Banach Spaces of Vector-Valued Functions (Lecture Notes in Mathematics)

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen

1676

S rin er BPerlin g Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Pilar Cembranos Jos6 Mendoza

Banach Spaces of Vector-Valued Functions

Springer

Authors Pilar Cembranos Jos6 Mendoza Departamento de An~ilisis Matem~itico Facultad de Ciencias Matem~iticas Universidad Complutense de Madrid E-28040 Madrid, Spain e-mail: cembrp @ sunam 1,mat.ucm,es mendoza@ sunaml .mat.ucm.es

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Cembranos, Pilar: Banach spaces of vector valued functions / Pilar C e m b r a n o s ; Jos6 M e n d o z a . - Berlin ; Heidelberg ; N e w York ; Barcelona ; Budapest ; Hong Kong ; L o n d o n ; Milan ; Paris ; Santa Clara ; Singapore ; T o k y o : Springer, 1997 (Lecture notes in mathematics ; 1676) ISBN 3-540-63745-1 Mathematics Subject Classification (1991): 46E40, 46B20, 28B05 ISSN 0075-8434 ISBN 3-540-63745-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10553403 46/3142-543210 - Printed on acid-free paper

Fox" Emilio, Dolores, Luis, Javier and Pablo Mendoza Cembranos

Table of C o n t e n t s

Introduction .................................................. Some N o t a t i o n s a n d C o n v e n t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 8

1.

Preliminaries ............................................. 1.1 B a n a c h spaces c o n t a i n i n g co . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 B a n a c h spaces c o n t a i n i n g ~1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 B a n a c h spaces c o n t a i n i n g fo~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 N a t u r a l subspaces of Lp(#, X ) a n d C(K, X ) . . . . . . . . . . . . . . . 1.5 T w o Radon-Nikod?~m Theorems. T h e dual of Lp(#, X ) . . . . . . 1.6 Some r e m a r k s on L p ( p , X ) spaces . . . . . . . . . . . . . . . . . . . . . . . . 1.7 T h e dual of C ( K , X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Notes a n d R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 12 14 17 27 34 37

2.

C o p i e s o f Co a n d ~1 i n L p ( p , X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Copies of Co in L p ( p , X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 W h e n does L p ( p , X ) c o n t a i n a copy of gl? . . . . . . . . . . . . . . . . . 2.3 Notes a n d R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 57 61

C(K, X) spaces ........................................... Copies a n d c o m p l e m e n t e d copies of ~1 in C(K, X ) . . . . . . . . . . C o m p l e m e n t e d copies of Co in C ( K , X ) . . . . . . . . . . . . . . . . . . . . Copies of foo in C ( K , X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes a n d R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 68 69 72

4.

Lp(/~, X) spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 W h e n does Lp(#, X ) c o n t a i n a c o m p l e m e n t e d copy of ~1? . . . . 4.2 W h e n does Lp(#, X) c o n t a i n a copy of ~o~? . . . . . . . . . . . . . . . . 4.3 C o m p l e n l e n t e d copies of co in Lp(#, X) . . . . . . . . . . . . . . . . . . . .

75 75 78 80

5.

The 5.1 5.2 5.3

6.

Tabulation

3.

3.1 3.2 3.3 3.4

space L~(Iz, X) ...................................... C o m p l e m e n t e d copies of co in Lo~(It, X ) . . . . . . . . . . . . . . . . . . . C o m p l e m e n t e d copies of g~ in L o ~ ( # , X ) . . . . . . . . . . . . . . . . . . . Notes a n d R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of Results

.....................................

83 84 91 101 105

VIII

7.

T a b l e of C o n t e n t s

Some

References Index

Related

Open

Problems

............................

....................................................

.........................................................

107 111 117

Introduction

"... there came wise men from the east ( . . . ) saying: ( . . . ) we have seen His s t a r . . . "

Matthew 2

The study of vector-valued function Banach spaces is a very active field of research, and it is already very old, too. It begins soon after Banach's book, in the thirties and early tbrties, with the classical works of Birkhoff, Boas, Bochner, Clarkson, Day, Duntbrd, Gelfand, Pettis, Phillips, etc., and continues its developing through last decades with the contribution of mathematicians as Grothendieck, Dinculeanu, Diestel and Uhl, Kwapiefi, Maurey, Pisier, Bourgain, Talagrand, and m a n y more. We focus our attention here on Lp(p,X) and C ( K , X ) spaces (tbr notations, see below "Some Notations and Conventions"), because we believe they are the most important and representative examples of vector-valued function Banach spaces. They are the points of reference in any study on the subject. We are interested in a problem (or collection of problems) which has deserved a lot, of attention in last years. To be precise we are interested in the following:

When does Lp(#, X ) contain a copy of Co, ~l or f.oo ? What about complemented copies? Same questions about C(K,X).

P r o b l e m 1:

The aim of this monograph is to provide a detailed exposition of the answers to these questions, giving an unified and self-contained treatment. It continues in some way a small part of the big work intiated in Diestel & Uhl's "Vector Measures" [39], which in turn was a continuation of some chapters of Dunfbrd & Schwartz's "Linear Operators" (Part I) [51].

2

0. Introduction

Of course, the questions in Problem 1 are just the classical ones concerning the study of the subspaces of a given Banach space. So we are led to the Bessaga-Pelczyfiski's and Rosenthal's old theorems characterizing the Banach spaces which have copies of Co or gl (see [35] or [93]). All these theorems give a very good insight in the internal structure of the Banach spaces involved. This is why we believe that Problem 1 is not only interesting by itself, but has also contributed very much to a better understanding of the structure of Lp(#, X) and C(K, X) spaces. In particular, it has made clear that their structure is much more than a simple addition of the structures of X and L~(#) or C(K). The problem has also motivated the development of useful techniques, which are the fruit of the work of many authors during the last twenty-five years. Let us add that we have now an almost complete solution to Problem 1. It is interesting to have in mind a standard way of thinking in vectorvalued function spaces. Since X and Lp(#) are (isometrically isomorphic to) complemented subspaces of Lp(#,X), it is clear that if either X or Lp(#) contains a copy of a Banach space Y, then so does Lp(#, X). Of course, tile same can be said about complemented copies. So, very often our main problem is to know whether the converse is true or not. The analogous situation also happens for C(K, X). This is why the first problem we usually have to face is the following: P r o b l e m 2: Assume F is any of the spaces: Co, Q 07" g~. Determine which,

of the following implications are true Lp(p,X)DF ~

Lp(p)DF or X D F

Lp(#,X)(D)F ~

L~(p,)(D)F or X D

Consider the analogous questions for C(K, X), that is, C(K,X)mF

C(K)DF or

~

C(K,X)DF ~ (~,)

?

XgF

C(K)DF or X D F (c)

(~)

When one of the implications in Problem 2 is true, we obtain a complete and natural solution to Problem 1 of tile following kind:

Lp(~,X)D~I ~

Lp(~)D~I or X D ~ I

0. Introduction

3

We will see that in m a n y cases, but not always, the answers are analogous to the preceding one.

Let us give a look to the content of the monograph. In Chapter 1, Preliminaries, we give some fundamental results which we will need. We have done an effort in the careful selection of these results. In the first three sections we recall the main characterizations of Banach spaces containing copies (or complemented copies) of co, 61 or ~ . In the remaining four sections we include different basic facts about Lp(#,X) and C(K, X) spaces. In particular, in Section 1.5 we study in detail the representation of the dual of Lp(#,X) spaces provided by lifting theory. Of course, the main references here are Dinuleanu's "Vector Measures" [40] and A. and C. Ionescu Tulcea's "Topics in the Theory of Liftings" [75]. However, in these books some results on the dual of Lp(#, X) spaces are quite dispersed. This is why we have preferred to provided quite a complete and unified t r e a t m e n t of the subject. The Second Chapter is devoted to Kwapiefi's and Pisier's Theorems characterizing when Lp(#,X) contains copies of Co and 61, respectively. These theorems were proved in the mid seventies and they were the first solutions to our problems. T h e y are deep and difficult, and for this reason they take an i m p o r t a n t part of our monograph. Concerning Kwapiefi's Theorem, we should remark the important contribution of Hoffmann-J0rgensen with his preliminary work. We will explain this better in the Notes and Remarks of the Chapter. To prove the theorem we do not follow Kwapiefi's approach but Bourgain's. This is very important for us because we will need Bourgain's results in subsequent Chapters. Chapter Three is devoted to C(K, X) spaces. Theorems of Saab and Saab, Cembranos-Freniche and Drewnowski are studied. They were proved in the eighties and are concerned with complemented copies of fl, complemented copies of co and copies of ~ , respectively. The easiest of them is CembranosFreniche's, which could be viewed as an exercise on Josefson-Nissenzweig theorem. However, it was quite surprising and had a particular influence when it was proved, because it provided the first negative answer to Problem 2 above. Notice that while Kwapiefi's and Pisier's Theorems were very difficult, they gave a natural answer. Chapter Four is devoted to the problems on Lp(#,X) spaces (1 < p < +oc) not solved by Kwapiefi's and Pisier's Theorems: complemented copies of Co, complemented copies of f-1 and copies of ~ . We give here contributions of Bombal, Emmanuelle and Mendoza, obtained between 1988 and 1992. A curious fact to be mentioned here is that we find a clear difference between the behavior of purely atomic and non purely atomic measure spaces, in contrast with the results of the preceding Chapters. This happens when we consider

4

O. Introduction

complemented copies of Co in Lp(#, X), and it will happen again in the next Chapter when dealing with L~(p, X). Chapter Five is devoted to Lo~(p, X). We wish to characterize when this space contains complemented copies of Co or fl. This is shown by the results of Leung and R/ibiger, Dfaz and Kalton. They were obtained from 1990 until very recently (see the "Notes and Remarks" of the Chapter). In the ~1 case, we find a connection between local theory and the problems we are studying. Wc believe that this is the last important finding in our subject. Next, we have included a table summarizing the results of the monograph, t~hat is, the solutions to Problem 1. We see that we have a complete solution to our problem with the only exception of the L~(#, X) case, and even in this case, for finite (or a-finite) measures the problem is completely solved. Finally, we have devoted a short Chapter to comment some open and related problems of the theory. It is clear that in our original Problem 1, we can change the spaces Lp(p,X) and C(K,X) for some other more or less similar spaces, like K6the-Bochner spaces or tensor products, to mention two simple examples. On the other hand, we can also change the candidate subspaces Co, ~1 or f.~ for some other else, like ~p (1 < p < +oc) or LI([0, 1]). There are also connections between the problems we have been considering and some other problems, related to classical properties of Banach spaces. In this final chapter, we comment some of these aspects.

This work is intended for those mathematicians interested in Banach Space Theory, Functional Analysis, and in general Abstract Analysis. The subject is crossroad of many branches in Banach space theory, and so one has occasion of applying many fimdamental and classical results. For this reason, we think that the techniques developed here are interesting tbr anyone working on Banach spaces and absolutely fundamental for those interested in the study of vector-valued function spaces and related fields. It is written at a graduate student level. In fact we have followed the first draft of this work to teach our 30 hours Seminar during the Course 96-97, and we have taken a great advantage of that experience. We think that it might be a first step to begin a research in the field. We assume the reader knows the basics on Banach space theory, and we mean for this to be familiar, for instance, with most parts of J. Diestel's "Sequences and Series in Banach Spaces " [35]. Of course, many of the proofs given here are different to the original ones. And we believe that some of them are much easier. In fact, we have done an effort trying to give the best, points of view and the best prooih to understand the theory. We would say that in some cases we give surprisingly easy proofs and one could think that we are actually proving almost trivial results. However, if one reads the original papers one does not have the same impression. We would say that this is so because of two main reasons.

O. Introduction

5

On one hand, as we have already mentioned, we have chosen carefully the preliminary results we need, and even the precise versions which are the most suitable for our purposes. Although these results are "well known", most of them were quite dispersed and very often they could not be found easily in the literature. On the other hand, it is normal that after several years we can see a theory in a much clearer way, that is, we understand better the results. When they were discovered, neither their relationship with other results nor the right tools to prove them were so clear. In our case for instance, Saab & Saab's Theorem 3.1.4 on complemented copies of 61 might seem now to be very easy. However, if we reflect a little bit about the main ingredients in its proof: lifting theory and Bourgain's Theorem 2.1.1, one realizes immediately that in the early eighties this theorem of Bourgain was not very well known, and it was not at all usual to apply lifting theory to get a theorem on C(K, X) spaces. Bourgain's name has just appeared. We would like to emphasize that our monograph contains a detailed exposition of some fundamental results of Bourgain which were not very accessible until now, at least for a beginner in the subject. His ideas are all over this monograph and are very important in our study. We must add that this is not just a compillation of known results. We have tried to put some important ideas into an accessible expository form and we have also included: (a) some results which appear here for the first time, and (b) some results which we believe have been quite unnoticed and are difficult to find in the literature. We give now a list of the most important of these results. The first three are in case (b), and the other two in case (a). 1. Proposition 1.6.3 on measurability. The result is quite an easy application of fundamental results about the Souslin operation, but we have not seen such an application in this context in the literature. 2. Lemma 2.1.3. It is a result due to Rosenthal which we have called KadecPetczyfiski-Rosenthal's Lemma because it is a refinement of known ideas of tile classical Kadec-Petczyfiski paper [77].We think that at the moment it is the best "subsequence splitting lemma" in L1 (#). 3. Theorem 2.1.4. It gives a large class of subspaces of Lp(#, X) in which the L~,-norm and the L l - n o r m are equivalent. We obtain the result as a quite easy consequence of the just mentioned Kadec-Petczyfiski-Rosenthal's L e m m a 2.1.3. 4. Dfaz-Kalton's Theorem 5.2.3 characterizing when Loo(#,X) contains a complemented copy of 61. When we write these lines, neither Diaz's contribution nor Kalton's have been published yet. However, they kindly have allowed us to use their ideas and preprints to give a complete account of their results. 5. Theorem characterizing when Loo(#,X) contains a complemented copy of L 1 ([0, 1]). It may be found in the Notes and Remarks of Chapter 5. It is

6

0. Introduction actually an immediate consequence of Diaz-Kalton's Theorem 5.2.3 and Hagler-Stegall's theorem [67] characterizing those duals which contain a complemented copy of C([0, 1])*.

ACKNOWLEDGEMENTS First of all we wish to express our gratitude to F. Bombal, our teacher, because he guided our first steps in the research and introduced us in this subject. We also have to thank B. Rodrfguez-Salinas, because he also had a great influence in our formation. J. Diestel's books, conferences, seminars and friendly conversations have been very important for us, too. We may say that the origin of this monograph goes back to November of 1990. Then, K. Sundaresan invited the second named author to deliver a talk at Cleveland State University. The subject of that talk coincided exactly with the one of this monograph and K. Sundaresan encouraged him to continue to work on this field. A few years later, the same author was invited by K. Jarosz to deliver a talk at the Second Conference on Function Spaces held at Edwarsville in the Spring of 1994. This was a new motivation to continue and bring to date the work which had been exposed at Cleveland. As a result he wrote the survey [98], which has become a very condensed preliminary version of this monograph. For these reasons, we are indebted with K. Sundaresan and K. Jarosz for their encouragement and contribution to the birth of this work. We are also indebted with J.M.F. Castill0 who, may be unaware, encouraged us to write this monograph. We have already mentioned above that N. J. Kalton and S. Dfaz have allowed us to include here some important results of them which will appear here for the first time. We have also had several interesting conversations with them on different aspects of this work. We are indeed very grateful to them. We are also grateful to many colleagues which have gently provided us some preprints and with whom we have had interesting conversations. Among others, we have to mention: L. Drewnowski, Z. Hu and B.-L. Lin. J. Bourgain and H. Rosenthal kindly told us about the history of the key lemma 2.1.3, since they knew we were wrong about its origin. H. Rosenthal gave us very interesting advices, too. They have our gratitude. We have to thank I. Villanueva, student of our seminar of 96-97, for several valuable suggestions and comments. It is a pleasure to recall also our colleagues and friends of our Department who have helped and encouraged us very often. J. Esquinas answered very patiently many questions about TEX we posed him. Also Springer-Verlag staff, and particularly, F. Holzwarth, gave us all the help we asked them.

0. Introduction

7

Finally, during the preparation of the manuscript we have asked D. H. Fremlin about different aspects of our work. He has always been extremely kind and his help can not be overestimated. We are very grateful to him. Of course, we are the only responsible of any possible mistakes in this work. This work has been partially supported by D.G.I.C.Y.T. grant PB94-0243

Madrid, September 1997

8

0. Introduction

S o m e N o t a t i o n s and Conventions In general, we will use s t a n d a r d notation as in [35], [39] or [93]. We will work with real or complex Banach spaces. Let X , Y be B a n a c h spaces. As usual, we say t h a t X has a (complemented) copy of Y if X has a (complemented) subspace Which is isomorphic to Y. We will denote X D Y and X D Y

(c)

Let X be a B a n a c h space, let (J'2, ~ , #) be a positive measure space , and let 1 _-integrable (p-essentially bounded, when p = oo ) functions with its usual norm. T h a t is, the vectors of Lp(#, X ) are (equivalence classes of) # - m e a s u r a b l e functions f such t h a t Ilfllv--

(/;

1

It/(c~)ll d#(oo)

< +oc

in the case 1 _< p < + o c , and Ilflloo = ess sup{Ill(w)I1: w 9 ~2} < +oo in the case p = + o c . In case X is the scalar field we simply denote Lv(#). If K is a c o m p a c t Hausdorff space, we denote by C ( K , X ) the B a n a c h space of all continuous X - v a l u e d functions defined on K, endowed with the s u p r e m u m norm. In case X is tile scalar field we simply denote C(K). In order to avoid trivial situations, we will always suppose that X , C(K) and Lv(p) are infinite dimensional. Notice t h a t in the case of C(K) this simply means t h a t K is infinite, and in the case of Lp(#), t h a t the corres p o n d i n g measure space (~?, Z , # ) is not trivial, i.e. it has infinitely m a n y disjoint measurable sets of finite positive measure. For a vector measure m : Z --+ X , Ilmll and r~ denote the variation and semivariation of rn respectively. T h e symbol cabv(E, X) stands for the B a n a c h space of all X - v a l u e d countably additive measures with b o u n d e d variation defined on Z , endowed with the variation norm. If X is the scalar field this space is simply denoted by cabv(Z). Recall t h a t if (s Z , #) is a measure space and we consider its completion (~2, 2 , s then the corresponding Lv(>, X) and Lv(#, X ) spaces coincide. For this reason we will always assume our measure spaces are complete.

1. Preliminaries

Since we wish to study when the spaces Lp(#, X ) or C(K, X ) contain copies or complemented copies of co, fl or ~ , we will devote the first part of the chapter to recall the main characterizations of when a general Banach space enjoys any of these properties. In the second part of the chapter we will give some general facts about the spaces Lp(#, X ) and C(K, X).

1.1 Banach

spaces

containing

Co

Bessaga and Petczyfiski's classical results [4] provide the main characterizations of Banach spaces containing co we will need in our work (see [35, Chapter V]). The basic sequences equivalent to the canonical basis of co will be simply called Co-sequences. Later on we will also use the expressions "61-sequence" and in general "s in the same obvious sense. We will say t h a t a sequence in X is complemented if its closed linear span is a complemented subspace of X. As usual, when we say that (xn) is seminormalized we mean that 0 < i n f [Ixnll< sup Ilxnll< +oc. Let us remember that a series ~ Xn in X is called weakly unconditionally Cauehy (or weakly unconditionally convergent), or in short w.u.C., if x*(xn) is absolutely convergent for each x* C X* (see [35, Theorem 6, Chapter V] for the main properties of w.u.C, series). The typical example of a non trivial w.u.C, series is the canonical co-basis, moreover we have (see [35, pages 42-45]): 1.1.1 ( B e s s a g a - P e l c z y f i s k i ) . Let (xn) be a seminormalized sequence and suppose that the series ~ xn is w.u.C., then (Xn) has a Cosubsequenee. Moreover, if (xn) is a basic sequence, then it is a Co-sequence.

Theorem

We also can give a characterization of the Banach spaces containing com-

plemented copies of co. We need first the following simple result 1.1.1 ( C h a p t e r V I I , E x e r c i s e 4 o f [35]). The boundedlinear operators from X into Co correspond precisely to the weak*-null sequences in X*, where each weak*-null sequence (x*) in X* has associated the operator x Proposition

10

1. Preliminaries

With the preceding proposition in mind the following result is trivial (as usual, 6..... = 0 if n r m and 5n~ = 1). P r o p o s i t i o n 1.1.2. Let (x,~) be a co-sequence in X , then (an) is complemented if and only if there is a weak*-nuU sequence (x~) in X* such that =

nm.

But the main criterion we will use to find complemented c0-sequences will be the following (versions of it may be found in [52] or [122]): T h e o r e m 1.1.2. Let ~ x~ be a w.u.C, series in X , then (Xn) has a complemented Co-subsequence if and only if there exists a weak*-null sequence (x*) in X* such that

x,i(xn)/+ 0. Proof. Let us show the non trivial implication. Let ~ x,~ be a w.u.C, series in X, and let (x~) be a weak*-null sequence in X* such that x ; ( x n ) /+ O. Then, without loss of generality, we may assume that there exists 5 > 0 such that for all n E N. Let T be the bounded linear operator associated to (x;) as in Proposition 1.1.1. Notice that ~ x~ and ~ T(x,~) satisfy the assumptions of ,Theorem 1.1.1, and then, taking subsequences if necessary, we may assume t h a t (x~) and (T(x~)) are both c0-sequences. Of course this says t h a t TIF is an isomorphism, where F denotes the closed linear span of the Xn'S. Observe now t h a t Sobczyk's theorem implies that the closed linear span of the T ( x ~ ) ' s is complemented in Co. Let P be a projection from Co onto this subspace. It is clear that (TIp) -~ o P o T is a projection from X onto F.

1.2 Banach

spaces

containing

~1

Of course the main characterization of Banach spaces containing gl is the classical Rosenthal's theorem (see Chapter XI of [35]). To give a criterion to find complemented gl-sequences we will use again w.u.C, series. The following result is quite easy. P r o p o s i t i o n 1.2.1 ( E x e r c i s e 3 o f C h a p t e r V I I o f [35]). The bounded linear operators from a Banach space X into ~1 correspond precisely to the w.u.C, series in X*, where each w.u.C, series ~ x n in X* has associated the operator x ~ ( Xn ( X ) ) . From the preceding result we get immediately the following:

1.2 Banach spaces containing .~l Proposition

11

1.2.2. Let (xn) be an fl-sequence in X , then (x~) is comple-

mented if and only if there exists a w.u.C, series ~-]x n in X* such that

=Sn n But to give a really useful criterion to find complemented fl-sequences we need another fundamental result of Rosenthal [110] which has m a n y different applications: the disjointification lemma. Its proof may be found in [35, page 82].

Let (#~) be a bounded sequence in eabv(Z). Then given e > 0 and a sequence (An) of pairwise disjoint members of Z there exists and increasing sequence (kn) of positive integers for which Lemma

1.2.1 ( R o s e n t h a l ' s D i s j o i n t i f i c a t i o n l e m m a ) .

II# ~ ( U An) < e j:#,,-,,

/or all n. We can now give the announced criterion. It is very similar to the one we have given for c0-sequences (Theorem 1.1.2) in the preceding section. 1.2.1 ( R o s e n t h a l [110, 111]). Let (xn) be a bounded sequence in X . Then (x~) has a complemented fl-SUbsequence if and only if there exists a weakly unconditionally Cauchy series ~ x n in X* such that

Theorem

o

Proof. The condition is of course necessary. For the converse take (x~) and (x*) as in the statement. We may suppose, without loss of generality, that both sequences are in the corresponding unit balls. Taking subsequences and multiplying the x~*~'s by adequate scalars if necessary, we may also assume that there exists ~ > 0 such that >

for all n. Let 7)(N) be the a-algebra of all subsets of N. For each m C N consider the countably additive scalar measure pm defined on 7)(N) by

A

~

~

x:(x. 0

nEA

(this is just the usual way in which an element of ~1 may be viewed as a member of cabv(7)(N))). Since ~ x~ is a w.u.C, series in X* and (Xn) is bounded, using Proposition 1.2.1 it is clear that (pro) is bounded in cabv(P(N)). Hence we deduce from the preceding lemma that there are subsequences of (xn) and (x~), which we continue to denote again in the same way, such that

12

1. P r e l i m i n a r i e s

I>,dl ( U {"}) = ~

I*;.(*~,,)1 < 5/2

for all m. Let R : X --+ ~1 be the bounded linear operator associated to ~ x,*~ as in Proposition 1.2.1. P u t z,,~ = R(Xm) = (X~(Xm))~. It is straightforward t h a t the bounded linear operator S:

~1

(t~,,)

> ~'1 > ~ trnZm

satisfies

I I Z - S l l < 1 - - 2' where I is the identity operator in gl. Hence S is an isomorphism and we can deduce t h a t (z,~) = (R(xm)) is an gl-sequence. Therefore (x,,) is an gl-sequence, too. Finally, if we define T : gl --+ X by T ( ( t ~ ) ) = ~ t,~xm, it is immediate t h a t T o S -1 o R is a projection from X onto the closed linear span of the x~'s.

1.3 Banach spaces containing s The main criteria to determine if a given Banach space contains goo are also due to Rosenthal [110]. We denote by (en) the unit vector sequence of goo. Theorem

1.3.1. The following are equivalent:

(a) X D e ~ . (b) There exists a bounded linear operator T : g~ -+ X such that

II T(~,~)II )40. (c) There exists a bounded linear operator T : ~ compact.

--+ X which is not weakly

Observe t h a t (c) implies (a) is the only non easy implication in this theorem (and it is just VI.1.3. of [39]). T h a t (a) implies (b) is trivial, and to understand why (b) implies (c) it is enough to recall that every weakly compact operator is unconditionally converging (that is, it transforms w.u.C. series into unconditionally convergent ones [35, Chapter V, Exercise 8]). Since ~ en is a w.u.C, series in g~, it is clear that the operator T of condition (b) is not unconditionally converging, and so it can not be weakly compact either.

1.3 Banach spaces containing ~

13

Remark 1.3.1. If A is a set, we denote by [A] the set of all infinite subsets of A. For M E [N], f ~ ( M ) is defined to be the subspace of g~ of all sequences ((,~) E g ~ with ~ = 0 for n r M. Of course it is isometrically isomorphic to go~. It is worth to mention that if T satisfies (b) of the preceding theorem then there exists M E IN] such that Tte~(M) is an isomorphic embedding (see R e m a r k 1 of Proposition 1.2 of [110]). A reformulation of (c) implies (a) in the preceding theorem is the following C o r o l l a r y 1.3.1. If X has no copy of ~ T : ~ -+ X is weakly compact.

then every bounded linear operator

And we get from this corollary the following classical result: C o r o l l a r y 1.3.2 ( P h i l l i p s , 1940). f ~ h,as no complemented copies of co. At this point it is interesting to know how are the operators from goo in X when X does not have copies of goo. This is our purpose now.

