623 long James space, 411 Lorentz function space Lw,q (I ), 524 Lorentz sequence space, 519, 955, 957, 971, 1642 – non-locally convex, 1666 Lorentz space, 1399 Lorentz spaces, isometries of, 526 low M ∗ -estimate, 749 lower p-estimate, 504 lower q-estimate, 504, 514 lower semi-continuous, 1021 Lozanovskii factorization, 1162, 1163 Luxemburg norm, 511 M-admissible, 1061 M-admissible families, 1061 M-allowable, 1061 M-basis, 1765 M-ideal, 310, 1046 M-ellipsoid, 759 Mackey–Arens–Katˇetov theorem, 1749 majorizing measures, 181, 338, 1625 Marchenko–Pastur distribution, 353 Marcinkiewicz interpolation theorem, 502, 503, 505, 510, 1134 Marcinkiewicz set, 879, 880, 887 marketed space, 115 Markushevich basis, 1104, 1765

Subject Index martingale, 6, 235, 253, 380, 401, 476, 630, 1610, 1612, 1616, 1628 – analytic, 263 – difference sequence, 6, 236, 476 – dyadic, 242 – inequality, 852 – simple, 239 – square function, 256, 257 – tangent, 264 – transform, 235, 237, 262, 880 martingale difference, 1610 matrix splitting, 859 Matuszewska–Orlicz indices, 514 Maurey extension theorem, 1710, 1735–1738 Maurey–Khintchine inequalities, 510 Maurey–Nikishin–Rosenthal factorization theorem, 872, 883, 884 Maurey–Pisier theorem, 51 maximal quasi-Banach ideal, 456 maximum principle, 426 Mazur map, 1347 Mazur’s intersection property, 1804 mean value estimates, 1542 mean value theorems, 422 measurable function, 36 measure – Banach space valued, 39 – concentration of, 321, 338, 346, 349, 735 – determination by balls, 1529 – differentiable, 1526 – ergodic, 609, 615 – Gauss, 475, 905 – Haar, 617, 873, 933 – Hausdorff, 1522 – Jensen, 637 – maximal, 609, 613 – quasi-invariant, 1521 – space automorphism, 1658 – stable, 901 – T -invariant, 615 – total variation of, 39 – unique maximal probability, 613 – Wiener, 476 measure preserving involution, 1609 Mergelyan theorem, 673 metric (see also approximation property) – entropy, 331, 338 – injection, 444 – π -property, 295, 296, 300 – surjection, 445 metric probability spaces, 1603 metric space – ﬁnite, 1612 Michael selection theorem, 1731

1857

Milne theorem, 676 Milutin lemma, 1552 Milutin theorem, 702, 1551 minimal and maximal structures, 1434 minimal extension, 1107, 1156, 1171 minimal mean width position, 725 minimal surface position, 724 minimal tensor product, 1432 Minkowski box theorem, 174 Minkowski compactum Mn , 1208 – diameter of, 1208 Minkowski functional, 918 Minkowski sum, 711, 844, 924 Mityagin–Pełczy´nski theorem, 688 mixed discriminant, 731 mixed homogeneity, 1411 mixed volumes, 726 mixing invariant, 1216 modiﬁed Schlumprecht space, 1063 modular space, 858 modulus of – continuity, 971 – convexity, 31, 413, 1480, 1607, 1608 – convexity of power type p, 413 – operator, 90 – smoothness, 789, 1400 – uniform convexity, 785 moment method, 344 monotone, 1618 Montgomery conjectures, 223 mountain pass theorem, 424 Muckenhaupt condition, 389 multi-index, 1364 multiplicity, 943 multiplier, 872, 1369 multiplier transform, 1369 N-function, 511 Nagasawa theorem, 702 Nagata–Smirnov theorem, 1793 Namioka property, 1792 Nash-Williams’s theorem, 1077–1080 near unconditionality, 1064 needlepoint, 1111 nest algebra, 339, 1496 Nikishin factorization theorem, 516, 1316 Nikolski˘ı inequality, 464 No Arbitrage (NA), 378, 384 No Free Lunch with Vanishing Risk (NFLVR), 379 non-commutative Λ(p)-sets, 1501, 1502 non-commutative Burkholder–Gundy inequalities, 1491 non-commutative Doob’s inequality, 1493

1858

Subject Index

non-commutative Grothendieck theorem, 1488 non-commutative Hilbert transform, 1498 non-commutative Kadec–Pełczy´nski dichotomy, 1509 non-commutative Khintchine inequalities, 1486 non-commutative martingale, 1490 non-normalized ideal norm, 448 non-smooth calculus, 414 non-trivial weak-Cauchy, 1022 nonlinear Schrödinger equation (NLS), 197 normal semiﬁnite faithful trace, 1463 normal structure, 515 normed vector lattice, 89 nuclear, 0, 1449, 1455 – representation, 449, 461 ω (f ; ·), 1400 .p Lp (Ω), 1367 .|α|k C(Ω), 1367 .|α|k |α|k M(Ω), 1367 oscα f , 1031 OAP, 1440 operator, 3 – 1-integral, 457 – 1-nuclear, 45, 449 – α-nuclear, 488 – γ -summing, 475 – k-normal, 331 – (k, β)-mixing, 1213 – Lp -factorable, 465 – lp -singular, 445 – p-concave, 27 – p-concavity constant, 27 – p-convex, 26 – p-convexity constant, 27 – p-integral, 71, 462, 488 – p-integral norm, 72 – p-nuclear, 461 – p-summing, 63, 220, 459, 475, 677, 929, 950, 957, 959–961, 1639, 1736 – p-summing norm (πp (T )), 63 – (p, 2)-summing, 959 – (p, q)-summing, 459, 677, 950 – (q, p, X)-summing, 693 – ρ-summing, 475 – A-universal, 443 – absolute integral, 106 – absolutely summing, 63, 677 – almost commuting, 320 – almost integral, 106 – approximable, 441 – Banach–Saks, 1581

– band irreducible, 110 – biquasitriangular, 332 – block diagonal, 324, 330 – compact, 4, 19, 94, 281, 316, 442, 535, 538, 542, 658, 943, 957 – compact friendly, 103 – completely continuous, 442, 686 – composition, 470 – conditional expectation, 1648, 1662 – convolution, 973 – creation/annihilation, 356 – diagonal, 469 – dilation, 1663 – dominated, 93 – essentially normal, 323 – factoring through, 14 – factorization of, 14, 96 – ﬁnite nuclear, 966 – ﬁnite rank, 441 – ﬁxing a space Z, 1580 – Fourier type p, 481 – Fredholm, 1264, 1645 – Fredholm operator, index of, 63 – Gaussian cotype q, 472 – Gaussian type p, 472 – Haar cotype q, 477 – Haar type p, 477 – Hilbert–Schmidt, 439, 470, 949, 950 – Hilbertian, 465 – Hille–Tamarkin, 969 – integral, 106, 457, 462, 475, 969 – lattice homomorphism, 21 – lattice isomorphism, 21 – lattice-factorable, 466 – lifting of, 17 – monotone, 1535 – nearly commuting, 320 – nearly dominated, 687 – nuclear, 45, 286, 449, 461, 881, 929, 959 – order bounded, 90 – Paley operator, 678 – positive, 21, 88 – power-compact, 943, 953, 959, 969 – quasi-p-nuclear, 878 – quasinilpotent, 90 – quasitriangular, 332 – Rademacher cotype q, 472 – Rademacher type p, 471 – regular, 90 – regular averaging, 1554 – related, 944 – Riesz, 943, 958, 959 – singular integral, 235, 244, 249 – strictly Lp -factorable, 465