Remark 1.3.2. We have mentioned above that weakly compact operators are unconditionally converging. Therefore, if an operator T : ~ --+ X is weakly compact then ~ T(e,~) is unconditionally convergent, and so ~ A,~T(e,~) converges for all ( k , ) E , ~ . Of course, this does not mean that T((A,~)) and ~,,'~__~ A,~T(e~) coincide. However, we will see in the next proposition that this is "almost" true. In its proof we will need to take a family {Nt}tE[oa] of infinite subsets of N such that tbr each pair of different real numbers s, t E [0, 1] the set iV, N Nt is finite. How to show that such a family exists ? : Of course it suffices to show this for Q instead of N. Well, for each t E [0, 1] take Nt the set of values of a sequence of different rational numbers converging to t. It is clear that the family {Nt}tc[0,1] enjoys the required property.

Proposition 1.3.1 (Drewnowski [46]). Let T:~oo --+ X be a weakly compact operator, then ther~ exists art i~finite subset M of N such that T((.(,~)) = ~ (.,T(en)

(1.1)

*~,--1

for all (r

e e ~(M)

Proof. If T : g~ --+ X is weakly compact we have just seen that ~ T(en) is unconditionally convergent. Thus tile operator S : g ~ --+ X given by S(({,,.)) = ~_, {nT(e,.) for ({~) E g~, is well defined. Let rn and mc be the representing measures of T and S respectively, that is, the X-valued measures defined on 7)(N) (the set of all subsets of N) by re(A) = T(XA )

and

mc(A) = S(XA) = ~ nff A

T(e~)

14

1. Preliminaries

for each A G 7~(N), where XA is the sequence (t~) with t~ = 1 if n E A and t~ = 0 otherwise. Both measures are strongly additive: the first one because T is weakly compact [39, VI.I.1.], and the second one because it is even countably additive (it is the "countably additve part" of m). Therefore m o = m - m e is strongly additive, too. Remember that the strong additivity of mo is equivalent to the following property of its semivariation r~0 [39, 1.1.18.]: r~0(A~) -+ 0 whenever (AN) is a sequence of pairwise disjoint subsets of N. By the argument given in the previous remark we know that there exists a family {Nt}tE[O,1 ] of infinite subsets of N such that Nt n Ns is finite if t ~ s. Since ~ is a non decreasing subadditive set function which vanishes on finite sets, it is immediate that ~ ( N 1 ) = r~0(N2) if N1 and N2 differ only in a finite set. So, for each e > 0 the set {t G [0, 1]: rn-o(Nt) > e} is finite, and consequently {t r [0, 1] : 77Eo(Nt) > 0} is countable. Thus, there exists s E [0, 1] such that m0(Ns) = 0. Of course this means that m0 vanishes on all subsets of N,, and taking M = Ns, it is clear that T and S coincide in ~oo (M) as required. [46]). Let (Tk) be a sequence of weakly compact operators from g~ into X , then there exists an infinite subset M of N such that

C o r o l l a r y 1.3.3 ( D r e w n o w s k i

=

n--1

for all (r

e g~o(M) and all k E N.

Proof. We just have to use the preceding proposition and a diagonal argumerit. Apply the preceding proposition to T1. We get M1 C [l~ such that T1 satisfies 1.1 in goo (M1). Now, apply again the preceding proposition to T2 (as defined in goo(M1)). We get 11//2 r [M1] such that T2 satisfies 1.1 in/~oo(M2). In this way we get a non increasing sequence (M~) of infinite subsets of N such t h a t each T~ satisfies 1.1 in goo(Mn). Let M be the subset of N consisting of the first natural number of Ma, the second one of M2, and so on. It is clear that M satisfies the required property.

1.4 Natural

subspaces

of L~(tt, X)

and

C(K,X)

Let us give our first glance to Lv(p, X) and C(K, X). Since we are interested in the study of subspaces of these spaces, let us begin recalling which are the obvious ones. Of course they are Lv(#), X and gp in the case of Lv(#, X), and C(K), X and co in the case of C(K, X). We devote this short section to remember these elementary facts. They are very easy, but we believe that

1.4 Natural subspaces of Lp(#, X) and

C(K, X)

15

some reflection on t h e m helps to u n d e r s t a n d better the spaces we are dealing with. For 1 _< p < +oo, take any vector z0 in the unit sphere of X , then the map

Lp( )

f

--+

----+ f(.)zo

is an isometric embedding. Moreover, its range is norm one complemented in Lv(#, X ) . To give a n o r m one projection it is enough to take any n o r m one functional x 0 on X such t h a t < x o , z 0 > = 1, and consider the m a p

Lp(#,X) f

~ Lp(~,X) >

Similarly, if f0 is any function in the unit, sphere of

X x

mo

L v(#),

then the m a p

----+ Lt,(# ,X) fo(.)x

is an isometric e m b e d d i n g and its range is n o r m one complemented, too. This time, to get a n o r m one projection, take a function 90 in the unit sphere of Lq(#) such t h a t

/nfo(t)go(t) d#(t) = 1, then it is enough to define Cp(~,X)

f

~

L,,(#,X)

--~

fo(.)./]n go(t)f(t) d#(t)

Tile situation for Loo (#, X ) and C(K, X) is completely analogous. We can see in the same way t h a t Loo (#, X ) contains m a n y n o r m one complemented isometric copies of Loo(#) and X , and C(K,X) contains m a n y n o r m one c o m p l e m e n t e d isometric copies of C(K) and X. On the other hand, Lv(#) does have m a n y complemented copies of gp. If we take a sequence (An) of pairwise disjoint measurable sets with finite positive measure, it is straightforward to show t h a t

Lp( )

f

> L,,(#)

>

.f(t) d#(t) )YAk(')

is a n o r m one linear projection onto the closed linear span of the Xak's, a subspace isometrically isomorphic to gv More generally,

16

1. Preliminaries

1.4.1. For 1 < p < +oc, if (fk) is a seminormalized sequence in Lp(#, X) of disjointly supported functions, then it is a complemented gpsequence in Lp(#,X). More precisely, the closed linear span of the fk's is isometrically isomorphic to gp and it is norm one complemented.

Proposition

Proof. It is straightforward to show that the closed linear span of the fk's is isometrically isomorphic to gp. On the other hand, if (rk) is a sequence of norm one functionals in Lp(p,X)* such that < fk,rk >=11 fk I1~, and we denote Ak = snpp(fk) = {w 6 Y2 : ft:(a;) r 0}, then one can easily show also that Lv(p,X)

~

f

~

Lp(#,X) <xakf,rk>

[[---f--k[[pfk

k=l

is a norm one linear projection onto the closed linear span of the fk's.

The situation for Loo(p,X) and Lo~(#) is quite similar. They contain m a n y isometric copies of ~o~. In fact, if (fk) is any normalized sequence in L ~ (#, X) of disjointly supported functions then the natural map g~

---+

(t~)

--~

L~(,,X)

Z

tkA( )

k=l

is an isometric embedding. Finally, let us see why C ( K , X ) (and C(K)) contains many isometric copies of co. Take any sequence (Gn) of pairwise disjoint non e m p t y open subsets of K , and a sequence (tn) of points in K such that tn E Gn for all n E N. Thanks to Urysohn's theorem, for each n 6 N there exists a continuous function fn on K , taking values in [0, 1], such that f~(tn) = 1 and fi~(K \ Gn) = 0. It is easy to show that the closed linear span on C(K) of the f ~ ' s is isometrically isomorphic to Co. Moreover, if (x~) any sequence of non null vectors in X, and we denote g,~ = .f,d.)x~, then the the closed linear span on C(K, X) of the g,~'s is isometrically isomorphic to Co. In fact, we have: Proposition

1.4.2. / f (gn) is a sequence in C(K, X) of non null functions

such that the sets {t C K : g,~(t) # O}

are pairwise disjoint, then the closed linear span of the g~'s is a subspace isometrically isomorphic to Co.

1.5 Two Radon-Nikod3~m Theorems. The dual of Lr,(#, X)

17

Proof. Of course we may assume the sequence (g~) is normalized, then it is enough to notice t h a t co

~

C(K, X ) .')0

7z

[

is an isometric embedding.

Remark 1.~.1. The preceding proposition in general does not provide a complemented copy of co. In fact, we should remember that C(K) very often does not have complemented copies of co (we will come back over this in Chapter 3).

1.5 Two

Radon-Nikod~m

Theorems.

The

dual

of

Lp(tt, X) We wish to present here in detail how to get a Radon-Nikod~m theorem for X*-valued measures using lifting theory, and how to apply it to obtain a representation of the dual of Lv(p, X). Both results are quite old. The first one is contained in [40] and the second one in [40] and [75]. However, one has to combine several results of those monographs to get precisely the versions we give (which are, of course, the ones we need). The inclusion of proofs will also help to understand better the theory we are studying. A few words to recall a well known situation (see [35, Chapter IV.1.] ). Assume the measure # is finite. Given F E LT~(#, X)* one can define a vector measure m : Z -+ X* by m(A)(x) = F(XA(.>) ibr all x E X and all A C 5 . This measure turns out to be countably additive, of bounded variation and absolutely continuous with respect to #. If we have a density function for m, t h a t is, a measurable function ~ : ~2 -~ X* so t h a t m(m) = j d ~(w)d~(w) for all A E E, then we have

F(f) = j ~ < f ( w ) , W ( w ) > dp(w)

(1.2)

for all f E Lp(#, X). Thus, the functional F is represented by the function qo (which, in this case, belongs to Lq(tt , X*), with ~, + 1 = 1).

18

1. Preliminaries

Nevertheless, we know t h a t in general there is no such a density function {'or all F ' s (this h a p p e n s only if X* has the Radon-Nikod:~m property). Now, w h a t h a p p e n s if we look for a density function in a weaker sense? For instance, what h a p p e n s if we look for a w*-measurable function ~ : s --+ X* such t h a t rn(A)(x) = ~

< x, ~(c~) > d#(a~)

for all x E X and all A E ~ ? Recall t h a t we say t h a t a function ~ : /2 --+ X* is w * - m e a s u r a b l e if the m a p aJ ~ < x, ~(a~) > is m e a s u r a b l e for all x E X. We will see, using lifting theory, t h a t such a w*-measurable function always exists. Of course, in general it is not m e a s u r a b l e but it shares m a n y of the good properties of the functions belonging to L q ( p , X * ) , and it represents tile functional /" in the sense of 1.2. Let us s t a r t with the definition of lifting in s and the main result of this theory: the existence theorem. As usual, for a positive measure A on 5 , EF,(A ) denotes the vector space of all scalar, A-measurable p-integrable (Aessentially b o u n d e d for p = +oc) flmctions defined on ~2 (we do not identify here a thnction with its equivalence class). D e f i n i t i o n 1.5.1. Let A be a complete and positive measure. A lifting in s is a linear map p : s ~ s such that: 1. For" each f E s p(f)(cv) = .f(cz) for A-almost all w E f2. 2. If f , g E s and f ( w ) = .q(w) for" A-almost all c~ E $2, then p(f)(cJ) = p(g)(a~) for all c~ E 12. 3. If f, g E s and f ( w ) <_ g(cv) dorA-almost allc~ E Xg, t h e n p ( f ) ( ~ ) < p(.q)(w) for all w E $2. ~. Let (~ E K. If f E s and f(cJ) = c~ ]'or A-almost alley E Y2, then p ( f ) (w) = a for all cv E 12. 5. For all f , 9 E s p ( f g ) ( w ) = p ( f ) ( w ) p ( g ) ( w ) for all w E (2 (multiplicativity). T h e m e a n i n g of condition 3 is clear if we work with R as scalar field. In the c o m p l e x case the meaning is the obvious one: if f, 9 E s are real valued and f(c~) _< .q(c~) for A-almost all cJ E /2, then p ( f ) and P(g) are real-valued and p(.[)(~) _< p(g)(~) for all w E /2. If we reflect fbr a m o m e n t on the preceding definition we see t h a t one might consider a lifting as a m a p p i n g from Loo(A) in s From this point of view, it would be a linear right inverse of the canonical quotient m a p 7r : s --+ Loo(A) with nice properties. T h e o r e m 1.5.1 ( E x i s t e n c e T h e o r e m [40, 75]). Let A be a complete, finite and positive measure, then there exists a lifting m s

1.5 Two Radon-Nikod:~m Theorems. The dual of L~(#,X)

19

Let A be a complete, finite and positive measure on ~ . We denote by s ) the vector space of all functions ~ : ~2 -+ X*, which are Aessentially bounded and w*-measurable, and by cabv~(Z,X*) the vector space of all measures m E cabv(Z, X*) for which there is (~ >_ 0 such that lira( A) II -< ~A(A) for all A E Z. Notice that, unlike most of the spaces we are dealing with, cabv~(Z, X*) is a non complete subspace of cabv(Z, X*). Our first step consists in applying the lifting existence theorem to get a density function for each measure belonging to cabv~(Z, X*). Subsequently, a standard application of this will allow us to find a density function for all A-continuous measures in cabv(Z, X*). 1 . 5 . 2 ( R a d o n - N i k o d : ~ m t y p e I). Let A be a complete, finite and positive measure on Z. Each lifting p in Coo(A) induces a linear and injective map fi : cabvx(Z,X*) ~ s (A,X*)

Theorem

such that for each measure m E cabv~ (Z, X*) the function ~ = fi(m) satisfies the following two conditions: 1. For each A E Z and all x E X m(A)(x) = ~ 2. The f nction

< x , ~ ( w ) > dA(w)

11 ( )II is easurable, and for each A E

I1- II (A) -- s II

dA( )

Proof. For simplicity we will give the proof in the real case. The result carries over in a standard manner to the complex case. Let p be a lifting in s Let m E cabvx(Z,X*) and c~ > 0 be such that

II (A)II

< c~A(A)

(1.3)

for all A 6 Z. For each x E X let us define rn:~ : Z ~ IR by r,~ (A) = r , 4 A ) ( x ) for all A E ~ . Of course, m~ belongs to cabv(Z). By 1.3, m , is A-continuous and so, by the Radon-Nikod3~m theorem, there exists f~ E s such t h a t

m~(A) = / ' f.(w) dA(w) ]A

for all A E Z. Now, we (:an define ~ : ~2 -+ X* by

20

1. Preliminaries

<~, ~(co) > -- ~(A)(co)

(1.4)

fbr all x E X and all co E g?. We m u s t show t h a t ~ satisfies all ore requirements. Recall t h a t two density functions of tile same scalar A-continuous measure differ only in a A-null set. W i t h this in mind, thanks to the linearity of p, we have t h a t ~(w) is linear for each co E ~?. Let us see t h a t it is also continuous. If we apply the R a d o n Nikod~m t h e o r e m to the measure II m II we get a flmction g E s with IIg I1~ < c~, such t h a t

I1-* II (A) -- .,../' .q(~) dA(~) tbr each A E Z. Then, for all A E K and all x E X, we have

~

fx(w) dA(co) =

m:,,(A) -- m(AD(x) <_ II'~(A) It Ilxll

5

II-~ll (A)I,=:ll = (~g(co)dA(co))Ilzll

Therefbre, fbr each x E X ,

A(co) ~ ,q(co)IIzll 5 ~ Ilxll for A-almost all w E D, and so < x , ~ ( c o ) > = p(.f~)(co) <_ p(g)(w)Ilxll

_< ~

Ilxll

(1.5)

whenever co E J2 and x E X . Thus, ~(co) E X* and II~(co)115 c~ for all co E D. In other words, p is well defined and bounded. Moreover, since ~ is clearly w*-measurable, we have t h a t ~ E s (A, X*). By its own definition it is also clear t h a t ~ satisfies condition 1. To see t h a t ~ satisfies condition 2 we will show t h a t [[ p(w) II = g(co) for A-almost all co E .(2. By 1.5 we have

ll~(co)ll _< pO)(co) for all co E .(2, and so,

II~(co)II _< .~(co) for A-almost all co E D. Let go be the s u p r e m u m in LI(A) (or Loo(A), see [51, IV.11.7.]) of the set of all functions of the form

Y~,

(.) I p(J:,)(.) I

i

with {Ai} a finite Z - p a r t i t i o n of D and {xi} C B(X). This function 90 E s (A) satisfies

1.5 Two Radon-Nikod~m Theorems. The dual of

Lp(#,X)

g0(~) ~ I1~(~)II ~ g(~)

21

(1.6)

for k-almost all w C S?. Given E > 0 take a finite ~ - p a r t i t i o n {Ai} of J~ and {:c~} C B(X) such t h a t

Ilmll (~) -e.

_< ~

rn.(A~)(x~).

Then

~ ( ~ ) dk(~)-~ = Ilmll (~)

< Em(Ai)(xi)

9

i

<_ /n go(-) d~(~). Since this is true for all e > 0, we have

/ g(w) d.~(w) <_/ go(w) d)~(w). Therefore, using 1.6, we get g0(w) -- II~(w) ll -- g(w) fbr almost all w E /2. Finally, from the definition of/3(m) ~ given in 1.4 and the properties of a lifting, it is clear that/3 is linear and injective. This completes the proof.

Remark 1.5.1. Given m E cabvx(Z, X*), the function ~ = /3(m) is actually bounded (not only A-essentially bounded). In fact, it is clear in the proof that if lira(A) II < c~k(A) for all A E ~ , then for all w C /2. In the first part of the proof it is explained how fi is defined given a lifting p in s (k). Actually the precise way in which fi is defined will not be i m p o r t a n t in our study.

22

1. Preliminaries

Of course fi in the preceding theorem is not onto, but it should be noticed that each ~ E s (A, X*) defines a measure m E cabv~(Z, X*) in the natural way: / *

m ( A ) ( x ) = JA < x , ~ ( w ) > d),(w) for each A E Z and each x E X.

T h e o r e m 1.5.3 ( R a d o n - N i k o d : ~ m t y p e II). Let )~ be a complete, finite and positive measure on ~ , and let m E c a b v ( S , X * ) be an absolutely continuous measure respect to )~. Then there exists ~ : [2 ~ X* w*-measurable satisfying the following three conditions: 1. The function w ~-~]l~P(w)I] (which we simply denote ]lO(.)I]) is measurable and belongs to ~1 ()~) 2. For all x E X and all A E Z m ( A ) ( x ) = /A < x, e ( . ) > 3. For alI A E Z

Ilmll (A)= /A 11 ( )11 Proof. Let m E c a b v ( Z , X * ) be absolutely continuous respect to A. Since the positive measure ]] m ]I is also absolutely Continuous respect to A, by the scalar Radon-Nikod:~m theorem, there exists g E s (A) such that IIm II (A) = L g(~) dA(w) for a l l A E 5:. For e a c h n E N p u t B~ = {w E ~ : n - 1 < g ( w ) < n } . The sequence (Bn) is a G-partition of [2. Consider for each n the measure rnn defined by m n ( A ) = m ( A N B,~) for all A E Z. We have IIm~(A) H - - I I m ( A n B ~ ) l l < _ l ] m l l ( A n B ~ ) = f

<_ n (A n

JA (1B~

g(w) dA(w)

< n (A)

for each A E Z. Therefore, mn belongs to c a b v x ( Z , X * ) . Let us fix a lifting p in s (A), and let us consider the map fi associated to it given in the preceding theorem. We can now define

1.5 Two Radon-NikodS, m Theorems. The dual of L~(/,, X) k0:~?

23

X*

>

(X3

> Z xoo (~)~(,,<)(~) n

1

It is clear that ~ is w*-measurable and that for all A E Z

II ~(.)II

is measurable. We also get

0(3

= ~ II-
I1-~11(A).

This means that II~(.)II belongs to/~1(/~), and condition 3 holds. Finally, it is cleat" t h a t for each x C X and all A E

/ i < x ' o ( w ) > dA(w)

=

~-~/A n=l

<x, qJ(w)>dA(w) ~B~

(2'O

=

~,~(A

n Bn)(x) = ,~(A)(x).

n=l

Remark 1.5.2. After Theorem 1.5.2, one could think that the preceding proof provides a linear m a p from the subspace of c a b v ( Z , X * ) of all absolutely continuous measures with respect to A into the vector space of w*-measurable functions from ~2 into X*. However this is not true. The point is t h a t our definition of q' given in the proof does depend on the choice of the Bn's. This might be surprising at first, but notice that if the definition of g' did not depend on the choice of the B~'s, the technique used in the proof could be used to show the existence of a linear lifting in s (a m a p from s into s with the same properties of a lifting with the exception of the multiplicativity condition 5). But it is well known t h a t there are no linear liftings in s for 1 _< p < oo (see [40, Proposition 9 of w or [75, Chapter III, Section 4]). From the preceding Radon-Nikod3~m theorem it follows in quite a standard way the following theorem giving a representation of the continuous linear functionals in Lp(p, X). T h e o r e m 1.5.4 ( T h e d u a l o f Lv(p, X) I). Let (s Z , p) be a finite measure space, let 1 , X)* there exists a function g" : ~2 --+ X* such that:

1. ~ is w*-measurable.

24

1. Preliminaries

2. The function w ~-~ll k~(w)II (which we simply denote by II ~(.)11) is measurable and belongs to s 3. F(.f) = fx? < f(w), ~(w) > d/z(w) ]'or all f E Lp(/Z, X ) , and

~. IIr I1 = II IIe(.) II II~. Conversely, each w*-measurable function ~P : g? ~ X* for which there exists g Es such that for ~z-almost all w E $2, induces by 3 a continuous linear functional F on L,~(/Z,X), whose norm is less or equal to IIgllq. Proof. It is convexfient to prove first the second p a r t of the theorem, which is straightforward. In other words, it is easy to show t h a t if you take ~ : ~2 --+ X* w*-measurable, for which there exists g E s such t h a t II e(w)II -< g(w) for /z-almost all w E ~ , then for each f E Lp(/Z, X ) the function w ~-> < f(co), ~(w) > belongs to s (/Z) and Lp(/Z,X) f

~

IK

---+

/ < f(co), tP(co) > d/z(w) Jo

defines a continuous linear functional on L p ( # , X ) , whose n o r m is less or equal to IIg IIq. Let us see now the first p a r t of the theorem. Given F E Lp(/Z,X)*, let us consider the m e a s u r e m : ~ ~ X* associated to F in the usual way (see p. 98-99 of [39]):

m ( A ) ( x ) = F(XA (.)x)

(1.7)

for all A E Z and all x E X. It is easy to show t h a t this m e a s u r e turns out to be an element of cabv(Z, X*). Moreover, it verifies I1-~11 (A) < p ( A ) ~ IlCll for all A E Z . Therefore rn is absolutely continuous respect to # and we can a p p l y the preceding theorem. We get k~ : F2 ~ X* w * - m e a s u r a b l e with

II ~(.)II belonging to s

such that

m ( A ) ( x ) = /A < x , k ~ ( w ) > d/z(w) for all x E X and all A E Z , and

(1.8)

1.5 Two Radon-Nikod3)m Theorems. The dual of

Lp(#, X)

25

/.

Ilmll (A)= / I1~(~)11 d~(~)

(1.9)

/A

for all A E ~ . Obviously 1.8 m e a n s t h a t for each simple functions .f

F(f) = fx? < f(w), g-'(w)> d#(co).

(1.10)

If we show t h a t II ]l#(.)ll lie -< [I F II then we will have t h a t condition 2 is satisfied. But, then we can apply the second p a r t of the t h e o r e m to the t u n c t i o n # and, t h a n k s to 1.10, we can assure t h a t F and the continuous linear functional associated to # coincide in the simple functions. Therefore they coincide in the whole Lp(>, X). Of course, this proves t h a t conditions 3 is satisfied and also the inequality II F I t - < II IIg'(.)ll Ilq. So, it only remains to prove II I1~(.)11 Ilq -< Ilvll, and to do this it suffices to see t h a t the m e a s u r a b l e flmction II ~(-) II defines in the usual way a continuos linear functional on Lv(p) whose n o r m is b o u n d e d by II V II. n Take a scalar Z - s i m p l e function t7. = ~,;=1 aiXA~. Given e > 0 there are, tbr each i r { 1 , . . . ,n}, Z-finite partitions (Bj)j of Ai, and vectors (x})j in the unit ball of X such t h a t Ilmtl (A~) _< ~-~ m(B~)(x}) + J Then, according to 1.7 and

.~ h.(w) ,,q~(w),,dp(w)

_e /t

1.9, we have

=

f i /A c~i ,,g'(w),, d#(w) i

l

i

i=1

<

t~1 i

~(B~)(x})

1

\i=l

+ -

n

"

\i=1

"

J

,,p

1

=

Ilvll

I~lPllxj I1~.(~}) \i=1

'

+~

26

1. Preliminaries 1

5

IlCll

Im I~ It(Ad

+c

i=1

--

Ilrll Ithll, +~.

Since this is true for all e > 0 and all simple functions h we conclude t h a t I1r II belongs to s and II I1r I1~ _< II F II. This completes the proof.