Subject Index – strictly p-integral, 462, 677 – strictly cosingular, 445 – strictly singular, 62, 445, 1263, 1645 – super weakly compact, 479 – transitive, 549 – translation invariant, 875 – UMD, 476 – uniformly p-smooth, 478 – uniformly q-convex, 478 – uniformly convex, 478 – uniformly convexiﬁable, 479 – uniformly smooth, 478 – universal, 443, 483 – weakly compact, 4, 95, 442 – weakly singular, 970 operator algebras, 543 operator Hilbert space OH, 1445 operator ideal, 440 – p-Banach, 448 – Banach, 448 – quasi-Banach, 448 operator of type 2, 1191 operator space structures, 1429 operator spaces, 354, 357, 1427 operators commuting with translations, 873, 879, 880, 883 optimal control or differential games, 419 optimal control theory, 426 optimal portfolio, 115 optimal sequence associated to f , 1041 option, 369 order continuity, 89 order continuous norm, 89 order of the derivative, 1364 ordered vector space, 87 ordinal index, 139, 154 Orlicz class, 511 Orlicz function, 855–857, 1153, 1155 Orlicz norm, 512 Orlicz property, 465 Orlicz sequence space, 140, 512, 518, 522, 658, 855, 1168, 1642 – non-locally convex, 1666 Orlicz space, 511, 523, 527, 855, 910, 1153, 1664 Orlicz spaces, isometries of, 515 Orlicz–Lorentz space, 527 orthogonal matrix, 1656 orthonormal system, 201 oscillation index, 1035 p-atom, 1685 p-Banach space, 403 p-concaviﬁcation, 30

1859

p-convex lattice, 504 p-convexiﬁcation, 30 ϕ-function, 514, 527 p-integral operator, see operator, p-integral p-nuclear, see operator, p-nuclear p-stable random variable, 861, 1627–1629 p-stable random vector, 906 p-subadditive, 402 p-summing operator, see operator, p-summing (p, 2)-bounded, 126, 147, 149, 150 PSH p -martingales, 403 p (X) , 284 π1 , 858 πp , 840, 861 π -property, 295–301, 307 πλ -property, 295, 307, 310, 312 Palais–Smale around F , at altitude c, 423 Paley inequality, 678, 679, 894 Paley’s projection, 1682 parabolic Hamilton–Jacobi equations, 428 Pareto optimal allocation, 114 Parseval equality, 460, 921 path of complemented subspaces, 866 paving problem, 334 payoff operator, 116 Peetre’s theorem, 1382, 1383 Pełczy´nski property, 685 perfectly homogeneous, 134 periodic boundary conditions, 218 permutation group, 1612, 1619 Perron–Frobenius theorem, 98 perturbed minimization principle, 395, 397 Pettis integral, 518 Pietsch factorization theorem, 64, 459, 840, 877, 950 Pisier’s lemma, 1406 PL-convexity, 1483 plank problem, 182, 183 plurisubharmonic functions, 397, 637 plurisubharmonic perturbed minimization principle, 403 Poincaré inequality, 350, 1620–1623 point – w ∗ -exposed, 628, 640 – w ∗ -support, 628 – denting, 634 – exposed, 601, 628, 656 – extreme, 601, 602, 605, 640 – farthest, 663 – nearest, 662 – PSH-denting, 638 – smooth, 30, 640 – strongly exposed, 628 – support, 601, 628

1860 point of continuity property (PCP), 636 polar decomposition, 66, 945 Polish ball, 1802 Polish space, 1009, 1801 polydisk algebra, 874 polynomial map on a Banach space, 806 polytope, 650, 924 – α, 659 – β, 659 – ε-approximating tangent, 659 portfolio, 116 positive cone, 87 positive curvature, 923 positive deﬁnite distribution, 919, 923 positive deﬁnite function, 901, 906, 909, 911 predictable σ -algebra, 375 preference relation, 113 price space, 113 primary, 133, 865, 1594 prime space, 1116 principal ideal, 90 probabilistic method, 358 product of ideals, 444 product space, 1614, 1615, 1623 projection, 4 – contractive, 255, 256, 1323 – partial sum, 7 – partial sum for a decomposition, 11 – Rademacher, 52, 1321 – Riesz, 9 projection constant, 71, 902, 931, 933 – absolute, 928 – relative, 928, 965 projectional resolution of identity (PRI), 1758 projective, 1452 projective tensor norm, 485 projective tensor product, 285, 882 Prokhorov’s inequality, 1610 property (u), 1028, 1048 property T of Kazhdan, 332, 359, 619 property C, 1750 property P(norm topology, weak topology) (P( · , w)), 1791 proximal subgradients, 412 pseudocompact, 1747 Ptak’s theorem, 1052 pure state, 333 put option, 116 q (X) , 284 quadratic perturbations, 396 quantum limit, 214 quartercircle law, 343

Subject Index quasi-p-nuclear operator, 878 quasi-Banach ideal – maximal, 456 – ultraproduct-stable, 456 quasi-Banach space, 402, 1099 quasi-Cohen set, 875, 889, 890 quasi-convex bodies, 769 quasi-interior point, 104 quasi-linear map, 1155 quasi-Marcinkiewicz set, 879 quasi-minimal Banach spaces, 1094, 1095 quasi-norm, 402 quasicomplemented, 1770 quasidiagonality, 320, 324, 330 quasiidempotent measure, 875 quasireﬂexive space, 646 Quermassintegral, 727 quotient of ideals, 444 quotient of subspace theorem, 752 quotient space, 3 ρX (τ ), 31 r-summing operator, 1389 RA-hierarchy, 1053 Rademacher – cotype q, 472, 1307 – functions, 16, 125, 126, 460, 848, 850, 853, 934, 1641 – projection, 482, 845, 1321 – theorem, 42 – type p, 471, 1306 Rademacher projection, 1327 Rademacher series, 1155 Radon–Nikodým derivative, 1610 Radon–Nikodým property (RNP), 35, 38, 154, 236, 259, 260, 402, 405, 414, 483, 601, 629, 1140, 1141, 1535 Rainwater–Simons theorem, 1749 Ramanujan graphs, 338, 359 Ramsey’s theorem, 1073 random n-dimensional space, 1206 – random space Xn,m , 1206 – random space Yn,m , 1207 random matrices, 319, 321, 341, 969 – edge of the spectrum, 345, 348 – global regime, 345 – local regime, 345 – spectral gaps, 345, 358 random matrix ensembles, 343, 347 random orthogonal factorizations, 766 random variable – r-stable, 6, 17 – Gaussian, 5, 16, 68 – Gaussian, standard, 5

Subject Index – symmetric, 6 random walk on the free group, 358 rank-one operator, 106 rapidly increasing sequence of n1 s, 1259, 1270 re-iteration theorem, 1140 real interpolation, 502, 1404 real interpolation method, 1135 real variable Hardy spaces, 879 rearrangement invariant space, 21, 257, 1637 reasonable crossnorm, 485 reﬂexive algebra, 338 regular – hull of an ideal, 445 – ideal, 446 – norm, 91 relative Dixmier property, 335 repeated averages hierarchy, 1053 replicate, 372 representable Banach space, 1016, 1017 representable, λ-, 53 representable, ﬁnitely, 53 representation – 1-nuclear, 449 – p-nuclear, 461 – dyadic, 453 – ﬁnite, 441 – Schmidt, 446 representing matrix, 625 reproducible, 132 restricted invertibility, 837–839, 854, 859–862 restricted unconditionality, 1055 retraction continuous afﬁne, 623 reverse – Brascamp–Lieb inequality, 171 – Hölder condition, 389 – isoperimetric inequality, 163, 169 – metric approximation property, 293 Ribe space, 1108 Ricci curvature, 1608 rich subspace of C(Q, E), 1396 Riemann ζ function, 345 Riesz projection, 1169, 1498, 1638 Riesz–Kantorovich formulas, 90 Riesz–Thorin interpolation theorem, 75, 1134 right inverse, 1382 risk-neutral, 373 Rochberg theorem, 703 Rodin–Semenov theorem, 519 Rosenthal 1 theorem, 18, 1079, 1080 Rosenthal compact, 1801 Rosenthal inequality, 128, 149, 521 Rosenthal property, 445 rough norm, 796, 1538 row vectors, 1431