Remark 1.5.3. Notice t h a t the canonical m a p considered in the preceding proof Lp(#;X)*

--+

F

---+

cabv(Z,X*) m

where m(A)(x) = F(XA(.)x) for all A E i : and all x E X , is an injective continuous linear o p e r a t o r from Lp(#, X)* into cabv(&, X*). T h e o r e m 1.5.4 gives us a good representation of each continuous linear functional in Lp(#, X ) and it is the result we will use in our work. If we now wish a r e p r e s e n t a t i o n of the Banach space Lp(#, X)*, t h a t is, if we wish to represent its linear structure, this is very easy. It is done in the following way. Let us denote by s (#, X*) the vector space of all w * - m e a s u r a b l e functions : Y? --+ X* for which there exists 9 C s such t h a t IIr

5 g(~)

for It-almost a2 E ~. Consider in s (it, X*) the equivalence relation * , where given ~Pl,~P2 E s X*), we say t h a t ~Pl * q~2 if for each z C X we have <x,r

> -- < x , r

>

for # - a l m o s t all w E Y2. Let us denote by L~.(It, q X *) the quotient space and by [~P] the equivalence classes. It is easy to show t h a t Lq~. (it, X * ) with the o p e r a t i o n s defined in the n a t u r a l way is a vector space (that is, * is c o m p a t i b l e with the linear structure of s (it, X*)). If we define Ill [r

Illq -- inf I l g L ,

where the infimum is t a k e n over all g E s (#) for which there is some q~o C [k~] such t h a t II r (~)11-< g(~) It-almost everywhere, it is also straightforward to show t h a t III - [[[q is a n o r m of B a n a c h space in L ~ . (it, X*) and t h a t the natural e m b e d d i n g from L~ (it, X*) into L,q. (it, X*) is an isometric isomorphism. Finally, let us consider the m a p

I : L ~q . ( p , X *)

[r

Lp(#,X)*

~

I([r

1.6 Some remarks on Lp(#,X) spaces

27

given by I([kP])(f) = / ~

< f(w), q~0(cv) > dp(a;)

(1.11)

for each f E L~,(#, X), if ~0 is any function in [~]. Standard arguments show that I is a well defined bounded linear operator. Moreover, the preceding theorem says that T h e o r e m 1.5.5 ( T h e d u a l o f Lp(#, X ) II). Let (/2, &, #) be a finite mea-

sure space, and let 1 < p < oc, then the bounded linear operator I defined in 1.11 is an isometric isomorphism onto Lp(p,X)*, and for each [q)] E Lq~.(p,X *) there exists ~o E [~P] such that the function a~ ~ll Oo(w) II (which we simply denote by IIr i~ measurable and belongs to Lq(#), and

III [r IIIq = II I1 0(.)11 L. 1.6 Some

remarks

on Lp(tt , X)

spaces

In this section we will give a few fundamental facts about Lp(#, X ) spaces. Our main aims are the following: (a) reducing our problems to the case of finite measures, (b) studying a question about measurability, (c) discussing briefly the differences between Lp (#, X ) spaces depending on the measure # is purely atomic or not, and (d) recalling the definition of uniform integrability. Many of the problems considered in Lp(#, X), when (/2, Z, #) is an arbitrary measure space, are solved as soon as we have the solution in the fnite measure ease. In particular, this is true for the problems studied here. Let us see why. For two positive measure spaces (/2, w #) and (/21, J21, pl) we will denote (/21, 221, ~1) C (/2, 22, p) if /21 E ~, 221 is a sub-a-algebra of 2' of subsets of f?l, and Pl = P 161- Of course we will identify Lp(pl, X) with a subspace of Lp(#, X), .just extending by zero the functions. Remember that a measure space (/2, 22, #) is said to be separable if the corresponding Banach space LI(p) (or equivalently Lp(p), 1 < p < oc) is separable. It is easy to prove that this happens if and only if there exists a sequence (Ak) of 22-measurable sets such that 22 is the p-completion of the a-algebra generated by the Ak's. Let us begin with the following simple result which can be found in [51]. L e m m a 1.6.1 ( L e m m a I I I . 8 . 5 . o f [51]). Let 1 <_ p < oo and let F be

a separable subspace of Lp(#,X), then there exists a a-finite and separable measure space (/21,~1,Pl) C (/2, 22,#) such that F lies in Lp(Pl,X). Of course, if the space F in the preceding lemma is complemented in Lp(p, X), then it is complemented in LB(#I, X), too (in fact, the converse is also true as we will see in Lemma 1.6.2). So the lemma provides a reduction

28

1. Preliminaries

to the or-finite case. Let us go hlrther, to reach the finite measure case. T h e next proposition is well known. P r o p o s i t i o n 1.6.1. Let (f21, ~1,1,1) be a a-finite measure space, then there exists a probability measure #o on (f21,~1) such that for 1 <_ p <_ + ~ , Lp(/zl, X ) and Lp(lzo,X ) are isometrically isomorphic. Besides, the measures 1,1 and #o have the same null sct,s, and consequently they have the same atoms. In particular, #~ is purely atomic if and only if #o is.

Proof. Let (A,~) be a sequence of pairwise disjoint measurable sets with finite and positive # l - m e a s u r e such t h a t Y?~ = [AnAn. Let us define #0 : S 1

--+

A

--+

[0,+~) 1

.=~ 2 n ~ (A,~)

# I ( A A A~)

It is clear t h a t / t o is a probability measure with the same null sets as #1, and so the identity

L~(#I,X)

f

)

L~(po,X)

~

f

is an isometric isomorphism onto. Moreover, for 1 

hi

where oo

h =

27p1(An)TXa~

is an isometric isomorphism onto~ too. As an immediate consequence of the preceding results we have: T h e o r e m 1 . 6 . 1 . Let (~2, Z , # ) be an arbitrary positive measure space, let 1 ~ p < oo, let F be a separable Banach space, and assume that Lp(#, X ) contains a copy (respectively, a complemented copy) of F. Then there exists a separable, finite and positive measure space ( Y2o, ~o, #o) such that Lv(#o, X ) contains a copy (respectively, a complemented copy) of F. Besides, if the measure space (Y2, Z , #) is purely atomic, we may assume that (f2o, Zo, Po) has the same property.

R e m a r k 1.6.1. In this m o n o g r a p h we are mainly interested in the spaces F = Co a n d F = g l .

1.6 Some remarks on Lp(p,X) spaces

29

Remark 1.6.2. Neither the preceding theorem nor L e m m a 1.6.1 can be extended to non separable subspaces F of Lp(p, X). To give simple examples showing this it is enough to take any measm'e space (Jg, Z, p) providing an Lp(p) non separable and non isomorphic to an Lp(#o) with finite measure #0. Then consider as X the scalar field and as F the whole space Lp(p). Although, as we have just remarked, the above results are not true for general non separable spaces, it is very interesting for us that they are true for g~. Our aim now is to show this. Let us begin with an easy result on complementability. 1.6.2. For" 1 < p < oc, if (~?l,Zl,#l) C (~?,~,#) is a-finite, then Lp(#l, X ) is norm one complemented in Lp(it , X ) . Lemma

Proof. Let us denote 5(f2~) = {A C 5 : A C f2~}. Clearly, 5(f2,) is a sub-a-algebra of 5 which contains 51. Let us denote by R the restriction operator from Lv(#, X ) = Lp(f2, ~, #; X ) into Lv(f2~ , s > I x ( & ) ; X). Since (f2s,5(f2s),Izbc(n,)) is a-finite we have a norm one conditional expectation operator E from Lv(f21 , 5(f2~), #IE(&); X) into Lv(f2~, 5~, #~; X ) = L~,(>I, X ) (see V.1. of [39]). It is plain that E o R is a norm one projection from Lp(#, X) onto L,,(#,, X). 1.6.2. Let 1 <_ p < oc, and assume that Lp(#, X ) contains a copy of ~ . Then there exists a cy-finite and separable measure space (o(21, ~F1, >1) C (f2, Z , p ) such that Lp(#l, X ) contains a copy of gc~.

Proposition

Proof. Sufficiency is obvious. Let us show necessity. Suppose that Lv(#, X ) contains a copy of go~ and let T be an isomorphic embedding from g~ into Lv(p, X ) . By L e m m a 1.6.1 we know that there exists a a-finite and separable measure space (f21, EI,#I) C (~, E, #) such that T(e,,) E Lp(pl,

X)

for all C N, where (e,,) denotes the canonical unit vector sequence in go~. Let P be the projection operator from L,,(I, , X ) onto Lv(#I, 2s given in the preceding lemnm. If we take To = P o T it. is clear that,

IIZ0(e,j II,, = liT(e,,) lip 74 0. Then, by Theorem 1.3.1, Lp(pl, X ) contains a copy of g~. This completes the proof. If" we put together the preceding proposition and Proposition 1.6.1 we obtain 1.6.2. Let (Jg, E , # ) be an arbitrary positive measure space, let 1 <_ p < oc, and assume that L~,(#,X) contains a copy of ~ . Then there ezists a separable, .finite and positive measure space (f2o, E0,/to) such that Ll,(ito, X ) contains a copy orgy:,.

Theorem

30

1. Preliminaries

Remark 1.6.3. In the preceding t h e o r e m no assertion a b o u t complementability of f ~ has been made. This is because such an assertion would have been superfluous. Recall t h a t ~ enjoys an i m p o r t a n t property: it is injective (see [35, page 71] or [93, page 105]). This means t h a t ~0~ is c o m p l e m e n t e d in every B a n a c h space containing it. In other words, XD~o~ if and only if X D g ~ . (c) W h e n one considers sequences in Lp(#, X ) spaces, one has very often to deal with different subsets of (2 associated to these sequences. In most cases it, is trivial to show t h a t such subsets are measurable, just because they can easily put as countable unions and intersections of sets which are obviously measurable. Nevertheless, there are cases in which the situation is not so easy, and in our work we will find one of these situations. Given a basis (e,,) in a B a n a c h space and a sequence (f,~) in Lp(p, X ) it is easy to see t h a t the set {cz C f2: (f~(c~)) is a equivalent to (en)} is measurable. However, can we also assure t h a t the set {w E /2 : (fi~(a~)) has a subsequence equivalent to (e,~)} is m e a s u r a b l e ? We will see that the answer is "yes". T h e key to prove this will be a beautiful old result a b o u t the so called Souslin o p e r a t i o n , also called operation (A) (see [109, C h a p t e r 1, Section 7], [118, C h a p t e r 2, Section 5]). This "operation" is defined as follows: D e f i n i t i o n 1.6.1. Given a family (A,~ .....'r,k) of subsets of a certain set Y2, where k runs N and ( n l , . . . , n k ) belongs to N k, we say that the set oo

where the union is over all infinite sequences of natural numbers, is obtained .f~vm application to the Souslin operation to the family ( A ~ ....... k). Our interest in Souslin operation lies in the following i m p o r t a n t property: T h e o r e m 1.6.3 (1.7 o f [109]). Given a finite (and complete) measure space (~2, Z , # ) , the Souslin operation applied to a family of measurable sets is a measurable set. Notice t h a t the point is t h a t although Souslin operation is applied to a countable family, it involves an union over the c o n t i n u u m m a n y infinite sequences of n a t u r a l numbers. Now we can deal with the question posed above.

1.6 S o m e r e m a r k s on

Lp(#,X)

spaces

31

P r o p o s i t i o n 1.6.3. Let (en) be a basis of a Banach space and let ( f , ) be a sequence of X - v a l u e d measurable functions defined on the finite measure space (Y2, r , #). Then the set {w C Y2 : ( f n ( w ) ) has a subsequence equivalent to (en)} i.s measurable. Proof. Given M > O, we say t h a t a sequence (xn) is M-equivalent to (en) if

1 It ~ ~,ne,~II-< II ~ ~,~.~,~ II-< M II ~ )',,~,, 11 tbr all finite sequences of scalars (A,~). Take any finite sequence n l , . . . , nk of n a t u r a l numbers. If nl < n2 < ... < nk put as An, ..... n~ the set of all co E ~2 such t h a t 1

k

k

k

II ~ ~,jej II _< II ~ ~Jf,,, (co)II < M II ~ :,vej II j=l

j=l

j=l

for all (Aj) E Qk. Otherwise, put A,z~....... ~ = ~. Of course all these sets are measurable, and so, by the preceding theorem, if we apply the Souslin o p e r a t i o n to the family ( A ~ ..... ,,~) we get a m e a s u r a b l e set. But notice t h a t

** 1 , * ~ 2 , - . . k = l

7~1 < n 2

<...

k=l

= {co E ~2: (fi,(co)) has a subsequence M-equivalent to (en)}, where the second union is over all infinite increasing sequences of n a t u r a l n u m b e r s . Therefore, the following equality finishes the proof {co C ~2 : (.f,~(c0)) has a subsequence equivalent to (e,~)} =

0

{co e /2 : (f,~(co)) has a subsequence m-equivalent to (e~)}

D1. ~ 1

As an i m m e d i a t e consequence we have the following C o r o l l a r y 1.6.1. Let ( ~ , E , # ) be an arbitrary measure space, let ( e , ) be a basis of a certain Banach space and let (.f,~) be a sequence in Lp(p, X ) , where 1 <_ p < +oc. Then the ,set {co C ~2: (f,,(co)) has a subsequence equivalent to (e,,)}

'is measurable.

32

1. Preliminaries

Proof. We use the standard reduction argument. By L e m m a 1.6.1, there exists a ~-finite measure space (g?~, Z ~ , # I ) C ($2, Z , # ) such that (f~) lies in LF(#I, X). Now, by Proposition 1.6.1, there exists a probability measure #0 on (~?1,2~1) and an isometric isomorphism from Lv(p~,X ) onto Lp(po,X ). Moreover, as we showed in the proof, this isometric isomorphism has the form f ---+ hf where h is a certain scalar function which does not vanish at any point. So it is clear that for each w E ~2~, (h(co)f,~(w)) has a subsequence equivalent to (e~) if and only if so does (f,~(co)). Now the conclusion follows from the preceding proposition. Now we wish to touch on another basic aspect of Lp(l.t , X) spaces. Let us remember that for m a n y purposes the most important scalar Lp spaces are gv and L~,([0, 1]). We will see that something similar happens also in Lv(#, X) spaces. To understand well this, one should study the isometric theory of Lv(p) spaces (see [84] and the Notes and Remarks of this Chapter), but we have preferred to give only a couple of fundamental results which will be needed in Chapters 4 and 5. We include simple and elementary proofs of them. The point is the following: when we have a finite measure p, it is well known that i.f it is purely atomic then the corresponding Lp(#) space is

isometrically isomorphic to gp, and otherwise, it has a complemented subspace isometrically isomorphic to Lp ([0, 1]). We will prove that the vectorial version of this result is also true. If we take as f2 = N, ~ tile v-algebra of all subsets of N, and > the counting measure, that is, the measure defined by ~(A) = c~..di,,al(A)

then, the corresponding Lp(#, X) spaces are the known gv(X) spaces of all sequences (x,,) of vectors in X such that

(zn)llp

=

IIx~ll ~ n

(:~)11oo

=

<+oo

for l _ < p < o o

1

sup ll,'r~ll < + o o n

with the [I [[p norm. Moreover we have tile following

Proposition 1.6.4. Let 1 <_p <_+oo, if ( f2, Z, #) is a o-finite purely atomic

measure space then Lp(#, X) is isometrically isomorj)hic to gp( X). Prvof. Let (A~) be a sequence of pairwise disjoint atoms of (~, 5 , #) whose union is ~2. Then the maps

1.6 Some remarks on Lp(p,X) spaces Lv(p,X)

--~

ep(X)

--+

(.(A~)Tx,~),~

33

OG

.f =

Z XA~ (.)x,.,.

rZ=I

for l g-o~(X)

O(3

are isometric i s o m o r p h i s m onto.

1.6.5. If (~2, ~, #) is not purely atomic and 1 < p <_ oo, then Lv(p, X) contains a complemented copy of Lv([0, 1], X ) .

Proposition

Prvof. Since (f2, ~P, #) is not purely atomic there exists a set ~21 C ~ with finite and positive m e a s u r e such t h a t no subset of O1 is an atom. W i t h o u t loss of generality we m a y assume p(f21) = 1. T h e n we can find a Z - p a r t i t i o n 1 Now for i = 0, 1 we can find a of Y?l in two sets A0 and A1 of measure ~. ~ - p a r t i t i o n of Ai in two sets Aio and Ai~ of m e a s u r e ~. Of course, in this way we construct a diadic decomposition {A~I.~ ~ : i l , . . . , i k ff {0, 1},k E N} of C2~ in Z - m e a s u r a b l e sets, exactly in the analogous way to the classical diadic decomposition of [0, 1] in intervals ,lii...ik with left side in 0 . i l . . . i k (the n u m b e r written in binary form) and length ~ , with k E N. Let 51 be the v - a l g e b r a generated by the A h .i~'s- Denoting by #1 the restriction of p to 5~, L e m m a 1.6.2 says t h a t Lv(p~ , X) is c o m p l e m e n t e d in Lp(p, X). So, to finish our proof it is enough to show t h a t Lv(pl, X) is isometrically isomorphic to Lv([0 , 1], X ) . But if we denote by 8 the subspace of Lp(pl, X) of all functions of the form ~ X~, ,~, (.)xi~ ...i~, where we only consider finite stuns and the x,~...~ k s are in X, it is very easy to show t h a t the m a p ,5

--+

Lv([0, 1], X )

is an isometry. Therefore, it can be extended to an isometric isomorphism from L,,(/,~, X ) onto Lv([0, 1], X ) . To complete the section let us recall a well known concept which plays a central role in Lp(#, N) spaces: uniform integrability. T h e subject deserves a h)t of a t t e n t i o n and we suggest the reading of Diestel's survey [36], which is entirely devoted to it. Here we only wish to give the definition and a couple of very i m p o r t a n t facts. To avoid useless difficulties we will assume t h a t ore" m e a s u r e space (Y2, Z , #) is finite, and so, fbr any p > 1 we have the continuous canonical e m b e d d i n g

34

1. Preliminaries

i : Lp(p,X)

r

.f

D e f i n i t i o n 1.6.2.

LI(p,X)

-+

f

We say that a subset A of L1 (#, X ) is uniformly integrable

if lira ~(E)-+0

/

]E

Il f l l d p = 0

.

uniformly in $ E A. Given p > 1, f E L v ( #, X ) and E E 5 , it follows from HSlder inequality that

s Ilfll dr= ~ Ilfllx~ d, <_IISIl~#(z)5 where 1_ p + tq = 1. As an immediate consequence we have that subsets of

Lt(#, X ) which are (contained and) bounded in Dr(v, X ) f o r some p, with 1 < p <_ +oo, are uniformly integrable. These are perhaps the simplest examples of uniformly integrable subsets of

L1 (#, X).

One of the most important applications of uniform integrability is DunfordPettis celebrated characterization of relatively weakly compact subsets of Lt (#). It says that they are precisely the bounded and uniformly integrable sets [39, T h e o r e m III.2.15] [51, Corollary IV.8.11.].

1.7 T h e d u a l o f

C(K, X)

In Section 1.5 we got a nice and useful representation of the dual of Lp(#, X). W h a t can we say about the dual of C(K, X ) ? The classical Riesz representation theorem establishes that the dual of C(K) may be viewed as a space of" measures via integration theory. Singer got the extension of this theorem to the vector-valued case. Before stating Singer's result let us remember a few things about vector measures. If K is a compact and Hausdorff space we denote by B ( K ) the c,-algebra of all Borel subsets of K . Given a finitely additive measure m : S ( K ) --+ X* of bounded variation, for each X-valued B(K)-simple function f = ~ { XA{xi, one can define the integral

./Kf

dm= ~

m,(Ai)(xi) i

1.7 The dual of C(K, X )

35

This integral is well defined and, using t h a t m has b o u n d e d variation, it can be extended to all X - v a l u e d functions which are uniform limits of B ( K ) - s i m p l e functions. In particular, the integral makes sense for all functions belonging to C(K, X), and we get in this way a continuous linear functional on C(K, X). Likewise in the scalar case, in general, m a n y additive X*-valued measures on /3(K) with b o u n d e d variation coincide in C(K, X) or, in other words, m a n y of t h e m define the same m e m b e r of C(K, X)*. However, only one of t h e m is regular (a vector measure m is regular whenever its variation II m II is regular). Let us denote by rcabv(K, X*) tile closed subspace of cabv(13(K),X*) of all its regular members. As in the scalar case it is also true t h a t each functional in C(K, X)* can be represented by a measure in rcabv(K, X*), moreover U(K, X)* m a y be identified with rcabv(K, X*). This is Singer's extension of the classical Riesz t h e o r e m representing the dual of C(K). It m a y be found in [125, L e m m a 1.6, Section 1.4., C h a p t e r II] (see also [70]). Theorem

Th.e map

1.7.1 ( R i e s z - S i n g e r ) .

rcabv(K,X*) 7rt

--+

C(K,X)*

~

%7~

where r,,~ is defined by

"~,,~(f) = ~ f dm for" each f E C(K, X), is an isometric isomorphism onto C(K, X)*. It is w o r t h to mention t h a t the simplest members of rcabv(K,X*) are the linear combinations of measures of the form #(.)x*, where # is a scalar regular measure on K , and x* E X*. In particular, we can take measures of the form fi,(.)x*, with t E K , where of com'se, for each A E B(K)

fit(A)= {

~

if t E A otherwise

Re'mark 1.7.1. If m : L~(K) --+ X* is a finitely additive measure, for each z E X , we can define the scalar measure rn:,: by ,,~:,:(A) = ,,,~(A)(,,) for all A E B(K). If m , is countably additive for all x E X, it is said t h a t 'm, is w*-countably additive. Analogously, rr~, is said to be w*-regular if m:~ is regular for all x E X. It: is easy to prove t h a t if m is of b o u n d e d variation then m is 'tu*-countably additive if and only if it, is countably additive, and 'm. is w*-regular if and only if it is regular. Moreover, thanks to Alexandroff t h e o r e m ( [51, T h e o r e m III.5. if]), each X*-valued finitely additive w*-regutar measure of b o u n d e d variation defined on K belongs to rcabv(K, X*). It is not difficult now to characterize weak convergence in C(K, X). T h e following result is very well known.

36

1. Preliminaries

P r o p o s i t i o n 1.7.1 ( T h e o r e m 9, [42]). Let (f,) be a bounded sequence in C ( K , X ) , and f E C(K, X). Then (f,~) is weakly convergent to f if and only

if (fn(t)) is weakly convergent to f(t) for each t E K. Proof. Assmne (f,~) is a sequence in C ( K , X ) weakly convergent to f E C ( K , X ) . Let to be a point of K. For each x* E X* let us consider the measure 5to (.)x* as a linear form in C(K, X). It follows that limz*(fn(t0)) =

lira [

f, d(ato(.)z*) = /" f d(ato(.)z*) = z*(f(t0)) /g

Theretbre, (f,.(to)) is weakly convergent to f(to). For the converse, assmne that (f.,J is a bounded sequence in C ( K , X ) such that (f,~(t)) is weakly convergent to f(t) for each t r K . Let us take a continuous linear form oil C(K, X), that is, a measure m in rcabv(K, X*). Let us consider the variation I]rn i] of m. Extending Hm H in the usual way we can assume the measure ]lm ]l is complete. Then theorem 1.5.2 implies that there exists p : K -~ X* bounded w*-measurable such that for each A E Z and all x E X

m(A)(z) = [

dllmll (t)

JA

Of course, this means that

/'K hdm= s < h(t),~(t) > dllmll(t)

(1.12)

fox' every X-valued LT(K)-simpte flmction h defined on K , and this implies that 1.12 is also true for all continuous functions h r C ( K , X ) . Nov+-, it fbllows from the usual Lebesgue dominated convergence theorem that lira f n

lim [ <.f,,(t),~o(t)> dllmll(t) 'n .It(

f'.,~d m =

J K

=

/i_ < f(t),~(t)> dllmll(t)=/_, f dm d/(

This completes the proof. Next corollary follows immediately from the preceding proposition and the following well known observation: a sequence (z,,) in a Banach space X is weakly Cauchy if and only if given two subsequences (y.,~) and (zn) of (z,~), their difference (71, - zn) is weakly null. C o r o l l a r y 1.7.1. Let (fi~) be a bounded sequence in C ( K , X ) , then (fi~) is

weakly Cauchy zf and only if (f,~(t)) 'is weakly Cauchy for" each t r K.

1.8 Notes and Remarks

37

1.8 N o t e s a n d R e m a r k s The results of the first section are due to Bessaga and Petczyfiski and are essentially contained in their classical work [4] of 1958. In our work, Theorem 1.1.2 will be the key to find complemented copies of Co. We believe that this Theorem has been "well known" for a long time (see for instance Exercise 8 of Chapter V of [35], which is a closely related result due to Petczyfiski [99]) and it is in the spirit of the classical BessagaPetczyfiski work. However we have not found explicitly stated in the literature the precise version we give. In any case, versions of this result may be found in [52] or [122]. The relatively recent paper of Rosenthal [114] must be consulted by anybody dealing with Banach spaces containing Co. Although we will not use Rosenthal's paper, it is worth mentioning at least one of its main results ( [114, Corollary 1.2]). In the statement the following definition is needed. D e f i n i t i o n . A s e q u e n c e (bj) in a Banach space is called strongly summing, or, in short, (s.s.), if (bj) is a weak-Cauchy basic sequence so that the scrics ~ c~ of scalars converges whenever

supll n

cj jll < j=l

A weak-Cauchy sequence is called non-trivial if it is non-weakly convergent.

Theorem ( R o s e n t h a l , 1994, [114]). Co does not embed in a Banach space X if and only if every non-trivial weak-Cauchy sequence in X has an (s.s.)-subsequence. It seems that this theorem, or some of the other results in [114], could have natural applications in the study of vector-valued function spaces, and in general in Banach space theory The results of Sections 2 and 3 on copies of gl and g~ are mainly due to Rosenthal. The key result to find complemented copies of 61 is of course T h e o r e m 1.2.1. It is well known and we believe that it is essentially contained in [110, 111], but we have not found it explicitly in the literature either. However, we must call the attention to a result in the literature which is actually an improvement of Theorem 1.2.1: Bourgain in [18, Proposition 11 of Appendix I] provides a characterization of sequences which are finite union of complemented gl-sequences. Proposition 1.3.1 and Corollary 1.3.3 are important to get later the results about copies of g~ in the spaces we are interested in. They were obtained by Drewnowski, but, as he points out, they have their roots in the celebrated work of Kalton on spaces of compact operators [78, Proposition 4].