1861

Ruan’s theorem, 1433 Runge theorem, 673 S # , 1408 S(Rn ), 1369 SS(X, Y ), 63 span K, 1746 span · K, 1746 ∗ span w K, 1746 σ -fragmented, 1803 ∼ - is isomorphic to, 1373 s-number ideals, 948 (s)-sequence, 1022 Saccone theorem, 686 scattered space, 1754 Schatten–von Neumann classes, 447, 895, 1446, 1465 Schauder bases, 7 – bases with vector coefﬁcients, 588–590, 594, 595 – – X-basis constant, 589 – – equivalent X-bases, 594 – – unconditional X-basis constant, 589 – negative results, 580 – non-explicit existence results, 581–583 – unconditionality in Lp , 569–579 Schauder decomposition, 11, 304 Schauder–Tichonoff theorem, 603 Schlumprecht space, 1062, 1256 Schmidt representation, 446 Schoenberg problem, 901, 906 Schreier classes Sα , 1351 Schreier families, 1051 Schreier unconditional, 1055 Schrödinger group, 197 Schur multipliers, 1500 Schur multipliers on S p , 1503 Schur property, 9, 443 Schwartz class, 1369 second order Hamilton–Jacobi equation, 430, 433 second order smooth minimization principle, 406, 424 second order subdifferential, 419 second order superdifferential, 420 security, 115 selection continuous afﬁne, 622 self-extension, 1157 semi-concave function, 420 semi-continuous functions, 1021 semi-martingale, 375, 376, 382 semicircular distribution, 343 separable complementation property, 311 separable perturbation of M, 1393 separably injective, 1454, 1709

1862 separating polynomial, 412 separation theorem, 601, 1011 sequence, 1022, 1028, 1030, 1046, 1054 – Sξ unconditional, 1055 – basic, 7 – boundedly convexly complete, 1059 – convexly unconditional, 1055 – nearly unconditional, 1055 – normalized, 7 – Schreier, 1585 – semi-boundedly complete, 1056 – seminormalized, 7 – series-bounded, 1056 – unconditionally basic, 10 – weakly Cauchy, 4 sequence ideal, 446 sequentially separable, 1758 set – 1-norming, 655 – Gδ , 608 – σ -directionally porous, 1533 – σ -porous, 1533 – Γ -null, 1534 – w-compact convex, 635 – antiproximinal, 664 – Aronszajn null, 1532 – Chebyshev, 662 – closed convex bounded (CCB), 601 – compact convex, 602 – compact convex metrizable, 612 – convex stable, 601, 660 – covered by δ-convex hypersurfaces, 1534 – dentable, 634 – Gauss null, 1532 – Haar null, 1531 – independent, 206 – interpolation, 644 – Korovkin, 640 – locally compact, 629 – norming, 646 – porous, 1533 – proximinal, 601, 662 – strongly antiproximinal, 664 – thin, 646, 647 – universally measurable, 631 – valued map, 622 – weak* compact convex, 611 – with dense extreme points, 661 short exact sequence, 1155 shrinking, 1765 Sidak Lemma, 176, 179 Sidelnikov inequality, 932 Sidon constant, 885 Sidon problems, 200

Subject Index Sidon sets, 205, 644, 872, 885–888, 891–893 sigma-martingale, 379, 380 simple function, 36 simple growth process, 112 simple predictable, 375 simplex, 613 – Bauer, 615, 616 – compact, 601, 613 – compact prime, 620 – ﬁnite-dimensional, 613 – Poulsen, 618, 619 simplexoid, 626 singular numbers, 446, 945 skipped block sequence, 141, 144 SLD map, 1791 Slepian lemma, 229, 849 Slepian–Gordon lemma, 350 slice, 35, 397, 634 slicing problem, see hyperplane problem small isomorphism, 156 small perturbations, principle, 8 Smirnov domain, 701 smooth minimization principle, 406, 417 smooth partitions of unity, 1797 smooth point, 30, 640 smooth variational principle, 1752 smoothness, 515 – n-dimensional, 1408 – anisotropic, 1408 – isotropic, 1408 – non-degenerate, 1412 – of Ascoli type, 1417 – of order k, 1412 – reducible, 1412 Šmulyan’s lemma, 1749 Sobolev embedding, 469, 1377, 1380, 1397 Sobolev projection, 1369, 1409 Sobolev space, 970, 1364 Sobolev spaces of measures, 1366 Sobolev–Besov imbedding theorems, 972 social endowment, 113 space – C(0, 1), 1642 – Hp (Tm ), 1666, 1667 – K-convex, 1320 – U1 , 1642 – Xp , 1652 – λ-injective, 1705 – λ-separably injective, 1707, 1717 – ζ -convex, 235, 237 – P1 , 1706, 1712, 1713 – Pλ , 1705, 1706, 1714 – ARNP, 236, 262, 263

Subject Index – AUMD, 236, 264 – Fréchet, 1667 – Gurarii, 626 – hereditarily indecomposable, 1263 – Hilbert transform (HT), 235, 245 – indecomposable, 1263 – injective, 1705 – James tree, 636 – locally convex, 603 – modular, 1650 – normed incomplete, 641 – nuclear, 1667 – of securities, 115 – polyhedral, 650 – quotient, 651 – rearrangement invariant, 21, 257, 500, 574 – RNP, 236, 259, 260, 263, 483, 601, 629 – separably injective, 1707, 1716, 1717 – UMD, 244, 250, 253, 264, 590, 894 space c∞ (Γ ), 1756 spaces of vector-valued functions, 588–595 – with values in a UMD space, 590 spaces with mixed norm, 107 spectral distance, 320, 327 spectral methods, 1619 spectral radius formula, 959 spectral theorem, 945 spherical design tight, 917 spherical harmonics, 926 – addition formula, 926 spherical isoperimetric inequality, 715 spherical Radon transform, 921 spin glass theory, 354 splitting of atoms, 846, 847, 850 spreading family, 1049 spreading models, 125, 136, 1035, 1036, 1076, 1310, 1339, 1340, 1353, 1650 – conditional, 1036 – unconditional, 1036 square function, 27, 863, 932 squares sets of, 206 stable, 137, 1479 – embedding, 524 – uniform algebras, 703 stable type p, 1628 star body, 918–920, 923 state space, 626 stationary case, 426 Stein restriction conjecture, 219 Stein theorem, 883 Steiner symmetrization, 712 Steinhaus variables, 934 Stieltjes transform method, 344, 357 stochastic exponential, 383

1863

stochastic interval, 375 stopping time, 375 Strichartz inequality, 197, 217 strict Aδ -functions, 404 strict Aδ -set, 404 strictly – convex norm, 784, 1773 – positive functional, 22 strictly singular operator, 1263 striking price, 115 strong Eberlein compacts, 1754 strong minimum, 398, 406 strong minimum at x0 , 420 strong type (r, s), 1369 strongly exposed point, 35, 628, 790 strongly summing, 1028 subadditive, 1618 subdiagonal algebra, 1495 subdifferentiable function, 795 subdifferential calculus, 414 submartingale, 401 submeasure, 1117 subspace – rich, 687 – smooth, 655 – tight, 682 subspaces of Lp – ﬁnite dimensional, 837 – Fourier transform characterization, 906 – inﬁnite dimensional, 140 successive approximation, 1627, 1629 Sudakov inequality, 756 sufﬁciently Euclidean, 306 summation operator – ﬁnite, 479 – inﬁnite, 443 summing basis, 1022 super – ideal, 455 – property, 56 – weakly compact operator, 479 super-property, 1306 support of a measure, 1751 support of a vector in a lattice, 23 surjection, 445 surjective – hull of an ideal, 445 – ideal, 446 symbol of an operator, 874 symmetric basis, 11, 298, 854, 1631, 1632, 1805 symmetric ideal, 441 symmetrically exchangeable random variables, 1653