38

1. Preliminaries

In Section 1.5 we give two old Radon-Nikod~m theorems and two theorems giving a representation of the dual of Lp(#, X) spaces. All these results are due to A. and C. Ionescu Tuleea and follow from the theory of liftings. Lifting theory seems a little bit old fashioned, at least for the general analyst. Many of us feel unease when we need consult the two main monographs which consider extensively the subject: Dinculeanu's "Vector Measures" [40] and A. and C. Ionescu Tulcea's "Topics in the Theory of Liftings" [75] (both written in the sixties). We may add that Diestel and Uhl, in their monograph "Vector Measures" [39], seem skeptical about the usefulness of abstract representations of functionals and operators on L p ( # , X ) using lifting theory. For instance, they say in the Notes and Remarks of Chapter IV: "... neither of which (they mean the abstract representations of functionals on Lp(#, X ) ) h,as found concrete applications to the structure of Lp(#, X) as yet". (see also their Notes and Remarks of Chapter III). Diestel and Uhl's point of view was right in the seventies, but now, applications of lifting theory to vector measures and to the representation of the dual of Lp(#, X) have already shown their usefulness and have to be used for any dealing with our subject. We can mention at least the following points in which, as far as we know, this theory is essential: 1. E. and P. Saab's theorem (theorem 3.1.4) on complemented copies of gl in C(K, X). 2. The Talagrand's fundamental paper on weak Cauchy sequences in L1 (E) [127]. 3. Theorem 4.1.2 on complemented copies of gl in Lp(p,X) (it uses the representation of the dual of Lp(p, X ) and it also uses E. and P. Saab's theorem of point 1). Although the results of this section in their greatest generality are due to A. and C. Ionescu Tulcea, they have a long history. Many authors contributed to get those results, proving them in more and more general conditions. In particular, fbr separable Banach spaces X the representation results of the section are much easier and do not need lifting theory at all. We believe that it is interesting to know how the situation is in this important particular case. Notice first t h a t in the whole section we have been working with functions instead of equivalence classes of functions, and this have been troublesome. However, we have to do that because we need the functions II O(.) II to be measurable for our representing functions ~, and it is easy to find examples (in non separable spaces X ) showing that this is not true for all functions ~P in s ()~, X*). So we have to be careful about which representing function we take among all possible choices. Or, in other words, if we consider equivalence classes of functions in s ()~, X*) we have that not all functions of a given equivalence class enjoy the aforementioned nice measurability property. When X is separable the preceding measurability problem disappears. It is easy to show t h a t in this case the functions II ~(-) II are measurable for all members ~ of s ()~, X*), for 1 _< p _< +oc. Moreover, the equivalence

1.8 Notes and Remarks

39

relation Z is just the usual "to be equal almost everywhere" equivalence relation. So, when X is separable everything works fine identifying functions with equivalence classes of flmctions as usual. On the other hand, for separable Banach spaces X the key result Theorem 1.5.2 was already proved in the classical Dunford and Pettis' paper [50] of 1940 (see theorem 2.1.1 and in general all Chapter II, Part 1, Section A of [50]). The techniques they used can be found in [51, Chapter VI, Section 8]. Observe that this Theorem 1.5.2 was the only one in the section in which lifting theory is used. In fact we deduce in quite a standard way the rest of representation results from it. A very general representation of tile dual of Lp(#, X ) spaces for separable spaces X was given by Dieudonn5 [41] in 1951. For simplicity we have been working in Section 1.5 in finite measure spaces because this is the most important case, and it is the case we will need in our work. However, it should be remarked that all the results are true in much more general conditions (see [40, 75]). In particular, it is very easy to extend the results to a-finite measure spaces. We could add that in the case 1 < p < +c~, tile representation theorems of the dual of Lp(#, X) spaces are true for arbitrary positive measure spaces (this has been noticed at least by Hu and Lin in [74]). The first aim of Section 1.6 is to show that, for 1 _< p < +oo, our study of Lp(p,X) spaces may be reduced to the finite measure case. It should be pointed out that we do not get a reduction for p = +ee. This case seems to be completely different and will be considered in Chapter 5. We make first a reduction to the a-finite case, and once this is done, we simply notice t h a t every Lp(#, X) space with a-finite measure is isometrically isomorphic to an Lp(p0, X) space with finite measure P0 (Proposition 1.6.1). We would like to recall another approach. Given a a-finite measure space (f2, Z, p) there exists a sequence (A,~) of pairwise disjoint measurable sets of finite and positive measure whose union is g?. Let us call #,. the restriction of # to ~(A,~) = {A C Z : A C A~}. Of course, the #~'s are finite measures. On the other hand, it is clear that Lp(#,X) is isometrically isomorphic to (~ | X))~, for 1 <_ p _< oo, where, as usual, ( ~ | denotes the sum of the Banach spaces X~ in the sense of.~p (see [93]). In this way, it would not be difficult to show, for instance, that if (.fk) is a c0-sequence (respectively an ~.l-sequence) in Lp(tt, X), then there exists n C N such that (XA~ (.)fk) has a c0-subsequence (respectively, an ~l-SUbsequence) in the space Lp(#~, X). The second goal of Section 1.6 is a problem of measurability (proposition 1.6.3 and corollary 1.6.1). Probably these results are well known but we have not been able to find them in the literature. We believe that the application of Souslin operation we give could be useful in some other similar situations. The third goal of the section is showing how the differences between Lp(#) spaces depending on the measure # is purely atomic or not, have a natural

40

1. Preliminaries

immediate translation into L~,(#,X) spaces. Actually, the natural place of the results in this direction (Propositions 1.6.4 and 1.6.5) would be the isometric theory of Lp spaces (see [84, Chapter 5]). We have preferred to avoid its technical difficulties, but let us say a few words. Recall that the important thing in the definition of Lp(#) (and in the same way, of Lp(#, x ) ) is not the measure space (~2, Z, #), but tile underlying measure algebra (~, t~). In other words, given two measure spaces (Y21, Z1, #1) and (Y22,Z2, #2), Lp(#l, X) and L~,(#2, X) are isometrically isomorphic whenever the corresponding measure algebras (~a, ill) and (~2, fi2) are isometric. In particular, we deduce from Caratheodory Theorem [84, Section 14, Theorem 5 ,and Corollary to Theorem 9] that if (Y2, Z, #) is a separable (and a-finite) measure space then Lp(ls , X ) is isometrically isomorphic to one of the following spaces: 1. Lp([0, 1], X) (purely nonatomic case), 2. gp(X) (purely atomic case), 3. (LI,([0, 1 ] , X ) | f~,(X))p (non purely atomic case, infinite number of atoms),

4. (Lp([0, I], X) e ~(X))~ where g~(X)

(non purely atomic case, m atoms),

denotes the sum of X in the sense of ~p, m times.

Last paragraphs of the section are devoted to the important concept of

't~ni[orm integrabilit y and we suggest the reading of the survey paper [36]. There is a minor technical difference between the definition of [36] and the one given here (see Theorem 1 of [36]), but this should not cause any problem. Finally, Section 1.7 deals with C(K, X) spaces. We give the classical extension of Riesz theorem found by Singer, and the characterization of weak convergence in C(K,X) (proposition 1.7.1). This last result has been very well know for a long time hut, as far as we know, it was noticed for the first time in its complete generality by Dobrakov in [42, Theorem 9].

2. C o p i e s of Co and

in

Lp(tt, X)

We can say t h a t the starting point in the search of copies of co and ~1 in vector-valued function spaces is in the pioneering works of HoffmannJ0rgensen [73], Kwapiefi [83] and Pisier [101] in the seventies. The results obtained then were the following: Theorem

( K w a p i e f i , 1974

[83]). For 1 <

a copy of co if and only if X does. Theorem

( P i s i e r , 1978

[101]). For 1 < p < 0% LR(P,X) contains a

copy of ~1 if and only if X does. It should be mentioned that Kwapiefi's theorem lies very much on the Hoffmann-J0rgensen's preliminary work [73]. This chapter is devoted to study these two theorems, but in our way we will have occasion of learning also some other interesting results such as Bourgain's theorem 2.1.1 on averaging of co-sequences, Kadec-PetczyfiskiRosenthal l e m m a 2.1.3, or Maurey-Pisier-Bourgain theorem 2.2.1 on glsequences in Lp(#, X). Since we are interested in the search of copies of Co and ~1 in Lp(#, X), let us r e m e m b e r first which is the situation in the scalar case. We can make the following quite elementary analysis. For 1 < p < oc, Lp(#) spaces are reflexive, and so it is clear that they can not contain copies of any of the spaces we are considering: Co, ~1, or foo. It is also clear that LI(#) does not contain copies of Co, because L I ( # ) is weakly sequentially complete and Co does not enjoy this property. Of course, if it does not contain copies of Co, it can not contain the bigger space fo~. Notice that an idea behind Kwapiefi's theorem is the following: "Since Lp(#) has no copies of co, if co lies in Lp(#, X ) then it must lie in X " . And a similar philosofy is underlying Pisier's theorem, too. As we mentioned in the introduction these ideas have been very important and fruitful in the study of vector-valued function Banach spaces. Soon after the works of Hoffmann-Jorgensen, Kwapiefi and Pisier, Bourgain [14, 16] provided new proofs of the preceding theorems, giving results and introducing points of view which turned out to be very important in subsequent works. This is why Bourgain's ideas lie all over this chapter. In particular, we will follow his approach to get Kwapiefi's theorem because it

42

2. Copies of co and gl in L;(#, X)

contains a result (theorem 2.i.1) which will be crucial in C h a p t e r 3 to get Saab and Saab's t h e o r e m 3.1.4.

2.1

Copies

o f Co i n L p ( t t , X )

If one thinks a little bit how to show t h a t if Lp(#~X) contains a copy of Co then X must contain a copy of co, too, a first n a t u r a l question arises: If (rio is a Co-sequence in Lv(p,X), can we take ~o E ~2 such t h a t ( f . ( ~ 0 ) ) is a Co-sequence in X ? However, one immediately realizes t h a t this question is not properly formulated, since to find a point in a measure space is in general meaningless, because singletons are very often null sets. On the other hand, t r y i n g to make such an assertion a b o u t the sequence (fn(WO)) itself is clearly too much, even in the simplest situation. W h e n we have a c0-sequence (Xn, y~) in X • X , for instance in e0 • Co, we can not hope t h a t either (x~) or (y~) to be a co-sequence, we only know t h a t either (xn) or (Yn) have a Cosubsequence. Of course, this can be easily translated to Lp(#, X). We can give for instance the following quite trivial example: Let A1, A2 be two disjoint measurable sets with finite positive measure, and let (en) be the canonical basis of Co, then we define

for each n E N, where i = 1 if n is odd and i = 2, otherwise. It is immediate t h a t (fn) is a c0-sequence in Lp(#, co), but (fn(~)) is not a co-sequence for ,any point ~ C ~2. T h e preceding a r g u m e n t s lead to a more accurate formulation of the question: if (fn) is a Co-sequence in Lp(#, X), can we find a subsequence (fn~) of (f~) and measurable subset A of ~ , with positive measure, such t h a t (f~k (w)) is a co-sequence in X for each ~ C A? The following simple example shows t h a t the answer is "No".

Example 2.1.1. For 1 < p < + ~ there is a co-sequence (fn) in Lp([O, 1],Co) such that for each subsequence (fnk) of (fi~), (fnk (t)) is not a co-sequence for

(almost) any t C [0, 1]. It is enough to define for each n E N f,~: [0,1]

t

--+

Co

---+ (rn(t) + 1)en

where (e~) denotes the canonical basis of Co, and (r~) is the sequence of R a d e m a c h e r functions (remember t h a t R a d e m a c h e r functions (r~) are defined by

r~(t) = sign(sin(2%rt) ) for all $ C [0, 1] and all n C N).

2.1 Copies of co in Lv(p,X )

43

On one hand, it is clear t h a t for each subsequence (fnk) of (f,~), and for almost all t C [0, 1] the sequence ( f ~ (t)) vanishes for infinite n a t u r a l n u m b e r s /c, and so it can not be a Co-sequence. On the other hand, let us show t h a t (f~) is a Co-sequence in Lp([O, 1], co). Let ( ~ ) be a finite sequence of scalars. We have 1

m

=

mffx t A= I

=

max

fl

= =

Ir,~(t)+lldt

I An(r~(t) + 1) I dt

n Jo

<

j~0

infix I An(rn(t) + 1) l d t

/0

I1 E ~ f ~ ( t ) I I

dt

n

II Zxnf ?l,

<_

II E A ' ~ f ' ~ lip. n

And we also have

_-(/o 1 II

(ff -< (ff

=

=

(m~x I .~(r~(t) + 1)I)Pdt) 1/p (max 21An I)Pdt) 1/p

2maxlAn n

IIp dt)

I.

Therefore (fn) is indeed a c0-sequence in Lp([0, 1], co). If we look at the sequence (fn) of the preceding e x a m p l e we see t h a t a l t h o u g h (fn~(t)) is not a c0-sequence for (almost) any t E [0,1], (fi~(t)) does have a Co-subsequence for (almost) all t E [0, 1] (the point is t h a t the subsequence depends on the t). This leads us to a new question: if (fn) is a c0-sequence in Lp(tt, X), can we assure t h a t the set of all co E (2 for which (fi~(co)) has a co-subsequence is non null ? Bourgain showed t h a t the answer is "Yes" in the case p = 1, and we will see t h a t for 1 < p < co the answer is "Yes", too. Of course, this provides an answer to the p r o b l e m posed at the beginning of the section: this guarantees t h a t if Lp(/*, X ) contains a copy of Co then X has the s a m e property. It m i g h t be surprising t h a t some results in this section are stated for

seminormed spaces. However this is not a more or less superfluous search of

44

2. Copies of co and ~1 in L~(p,, X)

generality. We will see in T h e o r e m 2.1.2 and in Saab and Saab's t h e o r e m 3.1.4 t h a t this is actually convenient for our purposes. We will consider c0sequences in s e m i n o r m e d spaces. Denote by Coo the n o r m e d subspace of Co of all sequences of only finitely m a n y non-zero terms. We will say t h a t a sequence (x.,~) in a s e m i n o r m e d space (X, II 9 It) is a c0-sequence if there exist two positive n u m b e r s M, d such t h a t

for all (a~) E Coo. Of course this simply means that, by passing to the quotient n o r m e d space, the corresponding sequence is a c0-sequence in (the completion of) this n o r m e d space, too. In the proofs of the few results we will state for senfinormed spaces the only facts of the theory of n o r m e d spaces which are applied are Bessaga-Petczyfiski t h e o r e m 1.1.1 and Bessaga-Petczyfiski selection principle. It is easy to show t h a t the usual proofs of these t h e o r e m s also work for seminortned spaces or, better t h a n this, one can see in the s t a n d a r d way, by passing to the quotient n o r m e d space, t h a t Bessaga-Petczyfiski results for s e m i n o r m e d spaces follow immediately from the corresponding ones in n o r m e d spaces. Before going into Bourgain's a p p r o a c h we will s u m m a r i z e some i m p o r t a n t properties of R a d e m a c h e r functions (r~) in [0, 1], because they will play a very i m p o r t a n t role in our study. These properties are of course quite well known, but for the sake of completeness we will include proofs. Proposition 2.1.2 is just a version of the "contraction principle" [37, 12.2]. ' P r o p o s i t i o n 2.1.1. Let (X, ]] are elements of X , then

oll

II)

be a seminormed space. If x, xl, x 2 , . . . , x,~

dt=2,, 1

x + i=l

IIx+ E 01=~l

for any collection of natural numbers nx < n2 <

II x

+

(t)x i=1

tl dt =

.

.

.

/o1 II x +

'

Oixi

II

i=l

< rim. Consequently,

r (t)x

II dr.

i=1

Proof. T h e result is an i m m e d i a t e consequence of the fact t h a t rnl, rn2, rn~ are i n d e p e n d e n t (symmetric) r a n d o m variables which take the values 1 and - 1 with probability 1 In fact, if for each O = (01, .,0m) C { - 1 , 1 } "~ we put 9

-

9

,

7Tt

A e = {t e [0, 1] : r,~(t) = 0i for 1 < i < m} = N { t E [0,1] : rn,(t) = 0i}, i=1

it is clear t h a t this set has m e a s u r e 2A~. Since, for a given m, the A o ' s form a partition of [0, 1], we have

2.1 Copies of co in Lp (#, X ) 91

7T~

nt

= 9

i

45

~ OG{--1,J

1

} ....

1 ~ 2"~

-

II ~ + ~

~ O

llx+ E

Oi=•

r~,(t)zi It dt

i=1

Oia:i

II

"

i=1

P r o p o s i t i o n 2 . 1 . 2 . Let ( X , II . II) be a s e m i n o r m e d space. I f zo, z~, z 2 , . . . , ~:,,~ are e l e m e n t s of X arm ao, al, a,.2, 9 9 am are real numbers, then

/o1II ,~0,0 § ~

....

'

~ m ( t ) ~ II at <- - Onlax ] ai I
II x0 +

i=1

•

~ ( t ) ~ II at.

i=1

In particular

II xo II _<

In

.•01

II :~o + ~ ,.,(t)** II ~t,

(2.1)

i=l

and 91

9

'n~--i

.I

i = 1

,@

i=1

"

Proo./: By

h o m o g e n e i t y , we m a y a s s u m e I ai I -< ] fo~' g E {0, ] , . . . , - , , } L e t us c o n s i d e r t h e real f u n c t i o n defined on R - -(,g2 m + l ,, t h e u n i t b a h of era+; by '30 /,

Ca0,-, .... , ~-~)'

>/

,

.l

~7~

II ,,o~:o + ~ airi(t)zi It dt

'/ 0

i

1

This function is obviously continuous and convex. Therefore, it takes its m a y i m u m a m o n g the extreme points of" the. c o m p a c t set --,~ocR(Un+l~/,which are the points of the tbrm ( e 0 , q , . . . , e , , ) with q = :t:1 for i = 0, 1 , . . . ,m. Notice now t h a t it follows easily from the preceding proposition t h a t in all these points tile function takes the same value.. To be precise, 9l

II 60"s

9 )

+

~{r~(t):,:~ II dt =

fi i

1

/0 l

II 2:0 + ~n), ri(t):ci {I dt

'

i=1

tbr any choice of signs Co, q . . . . , e,,, = :t_1. Hence, for any (ao, al . . . . , a,~) C B (\f' ~, ,5,o+ l )/ W C h a v e

II 9

+ ~ a~.~(t):~,~ II dt _<

'

/=l

/o' II

"

+

•

rdt)z,z II at

i=]-

This completes the proof of the first inequality. The others are just particular cases of it.

46

2. Copies of co and gt in Lp(#, X)

Now, let us begin to study Bourgain's approach. The key of this approach is L e m m a 2.1.2, which is an "averaging version" of Bessaga-Petczyfiski classical theorem 1.1.1 on c0-sequences. In its proof we will use the following simple fact about sequences of positive numbers. L e m m a 2.1.1. Let (Ai) and (6.,) be sequences of positive numbers and ass u m e that (Ai) is bounded and non decreasing, then (Ai) has a subsequence ( Ai,~ ) such that ~i,.+l _< (1 + &7,);~i.~ for all m E N. P r o @ Observe first that the sequence (Ai) must be convergent. Let us denote by A its limit. Consider the sequence ( ~x) . It is clear that

lim~A = 1 < 1 +(~m for all m C N. So, (Ai) has a subsequence (Ai~) such that A - -

<

Ai,,

1+din

for all m E N. Since (Ai) is non decreasing, we have Ai < A for all i. Therefore, in particular, we have ,ki,,~ for all m. L e m m a 2.1.2. /f (X, II 9 II) is a seminormed space and (xi) is a non null sequence in X such that f 1

sup I ~z

dO

n

II

II dt < i=1

then (:ci) has a Co-subsequenee. Proof, Let (zi) be a sequence in X such that limsup ]1 xi I1> 0 and

sup n

II .

r~(t)x~

dt = B < +oe.

i=1

Observe first t h a t according to Propositions 2.1.1 and 2.1.2, for each subsequence (zk~) of (zk), we have

2.1 Copies of c0 in L~(#,X)

sup

II

7~

n(Oxk, II dt

=

II

sup

"k~(t)xk~ II dt

n

i= 1

i=1 1

s

<

47

rn

sup/ nz

[I E

Jo

ri(t)xi II d t = B < +oo,

i=l

and therefore,

sup n

II .

n(t)*k~ II dt ~ B < +<~

(2.2)

i=1

T h e n we can assume without loss of generality t h a t there exists 7 > 0 such t h a t II xi I1_> ~ > 0 for all i E N. Let us see now t h a t (xi) must be weakly null. Given x* E X*, t h a n k s to the Khinchine's inequalities, for a certain A > 0 we have

A(Z

'/'

<

i=1

I 9

<xi,x*>r~(t)

=

ri(t)xi,z* >t dt

l< i=l

9

.1

_< s

ldt

i=1

m

!tY~(Ox., II II~*lldt<_BIIx*ll i= 1

for all m E N. Thus, } - ~i=i I < x i x* >l 2 converges and so lim < :r~, :1:* > = 0. i

Take now a non increasing sequence (6.,,,) of positive n u m b e r s such t h a t o~

cA3

1-[(1 + < ) < +oo

Z2' 6,, < +oo

n

~=1

1

Using the Bessaga-Petczyfiski selection principle (see [35, page 42] or [93, pages 4-5] ), we can extract a subsequence (Y0 of (xi) in such a way t h a t (Yi) is a basic sequence and satisfies

I[

a~> I1_<(1 + &,)II ~ ~y~ bl i--1

i

(2.3)

1

if ~, < m and a , , a 2 , . . . , a,,, are scalars. By 2.2 and Proposition 2.1.2, (.1~ II Z*j=~ r~(t)y~ 11 dt)~~176 is a b o u n d e d non decreasing sequence of positive numbers, so, it follows from l e m m a 2.1.1 t h a t there is an increasing sequence (i,,~) on n a t u r a l n u m b e r s such t h a t

48

2. Copies of co and gl in Lp(p, X ) 1

i.~+1

fo IIE ri(t)yi IIdr<_ (l+5,,j~ i

1

{,~

I[ Er~(t)Y,

1

Ildt

(2.4)

i=1

for all m C N. Our aim now is to show that (yi~) is a c0-sequence. This is the only point which remains to be proved, however it is tile hardest part in the proof. Since (Yi,,) is a seminormalized basic sequence, by Theorem 1.1.1, we only need to show that ~ Yi., is a w.n.C, series, that is, we have to show that

sup{ll

}-~O,,y~,, I1:-,,

c H, 0n = +1} < +oc.

(2.5)

n=l

To prove this let us define some quantities which relate the expressions involved in 2.3 and 2.4. For any finite choice of signs (01,02,..., 0,~) put

p(Ol,02,... ,Om) ['1 II

o~y~ + ~ ~(t)y~ II dt

=

aO

n=l

iCGm

where ~,,~ = { 1 , 2 , 3 , . . . , i , ~ } \ {it,i~,...,i,~}. follows from 2.1 of Proposition 2.1.2 that

First. of all, observe that it

O,~yi,, IIs p(Ol,O2,...,Om)

II s rz: l

Therefore, in order to prove 2.5, it is enough to show that the numbers t)(01, Oe,..., Ore) are bounded. Second, notice that it follows immediately from 2.3 that for each choice of signs (01,09,..., 0m) we have

p(01,02,... ,Ore)

II

= 9

s 91

<

O,~yi~ § }7_, ~(t)y~ tl dt n

1

iEO'm

'nz+]

/ (1 + <,) I1 ~ <,w~ + ~, /0

9

,.~(t)y~II dt

i~o-,,t+l

n:l

<_ (l+*m)p(O1,...,Om,Om+l). Therefore,

p(<,<,...,

e,~) < (1 + &,)p(<,..., r

e,,~+~)

(2.6)

We will need another property of the numbers p(01,02,..., Ore) which is contained in the following Claim. We will prove it at the end of this proof.

Claim: .[01 II ~<"~=1 ri(tDyi II dt = .1o,1 p(r] (t), r 2 ( t ) , . . . , r',~(t))dt With these properties of the p(01,02,..., 0m)'S in mind we only have to do some computations to finish our proof. By our Claim and 2.4 we have

2.1 Copies of co in Lp(#, X) 1

./i I D(rl ( t ) , . . . , r m (t), rm+l ( t ) ) d t

im+l

= fo II ~ '

<

49

,.~(t)y~

II dt

/=1

It~-~.ri(t)yill dt

(l+6m)~o

i=1

/o

=

(1 + 5r~)

p(r~(t),...,r,~(t))dt,

and thus

./i ~ (p(r~(t),... ,,',,,,(t),r,-~+~(t)) - (1 + 5,~)p(rl(t),... ,r,~(t))) dt <_ O. Then, given a finite collection ( 0 1 , . . . , 0 .... 0re+l) of signs and putting S = {t G [0, 1]: (l'l(t),... ,rm(t),Fm_l_l(t)) r (01,.

9,0,~,om+~)},

we have

.fo I (p(rl (t),. . . ,r,,(t),rm+l (t)) - (1 + (~,~)p(rl(t), 1

-

-

2,,+](p(O,,...,O,,,Om+,)-

§

/ (p(,.,(t),

<

0.

1

(1+Sin)p(01,..

,0,~))

.,,.,,~(t), I'm+ 1 (t)) - (1 + ~,dP(,'~ (t),

. , ,.,,~(t))) dt

Hence, using 2.2, 2.6 and the Claim, we have 1

2m,+1 (p(01,..., 0m, 0re+l) -- (1 -~- (~m)tO(0l,... ,0m)) ./S ((1 -I- (~m)p(rl(t),... ,r,n(t')) -- /9(rl(t),... ,rrn(t),"m+l(t)) ) d t

s

<_ .

(l+5,,)p(rl(t),.,.,r,,,,(t))

(5,, + 1

1 ~U(~).

,

l+6,

p(r](t),...,r,,,,(t))

p(ra(t'),... ,r,,,(t)) dt

.1

< 26,,,.~ p
=

2~,,~/i '

i,,~

II ~(t)w

II dt

< 25,,,.B

i=1

Therefore,

p(O,,.

,0,,,, 0,,~+1) _< (1 + (~m)p(01,..., 0m) + 4B2"~6,,~

)

dt

50

2.

Copies of co and

& in Lp(p,X)

Now, by induction, it is easy to see that

p(O1,...,Om,Om+l) <--n=lH(14-(~,z)p(01) +4B

nI~=l(l+~n)

(En=12nt~n)

and then

~(01,...,0,~) <

(p(1)+p(-1))H(I+6~)+4B

(1+6,~)

n=l

(

2~6~)<+oe n=l

for all m r IN and all finite collection of signs ( 0 1 , . . . , Ore). Hence we have shown that the numbers p(01,02,..., Om) are bounded. To finish the proof of the lemma it only remains to show the Claim.

Proof of the Claim: It is enough to observe that, using Proposition 2.1.2, we have .1

.~ p ( r , ( t ) , r 2 ( t ) , . . . , r , ~ ( t ) ) dt

=

j~0l(~l

~

r,~(t)yi. +

n=l

i~o-, n

2~m--.~----7 E

7~t II Er,~(t)yi.

el =-t- 1,iC or,.

:

=

1 1 2im-m 2m

1

2i,: II ~ 91

= This

e,:

E

•

/

+ E eiyi II at

n= l

0.=•

i E ~r.~

n=l

iCam

eiyi II

i=1 i.~

II ~ ~'i(t)y~ II dr.

s

'

i=1

completes the proof of the Claim and therefore, the proof of the

too.

It. is interesting to observe

i '

"n, i=1

that, since '17,

1

II ~0ix~ II, Oi 4zl

i=1

lemma,

2.1 Copies of c0 in L p ( # , X )

51

the preceding lelnma says that if (xn) is a non null sequence for which the set 1

{~

EII 0~=-F1

II: n E N, 0i = =t=1}

EOixi i=1

of averages is bounded, then it has a subsequence (Yi) for which the set n

{11 ~ O~y{ I1: ,~ E N, 0{

=

+1}

i=1

is bounded. Notice also that the preceding lemma provides the following particular version of one of the theorems we are trying to show (Theorem 2.1.2): Let (xi) be a sequence in X and assume that (ri(.)x{) is a cosequence in L1 ([0, 1], X ) , then (x~) has a co-subsequence.

We can finally state and prove Bourgain's theorem. T h e o r e m 2.1.1 ( B o u r g a i n [14], 1978). Let (f2, 2 , # ) be a finite measure ,space and suppose that for each co E f2, I . I~ is a s e m i n o r m on Coo such that co ~-~l x I~ is integrable on (f2, Z , p ) for each x E coo. Let us define the s e m i n o r m tl . IIo on Coo by

IIx Iio=/Ix

I~ d~(co)

for every x E coo. z] (xi) is ,~ sequence in Coo which is a co-sequence for II Ilo, then there exists a positive measure set A E ~ such that (xi) has a Co-subsequence for I. I~. whenever co E A. Proof. Assume that the hypothesis of the theorem holds. It is clear that thanks to the preceding lemma it, is enough to show that, the following is true: C l a i m : There exists a positive measure ,set A E Z such that .1

l i m s u P l x i I~> 0 and n,

sup/ 71,

.10

71,

I Zri(t)xi

I~ dt < +oo

i= 1

for all co E A.

Let M and 5 be two positive numbers such that 5max lai I < - II ~ - ~ a i x i IIo = / i l Z a i : r ' i I~; d#(w) <_ M m a x l a~ I

2. Copies of co and ~1 in Lp(tZ, X)

52

for all (ai) E coo. By our hypothesis, for each i E N, the function r defined by r = ]xi l~. for cc E g2, belongs to L1 (tL). Moreover, (r is a seminormalized sequence in L I ( # ) because

for all i E N. To prove t h a t our Claim is true we will show t h a t (a)

(r

(b)

p({w E ~2: l i m s u p i r

(c)

sup~z ~o1 I E i n

is uniformly integrable,

1

> 0}) > 0, and

ri(t)xi tw dt < + o c tbr almost all cv E f2.