1864

Subject Index

symmetrization, 1608–1610 – two point, 1609 systems of functions, 563 – Daubechies wavelets, 567 – Faber–Schauder system, 9, 564 – Franklin system, 564, 574, 575, 594 – Haar system, 9, 564 – Meyer wavelets, 566, 576, 577 – polynomial bases, 577–579 – Rademacher system, 16, 125, 460, 563, 848 – rational bases, 579 – spline wavelets, 567 – systems of analytic functions, 569 – tensoring, 568 – trigonometric polynomials, 577, 881, 889, 896 – trigonometric system, 13, 564 – Walsh system, 564, 575, 576, 1321 Szegö type factorization, 1496 Szlenk index, 802, 1018, 1559, 1719

transport of measures, 350 transportation cost, 1625 tree, 139, 143, 144, 154, 1009, 1583, 1754 – well-founded, 1583 triangular matrices, 1496, 1498 triangular truncation, 319, 329, 360 Tsirelson space, 276, 709, 1060, 1255, 1649 – 2-convexiﬁcation, 1649 – p-convexiﬁcation, 1649 twisted Hilbert space, 1157 twisted sum, 1155 two arrows space, 1756 two point symmetrization, 1609 type, 49, 125, 126, 137, 139, 472, 505, 526, 1177, 1181, 1188, 1194, 1306, 1315, 1339 – doubly generated, 1339 – Fourier, 481 – Gaussian, 472, 1710, 1738 – Haar, 477

T 2 , 279, 305 T p -smooth, 412, 413 tail distribution, 6 Taylor expansion of order p, 412 tensor – norm, 485 – – injective, 1563 – product, 153, 484, 916, 1123 – stability, 959 tensor products of C ∗ -algebras, 1449 the Brunel–Sucheston theorem, 1074, 1075 the Graham–Rothschild theorem, 1074 three space property, 1751 tight embeddings, 1628 tiling, 601, 661 – bounded, 661 – convex, 661 total variation of a measure, 39 totally disconnected compact, 1800 totally incomparable, 63 trace, 450, 1381, 1463 – duality, 44, 351, 456 – formula, 451, 463, 968 – matrix, 968 – spectral, 968 trace theorems, 1381 trace-class, 1170, 1171 transference theorem, 1371 transﬁnite oscillations, 1032 transitive algebra, 536–538 translation invariant – space, 873 – subspace, 872

Up , 279 ultraﬁlter, 55 ultrapower, 55, 455, 1307 ultrapower-stability, 455 ultraproduct, 55, 455, 1650, 1736 ultraproduct of operators, 55 ultraproduct, of Banach lattices, 55 ultraproduct-stability, 456 UMD Banach space, 253, 590, 894, 1494 unconditional – 1-unconditional over X, 1340 – basic sequence, 125, 128, 131 – basis, 9, 126, 251, 274, 277–279, 301, 302, 304, 855, 858, 1631, 1632, 1691, 1759 – constant, 250, 251, 256 – constant, complex, 250, 256 – convergence, 9 – ﬁnite dimensional decomposition, 295, 298, 301, 304 – ﬁnite dimensional expansion of the identity, 274, 278 – structure, local (l.u.st.), 59 unconditional basis constant, 1675 uniform – algebra, 673 – convexity, 413, 515, 527, 785 – Kadets–Klee property, 526 – retract, 876 – structure of a Banach space, 876 uniform convexity, 1607 uniform Eberlein compact, 1753 uniform homeomorphisms, 1805 uniform PL-convexity, 1483 uniformly