If (a) does not hold, then it is well known (see [35, pages 93,89]) t h a t there exist a sequence (Ai) of pairwise disjoint measurable sets, a subsequence of (r which we will continue to denote in the same way, and an e > 0 such that

'

i

for all i E N. Now, Rosenthal's disjointification l e m m a 1.2.1 says t h a t there is an increasing sequence (ik) of n a t u r a l numbers for which

fbr all k E N. Now, we reach the tollowing contradiction: for each n E N we have

k:l

dg2

k=l

m=l

'

im

k:l

> nt

Z

'

i.~

'

im

k=l ,kern

d)~k(aJ)) dp(w)) II?,~1

k=l,k@m

>_ 'n~- ~ /A ( ~ ~rJ 1

=

~

-

k=l

r

i,,~ k = l , k T ~ m

m

~/c

l;

li.,

r (~)

d~(~)

) > .,e - ~

k=l

,~=1,.~r

A~,,,

= 7t~.

2.1 Copies of co in

Lp(~,X)

53

Thus (a) is proved. Now, let us note t h a t (b) is an i m m e d i a t e consequence of (a): If we suppose t h a t #({cJ E s : l i m s u p i @(co) > 0}) = 0 then lim~@(co) = 0 for almost all c~ E (2, because the functions @ are non negative. Hence Vitali's l e m m a [51, IV.10.9.] says us t h a t limi II r It~ = 0, which is of course a contradiction. In order to prove (c), for each n E N put ~(co) =

/01• ]

r~(t)z~ I~ dt

i=1

for all co E f2. Clearly ~ belongs to LI(#) since, by proposition 2.1.1,

Oi=:t:l i = l

and by our hypothesis, each function co ~1 ~'~in=l OiXi [w belongs to L I ( p ) . According to P r o p o s i t i o n 2.1.2 it is also clear t h a t ( ~ ) is an increasing sequence. On the other hand, since for each n E N

/; 9

0~=-4-1

-

1

i=1

II

fi

0 x ll0 <M,

Oi=-b_l i = I

we deduce from the Lebesgue's m o n o t o n e convergence t h e o r e m t h a t the function co ~ sup~ ~,,(co) is integrable, and therefore, sup{~(co) = sup rt

n

I dO

ri(t)xi Ix at < +oc i

1

tbr almost all co E ~2. This finishes the proof of (c). Once we have shown (a),(b) and (c), to complete our proof it is enough to observe t h a t (b) and (c) imply t h a t our claim is true. Now, the following two results are i m m e d i a t e consequences. T h e o r e m 2.1.2. Let (f2, ~ , p ) be a finite measure space and let (fn) be a co-sequence in L I ( # , X ) . Then, the set of all co E f2 such that (fn(co)) has a Co-subsequence is a measurable set with positive measure. Proof. Let (f,~) be a c0-sequence in L1 (#, X ) . T h a n k s to Proposition 1.6.3, we only have to show t h a t the set of all a~ E s such t h a t (f~(co)) has a c0-subsequence is a non null set. By our a s s u m p t i o n , there are two positive n u m b e r s M and ~ such t h a t

54

2. Copies of co and

5 m a x l a~ I < II ~ a ~ f ~

~1

in Lp(p,X)

II~ = / ~ If ~~aifi(c~)II d~(~) ~ M m a x l a~ I '

i

for all (ai) E Coo. This suggests to define for each w E f? I x I~ = II Z

aifi(w)II i

for all x = ~ aiei 6 Coo. Of course the function co ~-~l x I~ is integrable on (f2, Z , # ) for each x E coo, and so we can consider the s e m i n o r m II 9 Iio on coo as defined in the preceding theorem. Moreover, the canonical unit vector sequence (e~) in coo is a c0-sequence for 11 9 Ilo because

i=1

i=1

i=1

n

i=1

Therefbre, the preceding t h e o r e m implies t h a t the set of all points w E ~2 for which (e~) has a co-subsequence for I . I~ is a non null set. To complete the p r o o f it is enough to observe t h a t "(e~) has a e0-subsequence for I 9 I~" simply m e a n s t h a t (fn(c~)) has a c0-subsequence.

2.1.3. Let (f?, Z, #) be a finite measure space and let X be a Banach space, then L1 (#, X ) contains a copy of Co if and only if X does.

Theorem

Proof. T h e condition is of course sufficient. Necessity follows from the preceding theorem. Our aim now is to show t h a t the preceding results are also true for Lp(#, X ) with 1 < p < +oc. To prove this we will use the following "subsequence splitting l e m m a " for L1 (/L)-bounded sequences. It was discovered by Rosenthal [113], as a refinement of results and a r g u m e n t s due to K a d e c and Petczyfiski [77]. As far as we know it is the sharpest result of its kind. 2.1.3 (Kadee-Petczyfiski-Rosenthal, [77, 113]). Let (f2, Z , # ) be a finite measure space. If (fn) is a bounded sequence in L I ( # ) , then there exist a subsequence (f,~) of (fn) and a sequence (Ak) of pairwise disjoint measurable sets such that (X,\Ak f'~) is uniformly integrable.

Lemma

Proof. Of course if (f,~) is uniformly integrable the result is trivial: it is enough to take ( f ~ ) = (fn) and AN = O for all n E H. So, let us suppose t h a t (.f~) is not uniformly i n t e g r a b l e This means t h a t there exists e > 0 satisfying

2.1 Copies of co in L p ( f , X )

55

there exist a sequence (Ej) of measurable sets with limj p ( E j ) = O, and a subsequence (fnj) of (rio such that

i

for all j 6 N Let ~0 be the supremum of all ds for which (*) holds. It is clear that we can take a subsequence of (f.T,), which we continue to denote in the same way, and a sequence (E~) of measurable sets such that 1 #(E,J < ~

and

js

If n l d # > e o -

1

2~

n

for all n C N. Take nl = 1. Since lim p(U~_kE,, ) = 0, k

by the absolute continuity of the indefinite integral of ] f,~, 1, there exists -,2 > ",a such that

L

1

, "~\u~=-2E" [fn, I d# > eo - 2,---T. Put

Now, applying the same argument

to

Ifn2

I, we get n3 > n2 such that

~: \u~=~3E~ ]fn2 ] dtt > eo - ~2n. Put,

A2 = En: \ LJ?z~

T~ 3

E,~.

It is clear that in this way we obtain a sequence (Ak) of pairwise disjoint measurable sets, and a subsequence (f,~) of (fn) such that /A

I fnk I d# > eo - 2n-~1

for all k 6 N. Let us finish the proof integrable. Suppose it is not the case. of (f,~), which we continue to denote of measurable sets with limk #(Bk) =

showing that (Xn\a~f'~k) is uniformly Then there exist e' > 0, a subsequenee in the same way, and a sequence (Bk) 0, such that

B ]Xr~\Akf~k I d # > d k

56

2. Copies of co and gl in Lp(#, X)

for all k e N. But this implies that 1

Xa\ak ]fnk [ d# > eo - 2n k + e' for all k E N. Since e0 - ~ 1- +e' > ~0 for k large enough, this is a contradiction with the fact that e0 is the supremum of the e's satisfying (*). We can now begin our study of Lp(#, X) spaces. Next theorem is a good example of how the preceding lemma can be used to deal with vector-valued functions. T h e o r e m 2.1.4. Let (s Z , p ) be a finite measure space, let 1 < p < +oo, and let i : L p ( p , X ) ~ LI(p,,X) be the canonical embedding. Let H be a subspace of Lp(#, X ) which does not contain a complemented copy of gp, then the restriction i IH of i to H is an isomorphic embedding.

Pro@ If i ]H is not an isomorphic embedding, then not all II 9 [11-null sequences in H are [I 9 lip-null. Thus, using the continuity of i, it follows that there exists a II. II1-null sequence (f~) in H such that 11f~ lip = 1 for all n E N. Taking a subsequence if necessary, we may suppose that a2 \ (fT~(J), and therefore (11 f~(c~)If), goes to zero for almost all aJ E ~2. Now we can apply the preceding lemma to the sequence (11 f ~ ( - ) I f ) , and we get a further subsequence, which we still denote (f~), and a sequence (A,~) of pairwise disjoint measurable sets such that (X,\A~ II f~(.) IIp) is uniformly integrable. An appeal to Vitali's lemma [51, IV.10.9.] allows us to conclude that (X,\A~ II fn(.)II t') is null in LI(#), that is, (X,\Anf~) is null in Lp(#, X ) . Therefore, taking again a subsequence if necessary, we have that ( X ~ fn) = (f~ - Xa\A~ f~) is a seminormalized sequence in Lp(#, X). Since the functions are disjointly supported, (X~ fn) is a complemented gpsequence in Lp(#, X ) (Proposition 1.4.1). At this point, the classical result on perturbation of basis (see [93, 1.a.9] or [35, Chapter V, Theorem 12]), guarantees that (/n) = (X~o f~ + X,\~o/,~) has an/?p-subsequenee which is complemented in Lp(#, X), and therefore in H. This completes the proof.

Remark 2.1.1. Notice that it is possible to give more precise versions of the preceding theorem. For instance, it is also true for the canonical injection from Lp(#, X ) into L~(#, X), with p > r > 1. We may also deduce from the proof that it also holds for all subspaces H which do not contain sequences arbitrarily close to normalized disjointly supported sequences, and in particular, which does not contain (1 +()-complemented (1 + c)-/?p-sequences for arbitrarily small ~ > 0.

2.2 When does Lp(#, X) contain a copy of gt?

57

As an immediate consequence of the preceding theorem we have: C o r o l l a r y 2.1.1. Let (~2, Z, >) be a finite measure space and let 1 < p <

+oc. Then we have: (a) Every Co-sequence in L1,(p , X ) is a co-sequence in LI(#, X ) . (b) Every gl-sequence m Lp(p, X ) is an fl-.sequence in L1 (p: X ) . (c) Every g<sequence in L p ( # , X ) is art g,--sequence in L I ( # , X ) , whenever l <_r < +oo, r ' r (d) If Lp(p, X ) contains a copy of H and H does not contain complemented copies of gp, then L1 (#, X ) contains a copy of H. We can now extend Theorem 2.1.2 to Lv(#, X ) spaces with 1 < p < +co. In the extension we also remove the finiteness assumption on #. 2.1.5. Let (f2, 5 , p) be an arbitrary measure space, let 1 <_ p < +oc, and let (f~) be a co-sequence in Lp(#, X). Then, the set of all co C g? such that (f,~(oo)) has a Co-subsequence is a measurable set with positive

Theorem

7Tteasurc.

Proof. Let (f,~) be a c0-sequence in Lp(#, X). Notice first that if # is a finite measure, we can apply the preceding corollary, and we deduce that (fi~) is a c0-sequence in L1 (p, X ) , too. Now, the conclusion follows from Theorem 2.1.2. For the general case, we use the standard reduction argument we have a h e a d y given in Corollary 1.6.1. By L e m m a 1.6.1, there exists a a-finite measure space (f~l, ~1,#1) C (f~, E,p) such that (f~) lies in L p ( # I , X ) . Now, by Proposition 1.6.1, there exists a probability measure P0 on (/21, Z1) and an isometric isomorphism from LF(#I, X) onto Lp(po, X). Moreover, as we showed in the proof, this isometric isomorphism has the form f ----+ h f where h is a certain scalar function which does not vanish at any point. Therefore, it is cleat" that (h(co)fi~(w)) has a Co-Subsequence if an only if so does (f,~(co)). So the conclusion follows from the first part. Finally, we get the following

2.1.6 (Kwapiefi). Let (~2, ~ , # ) be an arbitrary measure space and let 1 < p < +0% then L p ( # , X ) contains a copy of Co if and only if X contains a copy of co.

Theorem

2.2 When

does

Lp(lz, X)

contain

a copy

of s

Our initial point of view here is very similar to the one in the preceding section. In this way we may see that the natural modification of Example 2.1.1 show's t h a t f o r 1 < p < +oo there are *?l-sequences (f~) in Lp([0, 1],~1) such

58

2. Copies of co and ~1 in L~,(p, X)

that for each subsequence (fi~.k) of (f,.), (fn~(t)) is not an gl-sequence for (almost) any t E [0, 1]. So, likewise in the preceding section, it is natural to ask whether given an tl-sequence (f,.) in Lp(#, X), we can assure that the set of all cJ E f2 such that (f,,(co)) has an fl-SUbsequence is a non null set. The main theorem of the section states that we can. We begin our work with quite a general result about sequences in Lp([0, 1],X) due to Batt and Hiermeyer. Recall that (r~) denotes the sequence of Rademacher functions in [0, 1]. P r o p o s i t i o n 2.2.1 ( P r o p o s i t i o n 4.1 of [3]). Let (x~) be a bounded sequence in X and let 1 < p < +oo. Then, the following are equivalent:

(a) (x~) has no e~-subsequences. (b) (r,~(.)x,~) is a weakly null sequence m Lp([0, 1], X). Proof9 Suppose that (a) holds. In order to conclude (b) we will see that every subsequence of (r,~(.)xn) has a further subsequence which is weakly null in L~([0, 1], X). But Rosenthal's theorem says us that each subsequence of (x~) has a weakly Cauchy subsequence. Thus, we only need to show (*) If (yk) is a weakly Cauehy sequence in X and (rnk) is a subsequenee of (r,.), then (r,,~ (.)Yk) is weakly null m Lp([0, 1], X). So, assume we are in the hypothesis of (*) and let F be a functional in Lp([0, 1],X)*. By theorem 1.5.4, there exists a w*-measurable function ~ : [0, 1] -+ X*, such that H~(.)I] is measurable, it belongs to s (as usual, q denotes the conjugate of p) and

_r'(f) =

< f(t), g,(t) > dt

for all f E Lp([0, 1], X). The sequence (< Yk,~P(t) >)k converges for almost all t E [0, 1] because (Yk) is weakly Cauchy. Let us denote by h the pointwise limit of ()k. Since

I< Yk,C'(t) >l <_ ,up,~ IlYnll Ilk(t) II for all t E [0, 1] and all k E N, we can use Lebesgue's dominated convergence theorem. Therefore, h belongs to Lq([O, 1]) and (< Yk,~(.) >)k converges to h in Lq([0, 1]). Now, note that

] F(rnk(.)yk) ] =

'01

< r,,~(t)yk,~P(t) > dt

=

fo

(t) < y k ,

>

~-

~o 1 r'~(t)(--h(t)) dt + ~oo1 rnk (t)h(t) dt

dt

2.2 When does Lp(#, X) contain a copy of &?

59

!

<

(/o 1I

h(t)l q a t

+

/o 1r ~ ( t ) h ( t ) d t

Using the fact that (rnk) is weakly null in L~([0, 1]), it is clear that (F(rn~ (.)Yk)) is a null sequence. Thus we have shown that (a) implies (b). For the converse, it is enough to observe that, for 1 0 satisfies

for all finite sequences ()~,~) of scalars, then we have 1

1/p

for every finite sequence (A,0 of scalars. This finishes the proof. Our next lemma provides a very easy sufficient condition for a sequence in L1 (#, X) to be weakly null. L e m m a 2.2.1. Let (9, ~U, #) be a finite measure space, let (f~) be a bounded sequence in L ~ ( # , X ) , and let us suppose that (fn(co)) is weakly null for almost all co E ~2. Then (f,~) is weakly null in L] (#, X). Proof. By Theorem 1.5.4 we only need to show that if we take a w*measurable function q~ : ~Q --+ X* such that the function If~P(.) Hiis measurable and belongs to s (#), then

/<

A(co),~(co) > d# -~ 0.

But this clearly follows from the Lebesgue's dominated convergence theorem, because lim,~ < .f,.(co), k~(co) > = 0 for almost atl ca r /2 and

I< .f,,(co),~'(co) >I _< liA(~)ll lie(co) li _< sup li A(co)II II ~'(co)li. We can now give the main result of the section.

Theorem 2.2.1 (Maurey-Pisier-Bourgain, 1978). Let (~2,~,#) be an arbitrary measure space, let 1 < p < +oo, and let (fn) be an gl-sequence in Lp(/t, X ) . Let us assume that one of the following two conditions holds (a) It is .finite p = 1 and (f,O is uniformly integrable.

(b) 1 < p <

+no.

Then, there the set of all co C I2 such that (f,~(co)) has art g~-subsequence is a measurable set with positive measure.

60

2. Copies of co and ~71 in Lp(p., X)

Proof. By Corollary 1.6.1 we only have to show that the set of all co E S? such that (fn(co)) has an ~.l-SUbsequence is non null. First, let us consider case (a). By the hypothesis, there exist two positive numbers M, ~ such that

<~Zl~,~,l<_llZ<,,,,.f,~ll,<_ M Z la,~l for all finite sequences (a,<) of scalars. Since (11in(.)II) is uniformly integrable, by standard arguments (see for instance [35, page 111]), it follows that there exist N > 0 and a sequence (An) of measurable sets such that for every n E N we have i ~ \A. IIf,*(co)II d#(co) < 7~

and

II f,~(co)It < N for all w C An.

Take g~ = )lAnfn for every n E N. Notice that (gn) is bounded in LI(v,X) (even in Loo(#, X)) and for every finite sequence (an) of scalars we have

II ~ a~.q~ I1, _> II ~ anf,~ II, -II ~

an(fi~ - g n ) I I ,

Hence, (g~,) is an ~71-sequence. Let us proceed now by contradiction. If (f~(w)) has no Q-subsequences for almost all co E $2, then the same holds for the sequence (9,~). But (9~(co)),~ is bounded in X for almost all co E s and so, according to Proposition 2.2.1, (rn(.)gn(co)) is a weakly null sequence in "LI([0, 1], X) for almost all co E .Q. Let us denote by r~g~(.) the function

> C,([0,1],X)

Notice that the sequence (rngn(.)) is bounded in Loo(#, L1 ([0, 1], X)) because

IIr~g~(.)ll~

--

ess s u p ~

IIr,m,~(co)ll,--

ess s u p ~

/0 '

IIr~(t)gn(~)ll dt

= esssup~en Ilgn(co)ll _< N Thus, by the preceding lemma, it follows that the sequence (rng~(.)) is weakly null in L1 (#, L1 ([0, 1], X)). Since this space is isometrically isomorphic in the natural way to L1 ([0, 1], L1 (#, X)), we deduce that (rn(.)g~) is weakly null in this last space, where of course (r,~(.)g,~) denote the functions [0,1]

t

)

>

L,(#,X) rn(t).qn

Again, an appeal to Proposition 2.2.1 says us that (gn) has no gl-subsequences in L1 (#, X). This contradiction finishes the proof in the case p = 1.

2.3 Notes and Remarks

61

For the case (b), assume first the measure it is finite. In this situation, notice that ( f , ) is uniformly integrable, just because it is bounded in Lp(it, X) with p > 1. On the other hand, it follows fl'om Corollary 2.1.1 that (f~) is an f,~-sequence in L 1 (it, X). Therefore, our sequence is actually in the hypothesis of case (a). So the conclusion holds. For an arbitrary measure we just have to follow the same procedure we used in Theorem 2.1.5. As an immediate consequence of this theorem we get T h e o r e m 2.2.2 ( P i s i e r [101], 1978). Let (~2, Z , it) be an arbitrary measure space and let 1 < p < +oc, then Lp(it, X ) contains a copy of fl if and only if X does.

Remark 2.2.1. One could think that the finiteness assumption on it in part (a) of Theorem 2.2.1 could be removed. However, this is not the case as the following simple example shows: Take any uniformly hounded sequence (h~) of norm one non negative functions in LI([0, +co)) such that ]l,,~(t) = 0

for all t 9 [0, n]

for all n E N. Let (e~) be the canonical fl-basis. It is not difficult to to show t h a t if we take L, = h,,(.)e,~, then (fi~) is an uniformly integrable g~-sequence in L1 ([0, +co), ~1). However, for each t E [0, +co) the sequence (f,.(t)) is eventually null and so, it can not have any fl-SUbsequence.

2.3 N o t e s a n d R e m a r k s Tile importance of Hoffmann-J0rgensen's paper [73] in Kwapiefi's theorem 2.1.6, the main result of section 2.1, can not be overestimated. Kwapiefi's short note [83] was published in the same issue in which the aforementioned paper appeared, and even its title ("On Banach spaces containing Co. A supplement to tile paper by J. Hoffmann-J0rgensen 'Sums of independent Banach space valued random variables' ") reveals how it depends oil it. Hoffmann-Jorgensen's fundamental paper [73] is one of the pioneering works on probability in Banach spaces. It is focused on generalizing several classical theorems about sums of independent real random variables to the case of Banach space valued random variables. Among other results, it is showed that for a Banach space X the following are equivalent: (a) X satisfies: "If ( ~ ) is a sequence of independent, symmetric, X-valued random variables and their partial sums (Sin) = (}--]~;~" 1 P~) are bounded ahnost everywhere, then (S,~) converges almost everywhere".

62

2. Copies of co and ~1 in L~(#, X)

(b) L~,([0, 1], X) does not contain a copy of co for some p E [1, +oc). (c) L~,([0, 1], X ) does not contain a copy of co for any p ~ [1, +oc). These equivalences led Hoffmann-J0rgensen to conjecture t h a t the preceding conditions would also be equivalent, to the following one (d) X does not contain a copy of co. Kwapiefi [83] showed that the conjecture was true. In 1978, four years after the publication of the papers of HoffrnannJergensen and of Kwapiefi, Bourgain [14] provided an alternative proof of Kwapiefi's theorem obtaining a more general result. Actually, we must say that Bourgain's results are formulated in a more general setting than required now, but, as we have already remarked, later on his solution will be crucial for us (see Saab and Saab's theorem 3.1.4). This is why we present here Bourgain's approach. We do not use the probability language and no particular knowledge of probability is assumed. However, its flavour is all over Section 2.1. This should not be surprising because, as we have explained, the results considered here had their roots precisely in one of the pioneering works on probability in Banach spaces. We knew about Kadec-Petczyfiski-Rosenthal's lemma 2.1.3 through an unpublished paper of Bourgain [17], and we learnt there how to apply it in the study of vector-valued functions. Although it is mentioned by passing in some places in the literature (see for instance [19]), we are afraid that it is widely ignored. However, we believe that it is a very important result which deserves a lot of attention. We have known through Bourgain and Rosenthal that this "subsequence splitting lemma" was formulated for the first time by the second n a m e d author in the spring of 1979, in the Topics Cmtrse at the University of Paris VI. T h e o r e m 2.1.4 is perhaps well known for many specialist but, as far as we know, it. appears explicitly here for the first time. The same can be said about the case p > 1 in Theorem 2.1.5. Some of the particular cases of T h e o r e m 2.1.4, which we give in its Corollary, may be found included in .~ome proofs in the literature (see, for instance, [108], [69] or [10]). Also in 1978, Pisier, answering a question of Diestel, showed that, for 1 < p < +oo, L v ( p , X ) contains a copy of gl if and only if X does. He remarked in his paper: "soon after m,.] proof, B. Maurey gave another prvof of the result giving the following improvement (he is referring to theorem 2.2.1); I could verify that my proof gave the same extension, too'. Since Bourgain also asserts in [16] that he got independently the same result, we have called it Maurey-Pisier-Bourgain theorem. Bourgain's proof is quite complicated, via an averaging result for gl-sequences. Our proof follows very much Pisier's approach. We must say that Batt and Hiermeyer's proposition 2.2.1 has a clear antecedent in Proposition 1 of [101], which in turn, in Pisier's own words, is

2.3 Notes and Remarks

63

"just a r'eJormulation of a remark of Stegall and Odell" [112, Addendum, p. 377]. It is one of the main tools used by Pisier to prove his theorem. In our fbrmulations of Bourgain's Theorem 2.1.5 and Maurey-PisierBourgain's Theorem 2.2.1, we have taken advantage of the measurability results obtained in Section 1.6. In fact, in the original formulations of these theorems, no assertion about the measurability of the the set of all points a~ C J? for which (f~,(co)) has either a c0-subsequence or an (1-subsequence was made. We thought natural to study this, and so we were led to Proposition 1.6.3 All Section 2.2 is clearly connected with the problems of weak compacthess, weak convergence, and gi-sequences in Lp(#,X) spaces. We have to point out that Talagrand's fundamental paper [127] must be studied by anyone interested in such problems. In particular in [127] one can find much stronger resiflts than our simple lemma 2.2.1. Besides Talagrand's paper, the following papers should also be mentioned: [129], [38], [9], [30] and [33]. Once we have touched on weak compactness in Lp(#, X) spaces one can not help wondering about stron9 compactness in these spaces. In this line we have some quite recent papers: [62, 63], [2] and [32]. Finally, we would like to call the attention here to an important line of research which has some connection with the problems studied in this Chapter: vector-valued versions of results on basis properties of Lp (#) spaces. Recall that ibr 1 < p < +oc and # finite: (a) Lp(#) has an unconditional basis. (b) Every sequence of non zero elements in Lp(#) which is a martingale differen(:e sequence is an unconditional basic sequence. Diestel and Uhl [39] asked if Lp(#, X) has an unconditional basis whenever X has, and in general they asked for vectorial versions of these results. The first question was soon negatively answered by Aldous [1], who found a necessary condition for L~)(#, X) to have an unconditional basis. We believe that at the moment it is unknown whether it is sufficient or not. Concerning ~he second question, we (:an say that the Banach spaces for which assertion (b) is true have been extensively studied (see [21, 22, 23]). They at'(; the so-called UMD spaces, which turned out to be the natural context in m a n y problems in analysis.

3. C (K, X) spaces

After the flmdamental results on Lp(p, X) spaces of the preceding chapter, let us see now which is the behavior of C(K, X) spaces. We will s t u d y the contributions of E. and P. Saab, P. C e m b r a n o s and F. Freniche, and L. Drewnowski. T h e y provide a complete answer to the problems we are dealing with. T h e questions: "does Lp(#) have a copy of co, g-1 or g~o'?", "does Lv(#) have a c o m p l e m e n t e d copy of Co or 61?" have quite an easy answer, and it does not depend on the measure space (~2, ~ , p) (remember we are always assuming Lp(p) infinite-dimensional). T h e situation is completely different for C(K) spaces. We know t h a t C(K) always contains a copy of co (see Section 1.4), and we also know t h a t it never contains a complemented copy of 61 (this is not difficult and it will be shown in the next section), but if we wish to know whether C(K) contains a copy of gl, or a copy of 6oc, or a complemented copy of Co, then the answer will be: "It depends on the c o m p a c t space K. For some of t h e m the answer is yes, and for some of t h e m the answer is no". Of course, we would like to know for which c o m p a c t spaces the answer is "yes", t h a t is, we would like to have a charaterization of these facts t h r o u g h topological properties of K . In this line we have a satisfactory answer for copies of 61 ( T h e o r e m 3.1.1), but for complemented copies of co and for copies of too, the problems are still open.

3.1 Copies and complemented copies of

E1

in

C(K, X)

Let us begin characterizing when C(K, X) contains a copy of fl. We should first know the situation in the scalar case. For this reason we have to recall quite a classical and well known result due to Pdczyfiski and Semadeni [100]. To u n d e r s t a n d it we need to remember t h a t a c o m p a c t Hausdorff space is said to be scattered (or dispersed) if each of its closed subsets have isolated points (see [84] or [124]). T h e simplest example of a scattered c o m p a c t space is the Alexandroff compactification of the natural numbers, and in general any countable and c o m p a c t space. A typical example of a non scattered c o m p a c t space is [0, 1]. T h e tbllowing t h e o r e m is just [84, Section 13, Corollary to T h e o r e m 4] (see also [124, Section 8.5]).

66

3. C(K,X) spaces

Theorem

[100], 1 9 5 9 ) . Let K be a corn-

3.1.1 ( P e 8 9

pact and Hausdorff space, then C(K) contains a copy of ~1 if and only if K is not scattered Now we can s t u d y the vector-valued case. The following result is p e r h a p s well known and very old, but we have only found it in [25]. 3.1.2. Let K be a compact and Hausdorff space, then C ( K , X ) contains a copy of ~1 if and only if either X contains a copy of ~1 Of [( iS not scattered (equivalently, C(K) contains a copy of gl).

Theorem

Pro@ Let us show the non trivial iinplication. Assume K scattered and let us suppose t h a t C(K, X ) contains a copy of fl. Since a Hausdorff space which is the range of a continuous m a p defined on a c o m p a c t scattered space is also a c o m p a c t scattered space [124, II.8.5.3.], an old technique (see for e x a m p l e the p r o o f of [51, VI.7.6.]) allows us to assume also t h a t K is metrizable. B u t it is well known t h a t a scattered metrizable c o m p a c t space must be countable (see [124, 8.5.5.] or [84, Section 5, Corollary 2]), so K must be countable. Then, p u t K = {tin : m E N} and let (r be an fl-sequence in C(K, X). If X does not contain ft then Rosenthal's t h e o r e m says that, for each m E N, (r has a weakly Cauchy subsequence. But then, by a s t a n d a r d diagonal a r g u m e n t , we can deduce t h a t (r has a subsequence, which we continue to denote in the same way, such t h a t (r is weakly C a u c h y for all m E N. Therefore, by Corollary 1.7.1, (r is weakly Cauchy in C(K, X). Of course, this contradicts the fact t h a t it is an t~l-sequenee. T h e p r o b l e m of characterizing when C ( K , X ) contains a complernented copy of 61 was solved by E. and P. Saab [116]. To u n d e r s t a n d E. and P. Saab's proof let us begin thinking how to prove t h a t C ( K ) does not contain a complemented copy of f-1 ( r e m e m b e r t h a t m a n y C(K) spaces do contain copies of ~ , for instance C([0, 1]), see also Theorein 3.1.1 above). P e r h a p s the simplest proof of this is the following: Let us suppose t h a t C(K) contains a c o m p l e m e n t e d copy of ~1. T h e n C(K)*, the dual of C(K), which we identify with the space rcabv(K) of all regular measures of b o u n d e d variation on the Borel subsets of K (see Section 1.7), contains a copy of f-oo, and theretbre it contains a Co-sequence (p,,~). Take

k= 1

where II tLa. II denotes the variation of #k. R a d o n - N i k o d y m t h e o r e m allows us to consider (#,~) as a sequence in LI(A) and so we would have a copy of co in L1 (A). Of course this is a contradiction (use for example that, L1 (A) is weakly sequentially complete).

3.1 Copies and complemented copies of fl in C(K, X)

67

Observe now t h a t the preceding proof generalizes easily to prove that if

C(K, X) contains a complemented copy of fl then X must contain a complemented copy of 61, for spaces X whose dual X* enjoys Radon-Nikodym property. It is enough to apply K w a p i e f i ' s Theorem 2.1.3 and the classical Bessaga-Petczyfiski theorem asserting that a Banach space X contains a complemented copy of f-1 if and only if X* contains a copy of co (Theorem 10, Chapter V of [35]). E. and P. Saab realized that it was possible to overcome the restriction on X following carefully the representation of the dual of Lp(p, X ) provided by lifting theory (see Section 1.5), and using Bourgain's Theorem 2.1.1 instead of K w a p i e f i ' s Theorem 2.1.3. We believe t h a t these are the main points in Saab & Saab's approach. We have summarized it in the following result, which is not explicitly stated in [116], but it is clearly contained in tile proof given there. At first., one might be surprised by the statement of the next theorem just because it seems it does not have much to do with C(K, X). However, notice that the dual of C(K, X), which we identify with rcabv(K, X*), is just a subspace of cabv(B(K), X*), a particular case of the space involved in the statement. On the other hand, we must say that the full generality of our statement will be used in Section 4.1 (Theorem 4.1.1). 3.1.3 (E. a n d P. S a a b [116], 1982). L e t X be a Banach space, then cabv( ~, X*) contains a copy of Co if and only if X* does9

Theorem

Proof. Assume cabv(~, X*) contains a copy of co. Let, (rnn) be a c0-sequence in cabv(~, X*). Then there are C1, C.) > 0 such t h a t C1 max l ai i

I <- - II ~

a~,,,,i

It <- - C., max l(~i I i

i ~c

1

fbr all (ai) E Coo. Take A = ~ , . l p: II ~,~ II. It is clear that (m,~) is a sequence in cabvx(i2, X*), and we can assume A to be complete (if necessary we A-complete de a-field 2 and we extend by zero the m~'s). Take a lifting p in Lo~(A), and use Theorem 1.5.2. We get a sequence ( ~ ) = (fi(m,.)) in s (.~, X*) satisfying:

i

9~ s z

'

i

i

for all (a i) C Coo. Then we deduce from Bourgain's Theorem 2.1.1 that X* must contain a copy of co (recall the proof of Theorem 2.1.2). An immediate consequence of the preceding result is the following

T h e o r e m 3.1.4 (E. and P. Saab [116], 1982). C ( K , X ) contains a complemented copy of ~,1 if and only if X does.

68

3.

C(K,X) spaces

Pro@ If C(K, X) contains a complemented copy of ~1 then rcabv(K, X*), the dual of C(K, X), contains a copy of ~oo, and therefore, a copy of Co. Of course, this implies that cabv(g(K), X*) (Drcabv(K,X*)) also contains a copy of co. The preceding theorem implies that X* contains a copy of co. But, by the classical result of Bessaga and Petczyfiski mentioned above (Theorem 10, Chapter V of [35]), this means that X contains a complemented copy of ~1.

3.2 Complemented

copies

o f Co i n

C(K, X)

We have seen that C(K) (and therefore C(K,X)) always contains m a n y copies of Co (see Section 1.4). However, what can be said about complernerzted copies of Co? Notice first that in general C(K) does not contain complemented copies of Co. For example take g~, which is just C(~N), where as usual, /3N denotes the Stone-Cech cotnpactification of the natural numbers. Phillips classical theorem (Corollary 1.3.2) guarantees that C(r does not contain complemented copies of Co. In general, to get a C(K) space with no complemented copies of Co, it is enough to take any extremally disconnected compact space K , because the corresponding C(K) is a Grothendieck space (see [35] and the Notes and Remarks of this Chapter). On the other hand, we m a y find lots of Banach spaces not containing e0 at all. However, we will see in the next theorem that C(K, X) always contains a complemented copy of co (remember that we are excluding the trivial cases of finite dimensional spaces C(K) or X). Of course, as we have already mentioned, this was quite surprising and provided the first negative answer to Problem 2 in tile Introduction. The result was obtained independently by Cembranos and Freniche. It is worth to mention that the main ingredient in its proof is Josefson-Nissenzweig Theorem [35, Chapter XII]. We have found the proof we give in Bombal's paper [5]. Theorem

3.2.1 ( C e m b r a n o s - F r e n i e h e

[26]

[59], 1984).

The Banach

space C(K', X) always contains a complemented copy of Co. Proof. By Josefson-Nissenzweig theorem, there exists a normalized weak*null sequence (.z.,~) " "* in X * , Take a bounded sequence (x,~) in X such t h a t z~.(:~:,) = 1, and with this sequence (z~), let us repeat the construction of a c0-sequence we (lid just before Proposition 1.4.2. We get a c0-sequence (g,~) in C(K, X) such that 11.9,~ II = II.q,~(tn) II tor all n E N, where (t,~) is a sequence in K such that

g~(t,~) =

z,,,

if'n=m

0

otherwise

3.3 Copies of ~o~ in C(K, X)

69

Since (x;.) is w*-null, it converges uniformly to zero on the compact subsets of X, and so

c(/(,x)

--~ c(/(, x)

r

--~ ~ x;(r

is a well defined continuous linear projection onto the closed linear span of the (g,~)'s. This completes the proof.

Remark ,?.2.1. It should be pointed out that if we wish to use the language of Proposition 1.1.2 in the preceding proof, we just have to realize that (~t~(.)x,*~ is a w*-null sequence in C ( K , X)* satisfying <.q~.~t.,(.)zm > = < g . ( t , . ) , z ; ~ >

= ~.m < z n , z , ~ >

= d,~.~

for all n , m 9 N, where of course the g.,~'s, xn's, t,,'s and x m*'s are the ones of the preceding proof.

3.3 Copies of s

in

C(K, X)

Finally, we wish t o know when C(K, X ) contains a copy of ~ . The problem was solved by Drewnowski. We will need to apply very carefully all the power of the results of Section 1.3. T h e o r e m 3.3.1 ( D r e w n o w s k i

[46], 1990). C ( K , X )

contains a copy of

~.oo if and only if either C ( K ) or' X does. Proo.f. The condition is of course sufficient. Let us show that it is also necessary. To do this let us observe first the following fact: Given a compact HausdorJf space K and a Banaeh space X , if T " goo --4 C ( K , X ) is a bounded linear operator such that T((
(~T(e,~)(t)

rl=l

for' all t E K and all ((n) 9 e~, then lira T(~.,~) = O,

where (e~) denotes the canonical unit vector sequence in too.

70

3.

C(K,X) spaces

This fact may be proved using Remark 1.3.1, but instead of this result we will use Orlicz-Pettis Theorem ( [35, page 24]). Let T : go~ --+ C(K, X) be a bounded linear operator such that oO

T((r

= ~

r

for all t E K and all (r E go~. Then for each ((~) E goo, the bounded k k sequence ( ~ 1r = (T(~,~=I r is pointwise convergent to T((r and by Proposition 1.7.1, it is weakly convergent, too. This means t h a t the series ~ (.~T(e,~) is weakly convergent for all ((,~) E g~. Therefore, by Orlicz-Pettis theorem ( [35, page 24]), the series ~ T(e~) is unconditionally convergent. Of course this implies that limn T(e,,) = 0. Our aim now is to show that. if C(K, X) contains a copy of go~ but neither C(K) nor X does then it is possible to find a closed subset K0 of K and a bounded linear operator T : g~ -4 C(Ko, X) such that

T((r

=~ r 7~-- I

for all t E K0 and all (r

E goo, but

IIZ(e~) 1174 0. This is a contradiction with the fact we have just shown and our proof will be complete. So, let us assume that C(K, X) contains a copy of go~ and suppose that, neither C(K) nor X contain a copy of g-oo. Let J : goo --+ C(K,X) be an isomorphic embedding, and let us denote f,~ = J(e~). For each t E K consider the bounded linear operator

Jr: go~ (r

--9 --+

X J((r

Since we are assuming that X does not contain g~, it follows that all these operators are weakly compact and that ~ Jt(en) = ~ f~(t) is an unconditionally convergent series for all t r K (see Corollary 1.3.1 and remark 1.3.2), that is,

Z r

is convergent

(3.1)

for all (r r goo. Let D be a countable subset of K such that for each n E N there exists t , r D such t h a t

3.3 Copies of goo in

C(K, X)

71

Since (Jt)eeD is a countable family of weakly compact operators defined on f ~ , it follows from Corollary 1.3.3 that there exists M E [N] (let us recall that [ ~ is the set of all infinite subsets of N) such that oo

71,=1

for all ((.,~) E f ~ ( M ) and all t E D. Of course, goo(M) is isometrically isomorphic to g~o. So, without loss of generality, we may assume that this equality holds tbr every ((,~) E ~oo and all t E D. But this means that

J ((r

(t) = ~ (,~f,~(t) n

(3.2)

1

whenever (<,~) E goo and t E D. Let, K0 be the closure of D in K and let Xo 00 be the dosed linear span of U~=lfk ( K 0)- Given ((~) E goo, notice that 3.2 implies t h a t J (((~)) (t,) E X0 for all t E D. Hence, by density and continuity, we have J ( ( ( , ) ) ( t ) E Xo

(3.3)

for all t E /40. Since Xo is a separable subspace of X , we can take now a w*-dense sequence (x~) in B(X~)). Thanks to the Hahn-Banach theorem we can extend these functionals and assume (x~) C B(X*). For each j let us consider the operator Jj:

t,~

)

(r

,

C(K) .T; o J ((r

By Corollary 1.3.1, all these operators are weakly compact. And, using again Corollary 1.3.3, we deduce that there exists M E [N] such that

J, (((~)) = ~ (,~4j(en) for all ((,~) E g~(M) and all j E N. As belbre, may assume M = N, and with this in mind the preceding equality means oo

xj o J ((.(~)) =

(,. (xj o J(e,~)) n

1

for all ((n) E g-oo and all j E N, where notice that the convergence of the series is in the norm of C(K). In particular, using 3.1, it follows that given ((n) E g ~ and t E Ko

72

3. C(K,X) spaces oo

.,~ X"~ (J ((r

(~))

=

~

=

Xj

(,:j * (J(~)(t))

n

en

for all j C N. Therefore, it follows from 3.3 and the w*-density of the x~'s that J (((,d) (t) = ~

(,~f,dt).

(3.4)

Now, let R be the restriction operator from C(K, X) in C(Ko, X) and take ,I:foo --+ C(Ko, X). We have

T = R o

IlT(en) ll

= =

l]Ro,l(e,,,)ll= IlRfnll IIf,~(t,,)ll = Ilfnll = IIJ(en) H ~ 0,

and by 3.4, for each ((n) E ~o and each t C K0 we also have (DO

T (((,,)) (t)

=

R o J(((,,))(:)

=

(,~R(f,d(t)

Z n=l

oo

=

Z

oc

(,,(R o ,J)(en)(t)

n=l

=

~

(.r(e,,)(t)

n=l

This is the desired contradiction and the proof is complete.

3.4 N o t e s a n d R e m a r k s Saab and Saab's theorem 3.1.3 is not true for cabv(5, X): Talagrand in [127] provides an example of a Banach lattice which does not contain co such that cabv(Z, X) even contains goo. Ferrando in [57] studies when cabv(5, X) contains goo. Recall that if the Banach space X is a dual then it contains ~oo if (and only if) it contains co [35, Chapter V, Theorem 10]. Therefore, Saab & Saab's Theorem 3.1.3 may also be read as follows: "if X is a dual space then cabv(Z, X) contains goo if and only if X does". Of course, Talagrand's example just mentioned shows that the preceding result is not true for general Banach spaces. Several results contained in this monograph have been extended to tensor products. This is specially true with Cembranos-Freniche's Theorem 3.2.1. Since it was published several extension have been given (see for instance [117]).

3.4 Notes and Remarks

73

We have to point out that in the proof of Drewnowski's Theorem 3.3.1 some ideas introduced by Rosenthal in [110] and by Kalton in his famous paper [78] on spaces of compact operators are crucial (and part of them had already been included in section 1.6). No good characterizations of when C(K) contains a copy of f~o or a complemented copy of co are known. But we have some important partial results. For instance we have Theorem

(Rosenthal

[39, T h e o r e m

V I . 2 . 1 0 ] ) . If K is extremally dis-

connected then C(K) contains a copy of ~o. Recall that K is extremally disconnected (or Stonean) if the clausure of each open is open. Easy examples of extremally disconnected compact Hausdorff spaces are the Stone-Cech compactification of discrete infinite topological spaces. These spaces are very important in Banach space theory an in particular they are important in the problems we are considering. Grothendieck proved t h a t if K is one of t h e m then each bounded linear operator from C ( K ) in co is weakly compact [39, Corollary VI.2.12]. As an immediate consequence of this result we obtain the following

If K is extremally disconnected then C(K) contains no complemented copies of Co.

Theorem.

The preceding result is also an easy corollary of another important fact [84, Section 11, Corollary 2]: C(K) is injective whenever K is extremally disconnected. It is enough to observe that complemented subspaces of an injective space must be injective, while Co is not. The aforementioned Grothendieck's result as well as some other related ones led to the introduction of the tbllowing definition [39, pages 156, 179]: we say t h a t a Banach space X is Grothendieck if weak and weak* convergence of sequences in X* coincide. Notice that thanks to the identification between w*-null sequences and Co-valued continuous linear operators (see Proposition 1.1.1 and [35, Chapter VII, Exercise 4]), this condition just means t h a t each bounded linear operator from X in co is weakly compact. So Grothendieck's result may be read as follows: Theorem.

If K is extremally disconnected then C(K) is a Grothendieck

space. Clearly there are m a n y Banach spaces with no complemented copies of co which are not Grothendieck spaces (think for instance in ~1). Nevertheless this is not the case for C(K) spaces because we have

74

3. C(K,X) spaces

C(K) is a Gvothendieck space if and only if it does not contain any complemented copy of Co.

Theorem.

Pro@ If C(K) contains a complemented copy of co, then it can not be Grothendieck, just because dearly Co is not Grothendieck. Conversely, if U(K) is not Grothendieck, then there exists a w*-null non w-null sequence (p,~) in C(K)*. But, thanks to the identification between w*-null sequences and c0-valued continuous linear operators we have mentioned above, this means t h a t the continuous linear operator T associated to (#~) is not weakly compact. So, by [39, VI.2.17.], there exists a bounded sequence (f~) in C(K) such t h a t f,~. f,~ = 0 for m r n, and lim~ T(f,~) r 0. It is immediate that the series ~ f,~ is w.u.C. (it is clear that (f~) is even a c0-sequence). On the other hand, it is easy to show that the condition limn T(f,~) r 0 implies that there are subsequences of (#,.) and (fn), which we continue to denote in the same way, such that #,,(fi~) 7# O. So, Theorem 1.1.2 guarantees that (f,,~) has a complemented co-subsequence. This completes the proof. Those compacts K such that C(K) is Grothendieck are some times called G-spaces. Information about these spaces may be found in [123]. For a time it was not known whether each Grothendieck C(K) space must contain a copy of g~. Haydon [68] constructed a G-space K such that C(K) does not contain copies of goo. However, this C(K) does admit goo as a quotient. Talagrand, using continuum hypothesis, constructed a G-space K such that the corresponding C(K) does not even admit ~oo as a quotient. Extremally disconnected compacts satisfy the following two properties: (a) T h e y do not contain copies of goo, and (b) They do not contain complemented copies of Co. So, one could thing there is some relationship between these two properties in C(K) spaces. This is not the case. We have the following examples: 1. If K = [0, 1], then C(K) Dc0 and C(K)73eoo. (,) 2. If K = / 3 N x/3N, then C(K) Dc0 and C(K)Dgcr (,) 3. If K is any of the compact spaces given by Haydon or Talagrand, then C(K) 73 co and C(K)73goo.

(c)

4. If K =/3N, then C(K) 73 Co and C(K) = g.~. (c)

4. Lp(tt, X)

s p a c e s

We have already studied in Chapter 2 when Lp(t.t , X ) , for 1 _< p < +oo, contains copies of co or gl. We wish to consider now complemented copies of these spaces, and also copies of f-oo (remember that for this last space copies and complemented copies are the same). The case p = oo is postponed until next Chapter. Let us begin with the gl case.

4.1 When o f ~l ?

does

Lp(tt, X)

contain

a complemented

The first result is a consequence of Saab & Saab's theorem provides an answer to our question in a particular case.

copy

3.1.3 which

[6]). Let (~, Z, >) be a finite measure space and let 1 <_ p < +oo. If Lp(>, X ) has a uniformly bounded complemented glsequence then X contains a complemented copy of gl.

T h e o r e m 4.1.1 ( B o m b a l

Proof. Let (fn) be an uniformly bounded complemented gl-sequence in Lp(>, X). By the classical perturbation theorem (see [93, Proposition 1.a.9.] or [35, Theorem 12 of Chapter V]) we may assume that the f~'s are uniformly bounded simple functions. Let us denote f,~ = y'~.iXa~ (.)zp. We will assume the x?'s are in the unit ball of X. Let ~ Fn be a w.u.C, series in Lp(#, X)* such that F,~(f~) = 1 for all n EN. Let r be the canonical map defined in remark 1.5.3, and let us denote ~(F,~) = m~. Of course, y~. m,~ is a w.u.C, series in cabv(Z, X*), and for each n EN we have Ilrn'~ll

=

Ilrn~(A~)ll -> E

Ilrn,~H (D) >- E i

=

r 4

i =

=

mn(Ap)(xn)

76

4. Lp(#,X) spaces

Therefore c a b v ( Z , X * ) contains a copy of co. So, we deduce from E. and P. Saab's theorem 3.1.3 that X* contains a copy of co, too. Of course, by Bessaga-Petczyfiski's classical theorem (Theorem 10, Chapter V of [35]) this means that X contains a complemented copy of gl. Remark 4.1.1. It is interesting to know Bombal's original proof of the preceding result [6]. It is essentially the following: Assume our measure space is [0, 1] with the Lebesgue measure, and let (fn) be a a uniformly bounded complemented gl-sequence in Lp([0, 1], X). By Lusin's theorem and the classical perturbation of basis theorem, we may assume the f~'s continuous (and uniformly bounded). Considering the canonical injection from C([0, 1],X) into Lp([0, 1], X), it is clear that (fn) is also a complemented gl-sequence in C([0, 1], X). Now Saab & Saab's theorem 3.1.4 guarantees that X contains a complemented copy of gl, as desired. For simplicity, we have exposed the proof for Y2 = [0, 1], however we can use the same idea with any compact K instead of [0, 1], and in fact, using Kakutani's theorem [40, Section 18, Theorem 2], we have the result for all finite measure spaces. We have followed a different approach, because we have preferred to avoid such an structural theorem as Kakutani's. Now we wish to extend Bombal's theorem to all fl-sequences. To do this we will need the following lemma. L e m m a 4.1.1. Let (~7, Z , #) be a finite measure space and let 1 < p < oo. If (fn) is an gl-sequence in Lp(#, X ) then (fn) has a normalized block sequence (f88 such that sup IIf:,~(co)[I < oo ?lZ

for almost all co C ~2. Pro@ Let (fn) be an el-sequence in L v ( # , X ). The set {11 fn(.)I1: n e N} is bounded in Lp(#) and therefore it is relatively weakly compact. So, it is relatively weakly compact in LI(#), too. Let h E L I ( # ) be a w-cluster point of {li/n(-)ll: n e N}. Then h is in the w-closure of {11 f~(.)I1: n >_ m} for every m C N. But Mazur's theorem (see for instance Corollary 2, Ch. II of [35]) implies that for every m r N, h must be in the (norm) closed convex hull of {H f,~(.)II: n _> m}, too. Therefore, there are an increasing sequence 1 = k l , . . . , kin,.., of natural numbers and a sequence (r in LI(#) (norm) converging to h and such that 6m is a convex combination of {ll fkm (') l[, " " " , ll fkm+l-l

(') ll}

for each m E IN. Moreover, extracting a subsequence if necessary, we may suppose that (6~) converges to h almost everywhere. Hence sup I Cm(w ) 1< oo 79Z

4.1 When does Lp(#, X) contain a complemented copy of fl ?

77

for almost all co E .(2. Finally, if km + 1 -- 1

),~ It f{(.) II, X"~k''~+l-1

with Xi > 0, and z-,i=k,,~ _

Ai = 1, we define

kin+l--1

gm =

E

-Xifi

and

f,',~--

g,,~

i=k.~

for all ~rT.C N. Since (f,~) is an gl-sequence, it is clear t h a t Hgm lip is b o u n d e d away from 0, and so, it follows t h a t (f[~) satisfies the imposed requirements. At this m o m e n t we can answer the question of the Section. Theorem

4.1.2 ( M e n d o z a

[97], 1992). For 1 < p < oc, L p ( p , X ) con-

tains a complemented copy of gl if and only if X does. Pro@ By T h e o r e m 1.6.1, we m a y assume (~2, Z , #) is a finite m e a s u r e space. Let us prove the non trivial implication. Let (f~) be a c o m p l e m e n t e d flsequence in L~(t,,X ). By the preceding l e m m a and well known results on block basis (see for instance 2.a.1. of [93]) we m a y assume sup II f,~(co)l1 < c>o

(4.1)

?Z

for allnost all co E ~2. Since (f,~) is conlplemented, there is a w.u.C, series ~.,,. F,,. in Lj,(t*, X)* such t h a t < f,~,/2,, > = 1

(4.2)

for all 'H, E N. Using T h e o r e m 1.5.4 on the representation of Lv(#, X ) ~, we know t h a t there are w * - m e a s u r a b l e functions tp,~ : (2 --+ X* such t h a t for each 7~, E N, II~',,(.)II is measurable, it. belongs to 12q(#) and < . f , / ~ > = ./~ < f(co), ~P,~(co)> dp(c~) for all .f E L~,(I*, X). Therefore,

4.2 means t h a t

./~ < f,~(co),~,,(~) > dl~(w) = 1

(4.3)

ti)r all it, E N. Now we have two possibilities: (a) (< .f,~(.), q),,(.) >) C L 1 (tl.) is not uniformly integrable. In this ca.se, we deduce from 1.2.5. of [39] t h a t there exist e > 0, a sequence (A,.) of pairwise disjoint m e a s u r a b l e subsets of ~2, and a subsequence of (< f,,.(.), ~,,.(.) >), which we continue to denote in the s a m e way, such that.

4. Lp(p,X) spaces

78

fA

[.

< f~(~)' #~(~) > d , ( . )

=

[_ < XA. (~)f~(~), #.(~) > dp(w)

=

I< XA~(')fn('),Fn >l

n

d~2

> e

for all n E N. Since Y'~nFn is a w.u.C, series in Lv(#, X)*, the preceding inequality and Theorem 1.2.1 imply that (XA~ (.)fn(.)) has an gl-subsequence. But this is a contradiction because (XA~(.)f~(.)) is a seminormalized disjointly supported sequence in Lp(#,X) and therefore, it is an gp-sequence (Proposition 1.4.1). Thus we have to assume (b) (< f,~(.),kan(.) >) C LI(#) is uniformly integrable. Then, for each k E N put (2~ = { w E Q:llf,~(wDll_) it follows that there exists k0 9 N such that < f~(~), ~,~(~) > d#(cJ) > 1/2

/~ ko

for all n 9 N. Since ~ / 7 , ~ is a w.u.C, series in Lp(#, X)*, according to Theorem 1.2.1, (Xr~0 (.)f~(.)) has a complemented gl-subsequence in Lp(#, X). Now we deduce from Bombal's Theorem 4.1.1 that X has a complemented ~l-sequence.