Subject Index – A-dentable, 397 – F -smooth norm, 789 – convergent Fourier series, 895 – Gâteaux-smooth norm, 789 – integrable, 17, 143 uniformly convex, 1479 uniformly Gâteaux differentiable (UG), 1777 uniformly rotund in every direction (URED), 1781 uniformly smooth, 1479 unimodal, 178 unique – non-atomic lattices, 1665 – rearrangement invariant structure, 1658, 1662– 1664 – symmetric basis, 1637, 1641, 1642 – unconditional basis, 1637, 1639, 1640 – unconditional basis, up to a permutation, 1637, 1644 uniqueness in – ﬁnite-dimensional spaces, 1651 – general spaces, 1637 – non-Banach spaces, 1665 uniqueness of complements, 837, 865–868 uniqueness theorem for measures, 902 unit ball of np , 1624 unit in a Banach lattice, 90 unitarily invariant norm, 328 unitary ideal property, 66 unitary orbit, 326, 327 universality conjecture, 345, 346 upper p-estimate, 504, 514 upper semi-continuous, 1021 upper semicontinuous function, 608 Urysohn inequality, 728 usco map, 1761 v(μ), 1366 Vaaler theorem, 175, 176, 179 vacuum state, 356 vacuum vector, 356 Valdivia compact, 1801 valuation, 732 Vapnik–Cervonenkis class, 1193 viscosity solutions for Hamilton–Jacobi equations, 427 viscosity subsolutions, 427, 429, 430 viscosity supersolutions, 427, 429, 430 volume ratio, 169, 171, 172, 174, 748, 1224 volumetric invariant vk, , 1225 w∗ -Hδ -sets, 404 Walsh system, 1321

1865

Walsh–Paley martingale, 476 wavelet, 1691 wavelet basis, 565 wavelet set on Rd , 566 weak Lp , 500, 505, 523 weak Asplund space, 1537, 1752 weak cotype 2, 1230 weak Fatou norm, 89 weak Hilbert space, 277, 305, 313, 968 weak star uniformly rotund (W∗ UR), 1778 weak type (1,1), 879–881 weak type (r, s), 1369 weak∗ usco, 1535 weak∗ -locally uniformly convex, 408 weakly compact M-basis, 1765 weakly compactly generated (WCG), 1760 weakly complete dual, 1045 weakly continuous harmonic functions, 404 weakly countably determined or a Vašák space (WCD), 1760 weakly Lindelöf (WL), 1760 weakly Lindelöf determined (WLD), 1760 weakly Lindelöf M-basis, 1765 weakly locally uniformly rotund, 1792 weakly realcompact, 1803 weakly sequentially compact, 1747 weakly unconditionally Cauchy, 1023 weakly unconditionally convergent (wuc) series, 686 weakly unconditionally summing, 1022 weakly uniformly rotund, 1781 weight function, 524 weighted norm inequalities, 387 weighted shift, 322, 323, 329 Welfare theorems, 114 well founded, 1009 well founded closed tree, 1037 well-founded tree, 1038 well-posed, 395 Wermer theorem, 673 WET = Whitney Extension Theorem, 1372 Weyl inequality, 943, 948, 953, 957, 963 Weyl numbers, 945, 954, 956, 957, 970, 972 Weyl–von Neumann–Berg theorem, 324, 328, 330 Wiener criterion, 1410 Wiener measure, 476 Wigner semicircle law, 342 Wolf’s theorem, 1407 ξ -generate a spreading model, 1038 ξ -generating the 1 -basis, 1046 ξ -generating the c0 -basis, 1046

1866 ξ -th variation, 1040 (ξ, M)-convergent, 1054 Yan theorem, 380 Young inequality, 512

Subject Index ζ -function, 198 zonoids, 768, 844, 902, 911, 924, 1631 – approximating by zonotopes, 925 zonotopes, 768, 902, 924