Remark ~.1.2. We would like to point out that Bourgain's theorem 2.1.1 is on the basements of all this section. In fact, Theorem 4.1.1 (and therefore, Theorem 4.1.2) lays on Saab & Saab's theorem 3.1.3, but recall that this result in turn lays on the aforementioned Bourgain's theorem.

4.2 When

does

Lp(p, X)

contain a copy of ~?

The tbllowing result and its proof is strongly inspired by Drewnowski's Theorem 3.3.1 and so it borrows several Kalton's ideas. We will need again all the power of the results of Section 1.3. We will also use the preliminary work we have made in Section 1.6.

Theorem 4.2.1 (Mendoza

[96], 1990). If 1 <_ p < oc then Lp(p,X)

contains a copy of g.~ if and only if X does. Proof. The "if' part of the result is of course trivial. For the converse recall that we may assume (~?, ~ , p) separable and finite (Theorem 1.6.2). Let (Ak) be a sequence of measurable sets generating Z (see the beginning of Section 1.6). Let us suppose that Lp(#, X) contains a copy of goo and that X does not. Let ,] be an isomorphic embedding from go~ into Lp(p, X) and for each k E N define

Lp(tt, X)

4.2 W h e n does

Jk:

g~

---+

c o n t a i n a copy of g ~ ?

79

X

---+ /

J

a#

JA k

By Corollaries 1.3.1 and 1.3.3 there exists M r [N] such that, oo

Jk ((r

r

= E lzzl

tbr all ((~) C g ~ ( M ) and all k C N. We may assume, without loss of generality, M = N, and so we have oo

lZ z

i

for all ((n) E gee and all k C N. Let X0 be a closed separable subspace of X such that J(e,~)(t) C Xo #-almost everywhere for every n. Take ((.,~) r gee and k E N. We have ~J ((r '

Jk

~k

~'z=l

n=l

k

Therefore, L 9

E Xo

J ((~r~))d#

for all k 9 N

(4.4)

k

On the other hand, it is straightforward to show that r0={A 9

~: /' J((i,,,))dp 9 /A

is a sub-c,-algebra of 5 , and so, since 5 is generated by the Ak's, 4.4 means that, 50 = 5 . In other words, it follows that

./A J((C**))d# 9 Xo

for all A e Z.

Then we have (see for instance [85, X.5 Theorem 5]) J ((4,,,))(t) 9 X0

/t-almost everywhere,

and so

J

9 L,(#, Xo)

for every ((,,,) r foe. But this implies that J is an isomorphic embedding from ,0~ into the separable space Lp(#, Xo). This contradiction finishes the proof.

80

4. L v ( # , X ) spaces

4.3 Complemented

o f Co i n L p ( t t , X )

copies

In this section we wish to characterize when Lp(>, X ) contains complemented copies of e0. For the first time, it will be important to distinguish whether the measure space (27, ~ , >) is purely atomic or not. Let us begin studying the purely atomic case. L e m m a 4.3.1. Let 1 < p < +oc, let ~ xk be a w.u.C, series in [p(X), with (xk) = ((xkn)~)k, and let (%) = ((x~n),)k be a bounded sequence in eq(x*) = + 1 = 1). Then

lz - - T/~

Proof. If the conclusion were not true, then there exist e > O, an increasing sequence (m~:) of natural numbers and a subsequence of (Tk (Xk)), which we still denote in the same way, such that rnk+l

n~z

--i

k

for each k E N. Let us define the sequence (Zk)k = ((Zk~)n)k in [p(X) by

zk~ =

Xkn 0

if m k < n < mk+l otherwise

Notice that mA~+l --1

'lt:'fl~

,~,

Therefore the disjointly supported sequence (zk) is seminormalized. And so (see Proposition 1.4.1), it is an g~,-sequence. But Theorem 1.1.1 implies that (za:) has a e0-subsequence. Hence a contradiction.

Remark 4.3.1. Notice that the conclusion in the preceding lemma simply means that {(x;,.~(zk,~))~ : k r N} is a relatively compact subset of [l (see [51, IV.la.3.] or [35, Exercise 6, Chapter I]). T h e o r e m 4.3.1 ( B o m b a l [8], 1992). get (J2, Z , > ) be a purely atomic measure space and let 1 <_ p < oo, then L p ( # , X ) contains a complemented copy o/ Co if and only if X has the same property.

4.3 Complemented copies of co in Lv(#, X)

81

Proof. Let us show the non trivial implication. By L e m m a 1.6.1 we m a y assume (Y2, Z, #) or-finite, and by Proposition 1.6.4, Lp(#, X ) is isometrically isomorphic to gv(X). So all we have to show is the following assertion: "if (~v(X) contains a complemented copy of Co, then so does X " . Then, assume t h a t ~.p(X) contains a complemented copy of Co. In view of Proposition 1.1.2 there are a c0-basis (xk) in gv(X), where xk = (xk~)~ for each k E N, and a w * - n u l l sequence (%) in gq(X*) = gp(X)*, with rk = (x[m)n, such that X* rk(z.j) = E.,a:n(:r, jn) = 5kj

(4.5)

?Z

Notice that, *'or each n E N, ~ k xk,~ is a w.u.C, series in X and ( x ~ ) k is a w * - n u l l sequence in X*. So, thanks to Theorem 1.1..2, the only thing we must show is that limk xk, ~'* (zkm) # 0 for some m E N. But, if this is not the case, the preceding l e m m a implies that oo

lim%(xk)k

=

lip ~ z~.~(zk~) = 0, '~'z= 1

which clearly contradicts 4.5. This completes the proof. Let us see now which is the situation in the non purely atomic case. We have the following result of Emmanuele (see also [15]):

T h e o r e m 4.3.2 ( E m m a n u e l e [52], 1988). Let (g2, 5, p) be a non purely atomic measure space and let 1 <

= (Snm for all 7~, m E N, and let (r,~) be the sequence of Rademacher functions. It is immediate that (r,,(.)x,) is a c0-sequence in Lv(#, X ) , and in fact, in example 2.1.1 we studied a slightly more complicated situation. On the other hand, let us show t h a t (r',~(.)z*,.) is a w*-null sequence on L p ( # , X ) * . Notice first that if .f belongs to L1 ([0, 1], X ) , then / , .1

li~n / r.,~tt)ftt) dt = 0 /0

(4.6)

This is immediate *br simple functions because (r,~) is w*-null in L1 ([0, 1])* = s and then, by density, we can extend the result to all functions in Ll ([0, 1], X). Hence, for each .f E Lv(p, X ) we have

82

4. L~,(#,X) spaces

[ < f, r,~(.)z~*~ >1

=

Ii 1 < f(t), rn(t)z~ > dt

=

~

=

< /'01 rn(t)f(t) d t , z n* >

<

1

< r,~(t)f(t),z~ >

II:c,*~II

dt

fo 1 r~(t)f(t) dt

and the last sequence goes to zero, because of 4.6. Finally, it is clear that < r,~(.)x,,,,',,~(.)x;~ > = ~,,~ for all n, rn C N. So, it follows immediately from Proposition 1.1.2 that (rn(.)z,~) is complemented in Lp(#, X), and hence the conclusion. Of course, the preceding result leads to the following characterization. Theorem

4.3.3. Let (/?, Z, #) be a non purely atomic measure space and let

1 <_p < oo, then Lp(#, X ) contains a complemented copy of co if and only if

X contains a copy of Co. Proof. Sufficiency is the preceding theorem and necessity is just Kwapiefi's T h e o r e m 2.1.6 Finally, in the next theorem we summarize theorems 4.3.1 and 4.3.3. 4.3.4. Let 1 < p < co, ttten L p ( # , X ) contains a complemented copy of co ~f and only if one of the following two conditions holds:

Theorem

(a) (Y2, Z, #) is purely atomic and X contains a complemented copy of co. (b) (~?, Z, #) is not purely atomic aTtd X contains a copy of co. Re'mark ~.3.2. We will see in Section 5.1 that the preceding theorem is also true for p = +oo (at least tbr a-finite measures #).

5. The space L

(tt, X )

In the preceding chapters we have studied C ( K , X ) spaces and L v ( # , X ) spaces for 1 _< p < +oc. We devote this chapter to study Loo(#, X). For this space we restrict ourselves to consider a-finite measure spaces (#2, E, #). There are two important reasons which lead us to do this. The first one is the very definition of L ~ ( # , X). While for 1 <_ p < ec the definition of Lp(#, X) spaces is clear for arbitrary measure spaces and most (or all) authors give the same one, for Loo(#, X) the situation is completely different. There is no problem in the finite or a-finite case, but when we leave these cases some problems arise even when X is the scalar field: for Loo(#) spaces. We would like at least to attract the attention to the fact that the definitions of Loo(#) given by Dunford & Schwartz [51], A. and C. Ionescu Tulcea [75] or Rudin [115] do not coincide. Another important reason is that regarding the problems we are dealing with, for Loo(t,,X) we have a complete answer whenever the measure # is finite (or a-finite). However, for arbitrary measures, whichever definition of L~(I, , X) one takes, only quite partial answers are known at the moment. This is because the reductions to the a-finite case we made in Section 1.6 for Lv(t, , X) spaces, do not work for p = oo (or, at least, they do not work in an ingenuous way). We can add that some of the definitions of Loo(#, X) for arbitrary measure spaces involve some technicalities (for instance, the concept of locally/z-null sets) which, in our opinion, would hide the main ideas we are interested in. Therefore, in all this chapter (#2, 5,15) will be a a-finite measure space. In the Notes and Remarks we will mention some facts which are known in more general situations. Now, let us see which are our non trivial questions for L ~ ( # , X). We have already shown in Section 1.4 that Loo(t*, X) has many copies of foo, and so it has many copies of c0 a.nd fl, moreover, it has many copies of all separable spaces just because ~oo has this property. On the other hand, since t~ is a-finite, we know that Loo(#) is injective (see tor instance, [37, Theorem 4.14.]), and then it can not contain complemented copies of Co or g~, simply because they are not injective. We could also argue that Loo(#) is the dual of

84

5. The space

L~(p,X)

L I ( # ) and it is easy to deduce from Phillips' theorem (Corollary 1.3.2) that no dual can contain a complemented copy of Co. Alternatively, we could say that Loo(#) is isometrically isomorphic to a C(K) space with K extremally disconnected (see for instance [84, Section 11, Theorem 6]), and we have considered these compact spaces in Chapter 3. We mentioned there that the corresponding C(K) can not contain complemented copies of co. We also showed that no C(K) can contain a complemented copy of fl. Thus our goal here is to know when Loo(#,X) contains complemented copies of co or gl.

5.1 C o m p l e m e n t e d copies of Co in

Loo(tt, X)

When we studied complemented copies of Co in Lp(p,X) for 1 _< p < +oc (Section 4.3), we had to distinguish between purely and non purely atomic measure spaces. Now we have to do the same. Let us begin with the purely atomic case. For a purely atomic measure, Loo(#,X) is isometrically isomorphic to goo(X) (Proposition 1.6.4). Let us study this space. The first idea to do this is trying to apply the same arguments we used to study gv(X) for 1 <_ p < +oc (see Section 4.3). We will be able to apply some of them, however, it is plain that we will find extra difficulties. This is due to the fact that, likewise in the scalar case, for most purposes ~o~(X) is much more complicated than the others f.l,(X) spaces. This is particularly true when we deal with duals. In view of the results of section 1.1 (see Proposition 1.1.2 and Theorem 1.1.2), to characterize when goo(X) contains a complemented copy of Co it is convenient to know the w*-null sequences in foo(X)*. So, we need to work in g~(X)* and for this reason it is worth to recall first a few important facts about, the scalar case, that is, about g s We should remember that this space may be identified in a natural way with ha(N), the space of all scalar finitely additives nmasures with bounded variation defined on the a-algebra of all subsets of N. A particularly important subspace of ba(N) is ca(N), the subspaee of all countably additive measures. It is isometrically isomorphic to ~ and it, is norm-one complemented in t~o~ = ha(N). Thus we can decompose in a very nice way each member of ha(N) in sum of t w o measures, one of them countably additive, and the other one purely finitely additive. We will see that the same can be done in goo(X)* It is straightforward to show that Q (X*) is isometrically isomorphic in a canonical way to a subspace of ~ (X)*: each (x;) C f-1(X*) acts as a member of f ~ ( X ) * in the following way <

> =

fi

<

9>

5.1 Complemented copies of co in L~ (#, X)

85

for each (x,~) r g-oo(X). So, in all this section we will consider g-x(X*) as a subspace of g.co(X)*. Moreover, if i m : X --4 g.co(X) denotes the canonical injection given by ira(x) = ( 0 , . . . , 0 , x , 0 , 0 , . . ) m-1 times for each x e X, one can show easily that P : g-~(X)*

~

h(X*)

is a well defined norm-one projection onto h ( X * ) 9 This projection has an additional interesting property (see [102, Lemma 11.4] or [90, p. 53]): P r o p o s i t i o n 5.1.1. The projection P defined above is w*-sequentially continuous, that is, it transforms w*-null sequences in w*-null sequences (where we continue to consider g-l(X*) as a subspace of g-co(X)* in the canonical

way). Pro@ Let (Tk) be a w*-null sequence in fco(X)*. Take x = (xn)n E g-oo(X). We must show that < x, P(Tk) > --+ 0. Let us consider the continuous linear k

operator Tx : g-co

~

&o(x)

>

Since (Tk.) is w*-null in g-~(X)*, it is clear that (7k o T~)k is w*-null in g-~. Then, Phillips's lemma [35, p. 83] guarantees that oo

lira E k

<e'~'rk~

> =0,

n=]

where (en) is the canonical unit sequence in g-co (that is, the canonical c0basis). Therefore, using that T,(e~) = ( 0 , . . . , 0 , x , 0 , 0 , . . . )

= i~(x~)

~-1 times we have oo

< x, P(~-k) > rz~l

k

86

5. The space L ~ (p, X)

T h a n k s to the projection P defined above, gee(X)* is the direct sum of (in this space we have the "countably additive part" of the functionals) and k e r ( P ) , the kernel of P (in this space we have the "purely finitely additive part" of the functionals). Moreover, in view of the preceding l e m m a a. w*-null sequence in goo(X)* is sum of two w*-null sequences, one of t h e m in *71(X*) and the other one in ker(P). All we have to do is to study these two kinds of w*-null sequences. We will do this in the following two lemmas. We will see t h a t b o t h of t h e m have a "good" (although different) behavior. T h e first l e m m a m a y be found in [89, L e m m a 2.8] or [90, L e m m a 4]. T h e second one is implicitly contained in [90].

~1 ( X * )

L e m m a 5 . 1 . 1 . Let (Tk) = ((X~.,~),~)k be a w*-null sequence in goo(X)* which is contained in gl(X*) , that is,

,-k =

r el(X*)

lirn

[lxk~][ = 0

for" all k E N. Then

uniformly in k E N. Proof. If the conclusion were not, true, then there exist e > O, a subsequence of (rk), which we still denote in the same way, and an increasing sequence (ink) of n a t u r a l numbers such t h a t 77~k+ 1 --1

75= g/~,k

for each k C N. Therefore, there exists a vector (xn) in the unit ball of t?oo(X) such t h a t rnk+l--1

< x , ~ , x k*n > > e

E

(5.1)

for each k E N. On the other hand, since (rk) is w*-null, we have t h a t < t,~x,~, Xkn > = lim

lim < (t,,z~), rk > = lim k

k

k

n=l

t~ < x~, xkn > = 0 n=l

for each (tn) E t.oo. In other words, we deduce t h a t ((< x,,,x[.~ >)n)k is a weakly null sequence in gl. Hence, we deduce from Schur t h e o r e m [35, p. 85] t h a t ((<xn,:r:*k~ >)~)k is n o r m null, and so

lim Z

I<

>1 -- 0,

rz=l

which contradicts

5.1. This completes the proof.

5.1 Complemented copies of co in L~o (#, X)

87

5 . 1 . 2 . Let ~ Xk be a w.u.C, series in go~(X) and assume that (~-k) is a w*-null sequence in g~(X)* which lies in the kernel of P (t:' is the projection defined at the beginning of the Section), then Lemma

lira <xk,~-k > = 0 k

Pro@ Let ~ xk be a w.u.C, series in g ~ ( X ) and assume t h a t (vk) is a w*-null sequence in go~(X)* which lies in the kernel of P. We will suppose t h a t <xk,~-k > 74 0, and we will reach a contradiction. Since the elements of ker(P) vanish in every finitely s u p p o r t e d vector of goo(X), we have <Xk,Tk >

--~

< (Xkl,Xk2,...,Xkk--l,0,

O . . . ) , T k :>

+ < (O,...,O,xkk,Xkk+l,Xkk+2,...),Tk

>

k-~ times

=

< (O,...,O,xkk,Xkk+l,Xkk+2,...),Tk > k--~ times

Now, let us define

~:~oo

~ g~(X) n

(tk)

)

( E tkXkn)n k=l

Since ~ xk is a w.u.C, series, it is very easy to show t h a t r is a well defined continuous linear operator. Notice also t h a t if we denote by (ek) the canonical unit sequence in go~, then

~(ek) = ( 0 , . . . , 0, zkk, xkk+l, Zkk+2,...) k-1 times for all k E N., and so

< ek,Tk o4~ > = <xk,~-k > ~ 0 On the other hand, (rk o ~5) is a w*-null sequence in gc~. Therefore, we deduce from T h e o r e m 1.1.2 t h a t (ek) has a complemented c0-subsequence. But this is a well known contradiction (see Corollary 1.3.2). 5.1.1 ( L e u n g - R i i b i g e r , 1 9 9 0 ) . goo(X) contains a complemented copy of Co if and only if X contains a complemented copy of co.

Theorem

88

5. The space L~ (#, X)

Pro@ Let us show the non trivial implication. Assume that g~ (X) contains a complemented copy of co. Then, there exists a c0-sequence (xk) in go~(X) and a w*-null sequence (~-k) in g~(X)* such that <Xk,Tk > = 1 We can write and by Proposition 5.1.1 and the preceding lemma we have lim < x~,, Tk - P(~-k) > = 0, k

therefore, we conclude lim <xk, P(Tk) > = 1 k

(5.2)

Let us denote P(wk) = ( x ~ ) ~ E gl(X*) and xk = (Xk~)o C g o ( x ) . Since x~ is a w.u.C, series, it follows very easily that

Z Xnk k is a w.u.C, series for every n C N. Besides, by the w*-sequential continuity of P we have that (x~)~ is a w*-null sequence in X* for all n E N. On the other hand, it follows from Lemma 5.1.1 that

lira in

Ilzk, II

0

uniformly in k E N, and so

lira

x* > 1 = 0 ??,~m

uniformly in k E N. Therefore, we deduce from 5.2 that there exists no such that lim<xkn0, X*k~o > ~ 0 k

Now, it is enough a look to Theorem 1.1.2 to get the conclusion. Of course, another way to state the preceding theorem is the following: T h e o r e m 5 . 1 . 2 ( L e u n g - R ~ i b i g e r , 1990). Let ( ~2, Z , # ) be a a-finite purely

atomic measure space, then L ~ ( # , X ) contains a complemented copy of co if and only if X does.

5.1 Complemented copies of co in Loo(#,X)

89

Once we have solved the problem for purely atomic measures, let us consider the non purely atomic case. We will only need two results which are true for general Lo~(#, X) spaces. They were proved by Dfaz [27]. The first one is a characterization of w.u.C, series in Loo(#, X). Roughly speaking, it says that a series ~ f~ in L~(#, X) is w.u.C, if and only if it is "(essentially) uniformly w.u.C." L e m m a 5.1.3. Let (f2, X,p) be a or-finite measure space, then a series ~ f,~

in Lo~(p,X) is w.u.C, if and only if there exist a null subset Z of f2 and a positive number M such that

II E t,~f~(w)II <- M rt~ l

for all

all m

N and allc~E [ 2 \ Z .

Proof. The condition is clearly sufficient [35, Chapter V, Theorem 6]. Let us show necessity. Assume ~ fn is a w.u.C, series in L~(#, X). By [35, Chapter V, Theorem 6], there exists M > 0 such that

n=l

for all m C N. But the preceding inequality means that for each m C N and each (t~) belonging to B ( g ~ ) , the unit ball of g~, there exists a null subset Z(,~,(t~)) of f2 such that II ~

t~I~(co) II-< M

for all a~ E ~2 \ Z(,,~,(t~)). To get the conclusion it is enough to take Z = UZ(,~,(t~)), where the union runs over the denumerable set of all (m, (tn)) with m C N and (t~) C B ( g ~ ) with rational coordinates. T h e o r e m 5.1.3 (Dfaz, 1994). Let (fJ, Z , # ) be a a-finite measure space.

If Loo(#, X) contains a complemented copy of co, then X contains a copy of C0

.

Proof. Assume that Loo(#, X) contains a complemented copy of Co. Let (f~) be a co-basis in Lo~(#, X) and let (z,~) be a w*-null sequence in L~(#, X)* such that = 1 for all n C N. By the preceding lemma, there exist a null set Z of s and a positive number M such that

II E t~f,~(a~) II < M

(5.3)

90

5. The space L ~ ( p , X )

for all (t,~) E B ( f ~ ) , all m E N and all w E ~2 \ Z. Obviously, the series f,~(w) is w.u.C, for all c~ E ~? \ Z. If for some of these ~ ' s the series f,.(w) is not unconditionally convergent, then we deduce from BessagaPe~czyfiski's classical theorem [35, Chapter V, Theorem 8] that X contains a copy of co. Otherwise, let us show that we are led to a contradiction. We must assume that for all w E ~2 \ Z the series ~ f.,~(w) is unconditionally convergent and therefore [93, page 16],

Z t,~f~(w) ~,~1

is convergent for all (t~) E fo~. Thus, if (t~) E ~ the series ~ tnf,~ is pointwise convergent almost everywhere, although it is not in general convergent in L ~ ( p , X ) because Ilf~ II~ = 1 for all n. Then, for each (t~) E f ~ we can define

s((t,,)):~

x

-~

~=ltnfn(w) w

--+

0

ifw E ~ 2 \ Z otherwise

Now, 5.3 says that the measurable function S((tn)) is bounded, and moreover, it implies that

S :~ (t,~)

~ L~(it, X) ~

S((tn))

is a well defined continuous linear operator. Therefore, (~-n o S) is a w*-null sequence in g ~ , and, if we denote by (en) the canonical unit sequence in e ~ , we have <en, TnoS> = = 1 So, we deduce from Theorem 1.1.2 that ~o~ contains a complemented copy of co, a contradiction (see Corollary 1.3.2) which has appeared quite often here. C o r o l l a r y 5.1.1. If (f2, Z , it) is a a-finite non purely atomic measure space, then L ~ ( i t , X ) contains a complemented copy of co if and only if X contains a copy of Co.

Proof. Necessity is the preceding theorem. Sufficiency follows from Theorem 4.3.2 We can finally give a theorem summarizing the results of the section: T h e o r e m 5.1.4. Let (~2, E , # ) be a a-finite measure space, then L ~ ( # , X ) contains a complemented copy of co if and only if one of the following two conditions holds:

(a) (f2, Z, #) is purely atomic and X contains a complemented copy of co.

5.2 Complemented copies of ~1 in Loo(#, X)

91

(b) ([2, Z, #) is not purely atomic and X contains a copy of Co. Putting together the preceding theorem and Theorem 4.3.4 we get the following: T h e o r e m 5.1.5. Let 1 <_ p <_ ec and let (52, Z , # ) be a a-finite measure

space, then we have 1. If the measure p is purely atomic then Lp(p, X ) D c 0 if and only if X D co. (~) (c) 2. If the measure p is not purely atomic then Lp(p, X ) D Co if and only if (c) X Dco.

5.2 Complemented

copies

of

~1

in Lcc(tt, X)

Since Lc~(#) is quite different to the other Lp(#) spaces, it is not surprising we find a great difference between Loo(p,X) and L p ( # , X ) , too. However, taking in account tile previous results in the monograph, one could expect that, one of the following results be true: (a) Loo(#, X ) contains a complemented copy of 61 if and only if X does. (b) Loo(>, X ) contains a complemented copy of 61 if and only if X contains a copy of glThis is not the case. It was first noticed by Montgomery-Smith (p. 389 of [117]). He realized that an example given by Johnson [76, Remark to Theorem 1] provided a Banach space X0 with no copies of ~1 such that L ~ ( # , X0) contains a complemented copy of gl. Let us begin giving Johnson's example.

Example 5.2.1 ( J o h n s o n / 7 @ .

contains a complemented copy of

(~|

fl. In fact, if we denote by 5 the subspace of ( ~ | the forrn

of all sequences of

((a,),(al,a.2),..., ( a , , a 2 , . . . , a , , ) , . . . ) with (at.) belonging to ~1, we will show that ~ is a 1-complemented subspace

of (~-~.Q .1 )oo isometrically isomorphic to 61. It, is straightforward to show that .7" is isometrically isomorphic to t~l. To find a norm one projection from (~@f~)oo onto ff take a norm one contimmus linear functional L on foo such that k

for each convergent, sequence (at.). Now, let us define L1 : ( E @~r)oo ((a]),(a 2 ' a 2 ) , . . . , ( a 1., n -.,an),-..)n

> IK > L((al,al,a3,..))l 2

5. The space Loo(l~,X)

92

a n d for m _> 2,

L~: (E .e?)~ ((al), (a~, a ~ ) , . . . , (a~,

a'~

9 " " ~

n)~

K m

---+

" 9 ")

m+l

L((O,...,O,a,~,a

m

arn+2

....

...))

m-1 t i m e s C l e a r l y t h e L,~'s are n o r m one c o n t i n u o u s linear f u n c t i o n a l s a n d for any ((al),(al,a2),...,(al,a2,...,an),...) e 7=

.~/:, =

L ~ ( x ) = L(( O , . . . , O , a m , a , ~ , a , ~ , . . . ) ) m-- 1

= a,~

times

Now for each k ~ N define t h e c o n t i n u o u s linear o p e r a t o r

Pk: ( E e e ~ ) ~

~

x

e~~

~ (Ll(x),L2(x),...,Lk(x))

To c o m p u t e t h e n o r m of Pk t a k e

= (@), (~, ~),..., (a?,..., aD,...) 9 ( E ~e~)~, a n d d e n o t e Oj = s g n ( L j ( x ) ) for 1 < j < k. We have

II P~(x)I1~= [I (Ll(x),... ,Lk(x))I1~= I L~(x) I § 2 4 7 =

01Ll(X) §

(a 11 , a 21, a13,...)) + 02L ((0, a~,a3,...)) + . . .

01L

... + O k L ( ( O , . . . , O

=

<

Lk(x) l

L

,k ~kq-1 ,~k,'*k ,'''))

k-~ t i m e s

(

01(al a2,a 3,

.) +

+ Ok(O '

(I

~k ~kq-1

k-~ t i m e s

/)

II 01(al,a~,a~,...) § 02(O,a~,a*~,...) § ... + 0~(0,... ,0, a~, ~+1,...)I1~ k-1 t i m e s

=

II (01a~, 01a~ + 0 2 G . . . , . . . + Oka~ +1 ,

=

01~ + . . . + ok~L < ~ + 1

01a~ +~ + . . .

+ 0 ka k k+2

+...

,...)11~

sup{I Oxa{ l, I O~a~ + 6~'~ I,..., I 01ak q-'" @ Okak I, I eX~ +~ + . . . 9..+Oka~

+ l l l ,o l a k l + 2 +

...

+ 0 k a k+2 ~ I,...}

5.2 Complemented copies of gl in Loo(#, X)

<

sup{I a{ I, I a~ I + la~ I,.-., l a~ 1+... +la~ I, I a~+ll + ... 9 -.+1 ak+l k I, l a lk+2 I+ 9+la~ +2 I, .-}

<

sup{ll (al)I1~,1t (a~, aF,)II,,--. ..., I[ (a~:,... ,a~)I1~, I1 (a~+~,-

=

9

~k+l Ill, ,(*~:+I)

(ak+2,

.

'

9 , a kk++ 22, ~

93

II1,.-'}

II x 11~

Hence, we have shown that

Ilrkll _< 1 for all k E N. Now, it, is immediate that

P : (E~> 1)~ z

>

>

f

(PI(X),P2(z),P3(z),...)

is a well defined norm one projection onto 5c.

R e m a r k 5.2.1. Actually, in the preceding example it would have been natural to take as L a generalized Banach limit (see [51, II.4.22.]). In this case we could simply have taken m + l ,a,,~ m+2 , . . . ) ) L((a,~,a,~

instead of L ( ( O , . . . , O , a , ~.m, a .,,.,+i ... am m + i ,. ..~3 ]/

,,~

i

times

because both values coincide [51, II.4.22.].

R e m a r k 5.2.2. In [76] Johnson mentioned Example 5.2.1 while working in very general conditions. We have preferred to study the example from a very direct and elementary point of view. So we give an explicit form of the subspace isometrically isomorphic to ~1 and an explicit form of the projection, too. However we must point out that, the space ( ~ ~g~)oo even contains a complemented copy of L~([0, 1]). This is an immediate consequence of the following result of Hagler and StegalI [67]: A Ba'aach space X contains a copy of ( ~ (i~f~).~ iJ and only if X * contains a complemented copy of L1 ([0, 1])9

94

5. The space L~(#,X)

Remark 5.2.3. (Montgomery-Smith, 1992) There exists a Banach space Xo with no copies off1, such that Lo~(#, Xo) has a complemented copy off1. Consider X0 = ( ~ | that is the direct sum of (f'l~),~ in the sense of co ([931). On one hand it is clear that X0 has no copies of fl (notice for instance that Xg = (Y~'.| is separable). On the other hand, if we take f ~ ( X o ) , it is immediate t h a t this space contains a complemented copy of (~ | the space of the preceding example. Therefore, taking in account that L~(p, Xo) contains a complemented copy of foo(X0) (this is a very easy fact, and it is shown in the proof of Theorem 5.2.3), we have the following chain:

Lo~(#,Xo) (~)Deoo(Xo) (D) (~-~|

(c)Del

So, we conclude that Loo(p, X0) contains a complemented copy of gl. Montgomery-Smith's remark was very important because it showed what one can not expect. However, it seemed the example involved was a very particular case and it made difficult to give a reasonable conjecture concerning which Banach spaces X provide Lo~(p, X) with complemented copies of fl. In fact, some people thought it was not possible to give a useful condition describing the Banach spaces X enjoying this property. For this reason, it was surprising when S. Diaz [28] realized that the crucial condition which is satisfied by the space X0 in Montgomery-Smith's remark is quite simple. Dfaz discovered that the point was that X0 c o n t a i n s / ~ ' s uniformly complemented (see the definition below). This was a completely new point of view, because it related for the first time the theory we are considering with the local theory of Banach spaces (the theory mainly concerned with the structure of finite dimensional subspaces of the Banach spaces). Let us give a look to a few well known and important results on the finite dimensional structure of Banach spaces. Our aim is to understand the main facts about the Banach spaces X which c o n t a i n / ~ ' s uniformly complemented. Given a Banach space X, we will denote by idx the identity operator on X. If X and Y are Banach spaces and a bounded linear operator T : X --+ Y is an isomorphism onto T(X), we will denote by T -1 de inverse of T, defined on

T(X).

For two isomorphic Banach spaces X and I/ we denote by d(X,Y) the Banach-Mazur distance between X and Y, that is

d(X, Y) = inf{I IT IIII T -1 I1: :F is an isomorphism from X onto Y}. Let 1 _< p _< +oc. We say that a Banach space X contains fpn ~ s uniformly if there exists a sequence (E~) of finite dimensional subspaces of X such that

5.2 Complemented copies of gi in Loo(#, X )

95

s,~p d(E,,, e~) < + c ~ If there are also projections P,~ from X onto E~ with sup IIP~,II < +o~ ?z

we say that X contains g.p'S uniformly complemented. Thanks to the injectivity of all g~'s it is clear that in the case p = +oc both definitions coincide. Notice that another way of saying that X contains g~'s uniformly complemented is to say that for each n C N there are bounded linear operators Cvn

I. >

X Q% ~pn

satisfying Q,~oI,~=ide?,

and

sup ItQ,~It {lI,~ II < +oo lz

If' one takes adjoints, it is clear that if X contains g~'s uniformly complemented then X* contains gq'S uniformly complemented, where, as usual, q is the conjugate of p, that is, 71 + 1 = 1. Therefore, applying the argument again, we deduce that X** contains g'~"s uniformly complemented, too. The converse is also true, that is, if X** contains g~'s uniformly complemented then X has the same property. This follows easily from the tollowing version of the Local Reflexivity Principle [92, Theorem II.5.1]: T h e o r e m 5 . 2 . 1 ( L o c a l R e f l e x i v i t y P r i n c i p l e ) . Let X be a Banaeh space and let E , G be finite dimensional subspaces of X** and X * , respectively, and let e > 0. A s s u m e that P is a projection from X** onto E. Then there are (zn i'rl:'jective linear operator T : E > X and a projection Po from X onto T ( E ) such that 1. T I~':nx = i d E n x . 2. < T ( x * * ) , x * > = < x * , x * * > for allx** C E and allx* E G. 3. IITll IIT-111 < 1 + e (and so we can assume IITII <_ 1 + e), and

~. lIPs, II _< (1 +~-)IIPII. T h e following p r o p o s i t i o n sumInarizes our remarks given above.

P r o p o s i t i o n 5 . 2 . 1 . Let X be a Ba'naeh, ,~pace, let 1 < p <_ + o o and let "as 1 = 1). Then the following are &:'note by q the co'r~:j'u,gate of p (that is, ~1 + 71 equivalent: (a) X eontain.s ~"~i's '~mi,formly complemented. (b) X* contains fl~ 's "wrhformly complemented. (c) X** contains f;~ 's uniformly complemented.

96

5. The space Loo(#,X)

Now, let us concentrate our attention in the case p = 1 (and by duality, in the case p = +oc, too). We need a good characterization of the Banach spaces containing f~'s uniformly complemented. This will follow from the following particular version of the "Great Theorem" of Maurey-Pisier (remember that the "only if' part in the theorem is trivial, the difficult part is the converse):

T h e o r e m 5.2.2 (Maurey-Pisier, T h e o r e m 14.1 of [37] ). X contains f ~ 's uniformly if and only if X does not have finite cotype. With the two preceding results in mind the following characterization fbllows immediately. As usual, let us denote by co(X) the Banach space of all null sequences in X, endowed with the supremum norm. We only have to recall that f l ( X * ) is the dual of co(X) and that X and gl(X) has the same cotype [37, Theorem 11.12]. P r o p o s i t i o n 5.2.2. The following are equivalent: (a) X contains PZ's uniformly complemented. (b) co(X) contains e[~'s uniformly complemented. (c) X* does not have f n i t e cotype.

(d) e~(X*) does not have finite eotype.

Once we know the main facts concerning Banach spaces which contain f~'s uniformly complemented, we need two more ingredients to get the characterization of Banach spaces X such that Lo~ (#, X) contains a complemented copy of fl. The first one is a very well known perturbation lemma. We will include the proof for the sake of completeness. As a consequence of it we will show the close relationship between the finite-dimensional subspaces of L ~ (p, X) and foo(X). The second ingredient is Kalton's contribution. It is a lemma which relates g~(X**) and f~o(X)**. It is crucial in the proof of the main theorem 5.2.3 (see also the Notes and Remarks).

L e m m a 5.2.1 ( P e r t u r b a t i o n l e m m a ) . Let F be a finite dimensional sub.space of X and let P be a projection from X onto F. Suppose d i m ( F ) = n and let { x t , . . . , x T ~ } be a unitary basis of F. Then, given c > 0 there exists 5 > 0 such that if {Yl, 999 Yr~} are vectors in X satisfying max II

l
z~ -

:~;~ II

< ~,

and we denote by G the linear span of the Yi 's, we have (i) d( F,G) < 1 + e, and (ii) There exists a projection Po from X onto G with IIPo II <- (1 + e)IIPll.

5.2 Complemented copies of gl in Loo (#, X)

97

Pro@ First, r e m e m b e r t h a t i f Y is a subspace of X and T : X + Y is a b o u n d e d linear o p e r a t o r which satisfies II idy - T II<_ r / < 1, then T is an isomorphism onto its range T(Y). Moreover, since for any y E Y II Z(y) II _< II Y - r ( y ) II + II y It _< (1 § ~j) II Y II, and 11T(y) II > II y II - II y - r ( : A II > (1 -,7) II y II, it, follows t h a t

II T II I1 T -~ II _< 1 + ~ l-r/

Suppose now the assumptions of our l e m m a are satisfied. Then, there exists M > 0 such t h a t

M 3-~ I Xi I_< II i=1

Aix~ II i--1

for all scalars (ki). Given e > 0 take rl ~ (0, 1) such t h a t

l+r]

--
11

M(IIPII § satisfies the required properties. Take { Y l , - . . , Y,~} vectors in X verifying

max II x~ - :,j~ II < d,

l
n

and denote by G the linear span of the yi's. For x = }-~i=1/~ixi E F, define S(z) = ~ i = 1 Aiyi E G. Since

II : , : - S(x) I1 i=1

<

i

1

i=1

i=1

a M II * II,

we have

II'idr

_

S{I<_ a M

rl

_

< 7 / < 1

(IIPII +1)

Therefore, S is an isomorphism from F onto G witch

IlSll tlS < II < l § -

Thus, d(F,G) < IlSlt IIS-' II ~< 1 §

l-r/-

< 1+~

98

5. The space Loo(p,X)

In order to get the projection P0, consider the operator T : X --+ X defined by T = idx + (S o P) - P. It verifies r] II r - i d y II = II (S - i d a ) o P II <

II P II

(11 P II §

< ,1 < 1.

Hence, it is well known that T is an automorphism on X. Moreover,

11T II II T - ~ II _<

1 + r/ < 1 + e. 1 - 7/

Since T ( F ) = G, it is clear that P,, = T o P o T -1 is a projection from X onto G with H P0 II _< (1 + e) II P II, as required. P r o p o s i t i o n 5.2.3. Let ($2, 5, p) be a c,-fi'nite measure space, let F be a finite dimensional subspace of Lar and let P be a projection from L o c ( # , X ) onto F. Then, given e > 0 there exist a finite dimensional subspace G of g.oo(X) and a projection Po from goo(X) onto Cj such that

d(Jz, G) <_ 1 § e arm

II 5, II _< (1 +

e)

II P II

Proof. Assume the hypothesis of the proposition holds, let e > 0, and apply the preceding l e m m a to the finite dimensional subspace r of Loo(>, X ) , taking any unitary basis { f l , . . . , f,~} of S. Given the positive number 6 provided by the lemma, take {g~,... ,g,~} in L ~ ( p , X ) countably valued such that max

l
II .f,

- .~t~ I1~ <

Let (A,,~) be a sequence of palrwise disjoint measurable sets with positive measure such that each gi is constant in the Am's and >(F2 \ u,,A,~) = 0. Let us denote by 7 / t h e subspace of Loo(p, X ) of all functions which are constant in tile A,,,'s. Of course, ~ is isometrically isomorphic to ,~ If we denote by g tile finite dimensional subspace of L ~ ( p , X ) spanned by the 9i's, we have

g

c ~ c

Loo(#,X),

and tile preceding l e m m a guarantees that d(m,g)_< l + e , and t h a t there is a projection P0 flom Loo(#, X ) onto g such that

II Po II _< (1 + ~) II P II. If we consider the restriction of Po to 7{, we get the desired conclusion.

5.2 Complemented copies of g-1 in Loo(#, X)

99

As usual, in the next l e m m a we identify in the canonical way X as a subspace of X**. In this way g.oo(X) is a subspace of goo(X**) and a subspace

of ~oo(X)**, too. Lemma

Given e > 0 there exists a continuous linear op-

5.2.2 ( K a l t o n ) .

eratoF

J: e~(x**) ~

:~(x)**

such that It J II _< 1 + e and Jle~(x)= ide~(x). Pry@ Let ~: be a finite dimensional subspace of goo(X**). Then there exists a sequence (F,-~) of finite dimensional subspaees of X** such that

By the local reflexivity principle (Theorem 5.2.1), tbr each n E N there exists an injective operator u~. : F , -+ X such that

u,dxnv~= i d x n f ,

and

II u~ II < 1 + e.

Now we can define

Ia:

a ---+ (.,~) ---+

:oo(x) (~,.(~,~))

It is clear that I a is a well defined continuous linear operator which satisfies II Ia II -< 1 + e and /,xlane~(x)= idane~(x), Now we will use a classical Lindenstrauss compactness argument. Denote by B(goo(X**)) and B(foo(X)**) the unit balls of foo(X**) and goo(X)** respectively, and define .:a : B(goo(X**))

x

>

>

:o~(X) Ia(x) 0

ifzE otherwise

Of course Ja neither is linear nor continuous, but notice t h a t

.Ia(B(.:~(X**)) ) C (1 + e)B(g~(X)) C (1 + e)B(goo(X)**)

(5.4)

Now we can consider the net (Ja)a, where ~ runs on the directed set of all finite dimensional subspaces of goo (X**). By 5.4, this is a net in tile compact space

( (1 + ~)B(e.~(x)**), ,,,:).le~(x"ll Therefore immediate to denote immediate

(Ja)a has an accumulation point J in this compact space. It is that J can be extended to a linear operator, which we continue J, fl'om the whole space foo(X**) into goo(X)**, and it is also that this linear operator J has tee required properties.

5. The space L~o(#,X)

100

We can finally state and prove the main result of the Section. T h e o r e m 5.2.3 ( D f a z - K a l t o n ) . Let (Y2, ~, >) be a or-finite measure space,

then the following are equivalent: (a) (b) (c) (d) (e)

Loo(p,X) contains a complemented copy of gl. L~o(p,X) contains g[~'s uniformly complemented. ~.oo(X) contains a complemented copy of gl. ~.oo(X) contains g.[~'s uniformly complemented. X contains g.~ 's uniformly complemented.

Proof. We will show the following chain of implications: (a) ~ (b) ~ (d) (~-) ~ (6 ~ (a). T h a t (a) implies (b) is trivial, and (b) implies (d) is an immediate consequence of Proposition 5.2.3. Let us show now that (d) implies (e). We assume that goo(X) contains g~"s uniformly complemented. So there exist M > 0, a sequence (F~) of finite dimensional subspaces of goo(X) and a sequence of projections ~

: coo(x) + G

such that

d(G,q~) < M and for all

n C N.

IIP~II-<M

Kalton's lemma 5.2.2 provides a continuous linear operator J : Go(X**)

go~(X)**

such that

Jigs(x) = idg~(x) Of course we may consider the F~,'s as subspaces of goo (X**), and if we denote by P,~* the second adjoint of P,~, it is clear that the operators

R,F o j : e~(x**) + G provide a bounded sequence of projections. Therefore, goo(X**) = co(X)** contains g~'s uniformly complemented. Then, by Proposition 5.2.1 this means that c0(X) contains f~'s uniformly complemented, too. But now, thanks to Proposition 5.2.2, we deduce that (e) holds. To see that (e) implies (c) is very easy. Assume that there exist M > 0, a sequence (F,,,) of finite dimensional subspaces of X and a sequence (P~) of projections from X onto F~ such that

d ( G , q ~) <_ M and

IIP,~II<M

for all ,~. C N. It is clear that (}-~ (~F,,)oo is isomorphic to (y~. Og.~)oc, and we know (,Johnson's example 5.2.1) that, ( ~ Og.~)~o contains a complemented

5.3 Notes and Remarks

101

copy of gl. Therefore (}-~.| contains a complemented copy of gl, too. On the other hand, it is clear that

(P,~) : e~(x)

--+

x

---+

( ~ eF,,)oo (p~(x))

is a projection onto (y~ | So, it follows that goo(X) contains a complemented copy of g~. Finally, note that goo (X) is isometrically isomorphic to a complemented subspace of Loo(p, X). Indeed, likewise in the proof of proposition 5.2.3, take a sequence (Am) of pairwise disjoint measurable sets with finite and positive measure, and consider the subspace 7 / o f Loo(#, X) of all functions which are :onstant in the A,~'s. Then the map

Loo(#,X) f

~

7/

1(s

> ~-~.X,~(.),-777-77 ?1,

fd#) n

ts a projection (it is a conditional expectation). Since 7t is clearly isometrically isomorphic to goo(X), our statement is proved. Hence (c) implies (a), obviously. This completes the proof.

Remark 5.2.4. It should be pointed out that the preceding result provides immediately examples of Banach spaces X which are even separable and reflexive for which L ~ ( p , X) contains a complemented copy of ~1. Take for instance X = ( ~ | )2. 5.3 N o t e s a n d R e m a r k s As we have already mentioned, in this chapter we have considered only orfinite measure spaces, while in the rest of the monograph we have been working in arbitrary measure spaces. In the introduction of the chapter we have explained why, but we would like to call the attention here to a significant fact. At the moment, Banach space specialists usually study almost exclusively L ~ ( # ) spaces for finite (or ~-finite) measures >, with only one i m p o r t a n t exception: spaces of type goo(F). Here, as usual, s denotes the Banach space of all bounded scalar functions defined on a certain set F, endowed with the supremum norm. The vectorial version of this space is goo(F, X), which is d e f n e d in the natural way. We believe t h a t while considering the finite measure case, we have exposed most of the main ideas related with our problems, but we would like to mention also the results which are known in more general situations. Concerning the search of complemented copies of co in Loo(/~, X ) there are two results. One for the purely atomic case, and the other one for the non purely atomic one:

102

5. The space Loo(#,X)

(a) Leung and Rgbiger [90] showed that if the set F of indices is measurable (which includes of course the denumerable case), then Coo(F, X) contains a complemented copy of c0 if and only if X does. They left open the general problem for arbitrary sets of indices. (b) Dfaz proved in [27] his theorem 5.1.3 (and corollary 5.1.1) for arbitrary measure spaces. About complemented copies of Cl in Loo(/~, X) we can say that if X contains C~"s uniformly complemented then Loo(/z, X) contains a complemented copy of g~ (this is just because if X contains C~'s uniformly complemented, by theorem 5.2.3, g~(X) contains a complemented copy of Cl, and this space is clearly complemented in Loo (p, X)). However, we do not know if the converse is true or not. We must say that actually Leung and Rgbiger [90] work in spaces more general than Coo(F, X). They study sums in the sense of Coo of families of Banach spaces. However, we would add that, concerning the problem we are dealing with, these two kind of spaces present a completely analogous behavior. Now, let us give a look at the results of the chapter. At the beginning of Section 5.1 we recall that we can decompose in a very nice way each member of ba(~ -- go~ in sum of two measures, one of them countably additive, and the other one purely finitely additive. Of course, this is just a particular case of Yosida-Hewitt decomposition theorem [51, III.5.8]. When we move the above decomposition from C~ to Coo (X)* we introduce the projection P in t~oo(X)* which appears at the beginning of the section, and which is crucial to get Leung and RSbiger' theorem 5.1.1. It is worth to observe that this projection could also be viewed from a more general point of view. One can easily show that if X is a closed subspace of Y and there exists a bounded linear operator S : Y -+ X** such that S Ix = idx (where, as usual, we identify X with a closed subspace of X** in the canonical way), then there exists a continuous linear projection P : Y* -+ Y* whose kernel is X • = {x* E Y* : x*(X) = {0}}, and therefore, X* = Y*/X • is complemented in Y*. In particular, our assumption is satisfied whenever X C Y C X**, taking as S the natural embedding. If we apply this to

co(X) c coo(x) c coo(x**) = co(X)**, we get a pro.jection P from g.oo(X)* onto its subspace co(X)* = Cx(X*). However, we do not conclude from the general theory that this projection is w*-sequentially continuous, which is very important for us. This is true in our particular situation, as we show in Proposition 5.1.1. In this monograph we have hardly dealt with (~-~| spaces, but one should notice that Johnson's example also shows that the behavior of ( ~ @X,,.)oo is quite different to the one of ( ~ | for 1 < p < +oo.

5.3 Notes and Remarks

103

We would like to point out that the classical perturbation lemma about, basis ( [93, 1.a.9.] or [35, Chapter V, Theorem 12]) is just an infinitedimensional version of the well known iemma 5.2.1. In Section 5.2, we explain in general when we say that a Banach space X contains ~l]"s uniformly (complemented) for 1 _< p _< +oc. Actually we are only interested in the case p = 1 (and by duality, in the case p = +oo), but, we believe that to consider the general situation helps to understand better these concepts. Notice that Theorem 5.2.3, the main result of Section 2, lies on the "Great Theorem" of Maurey-Pisier (Theorem 5.2.2). This is because this theorem provides a good characterization of when X contains ~ ' s uniformly complemented, and by duality and the Local Reflexivity Principle (Theorem 5.2.1) this provides a good characterization of when X contains g~'s uniformly complemented. We do not know of any "elementary proof'. Theorem 5.2.3 is due to S. Dfaz [28] and N.J. Kalton. S. Dfaz in [28] proved that conditions (a), (b), (c) and (d) are equivalent, and that they hold whenever (e) is satisfied. He also showed that the five condition (a), (b), (c), (d) and (e) are equivalent in quite general conditions (for instance, for Banach lattices). Therefore, the only open problem left by him was whether (a) (or any of the other equivalent conditions (b), (c) or (d)) implied (e) or not. In the Spring of 1996 we asked N.J. Kalton about this. He showed us how to prove that (d) indeed implies (e), and kindly allowed us to include his solution here. We have summarized it in lemma 5.2.2 which is a reformulation of Kalton's ideas contained in [79]. We believe that the lemma is much more subtle than it seems and provides a very interesting connection between 5oo (X**) and f-oo(X)**. In fact, we have seen several times people proclaiming that necessity of condition (e) was very easy and later on they were not able to give a proof. We may add that we. have read several wrong proofs of the assertion (some of these wrong proofs were made by us). To show necessity of (e) in Kalton-Dfaz Theorem 5.2.3 one tbllows more or less the following idea: if too(X) contains g)'s uniformly complemented, then its second dual, g~(X)**, does either. From this fact we wish to conclude that go~(X**) = CI(X*)* = co(X)** has the same property, but these spaces are quite well known, and thanks to Maurey-Pisier theorem we may conclude our thesis. The point is that although too(X)** and goo(X**) might seem close in a first view, a careful look shows us that they are indeed very different. This is the problem solved by Kalton's lemma. We wish to point out that Kalton's lemma is closely related with the important notion of locally complemented subspace. From this point of view it just says that too(X) is locally complemented in too(X**). The notion of locally complemented subspace has been studied by several authors and has found many interesting applications in Banach space theory (see [55], [79]

104

5. The space L ~ ( p , X )

and [64]). Its isometric version has been considered recently in [106].

[91] and

It is worth noticing that Diaz-Kalton Theorem 5.2.3 and Hagler-Stegall result mentioned in remark 5.2.2 can be combined fruitfully to obtain a significant consequence: T h e o r e m . Let (Y?, Z, #) be a a-finite measure space, then the following are equivalent:

(a) (b) (c) (d) (e)

L ~ ( # , X ) contains a complemented copy of L1 ([0, 1]). got(X) contains a complemented copy of Ll([O, 1]). L ~ ( p , X ) contains a complemented copy of ~1. go~(X) contains a complemented copy off1. X contains g~l~'s uniformly complemented.

Pro@ We know by Theorem 5.2.3 that (c), (d) and (e) are equivalent. On the other hand, goo(X) contains a complemented copy of (y~ | whenever (e) holds, as shown in the proof of Dfaz-Kalton Theorem 5.2.3. Thus, HaglerStegall's result mentioned in remark 5.2.2 proves that (e) implies (b). To see that (b) implies (a), it is enough to observe that L ~ ( p , X ) contains a complemented copy of ~o~(X) (this easy fact was shown in the proof of Theorem 5.2.3). Finally, (a) implies (c) is trivial. Dfaz [28] has also proved that for 1 < p < +o% Loo(p,X) contains a complemented copy of gp whenever X contains g~ 's uniformly complemented.

6. T a b u l a t i o n o f R e s u l t s

Here is a t a b l e w h i c h s u m m a r i z e s all t h e r e s u l t s of o u r m o n o g r a p h .

C(K,X)

LI(/Z, X )

Lp(It, X )

Lo~(#,X)

l
D co

D co (c)

always

XDco

XDco

always

(trivial)

(2.1.6)

(2.1.6)

(trivial)

X~co (~)

always

X Dec (c) or XDco and # is not purely atomic

# is not purely atomic

X Dco (c) or XDco and p is not purely atomic

(4.3.4)

(4.3.4)

(5.1.4)

or XDgl

always

XDCl

always .

(3.1.2)

(trivial)

(2.2.2)

(trivial)

XDgi (~)

always

XDg~ (~)

xJ~ uniformly complemented

(3.1.4)

(trivial)

(4.1.2)

(5.2.3)

or XDg~

XDgoo

XDg~

always

(3.3.1)

(4.2,1)

(4.2.1)

(trivial)

(3.2.1)

or

XDco and

C(K)~el D gl

D gi

(A

D g~

T h e m e a n i n g of t h e T a b l e is clear. In t h e first row we h a v e t h e spaces we a r e s t u d y i n g , in t h e first c o l u m n we see t h e different c o n t a i n m e n t s we a r e

106

6. Tabulation of Results

considering, and in each box we put the condition which must be satisfied and the n u m b e r of the corresponding theorem if it is not trivial. "Trivial" means t h a t it follows from the following easy containments

C(K, X) D C(K) D co (c) (c)

L1 (#, X) D L1 (i.t) D el (~) (c) (c)

Recall t h a t we are assuming C ( K ) , Lp(#) and X infinite-dimensional. For 1 <_ p < + c o the measure # in Lp(#, X) spaces is arbitrary, in the case p = + o c it is assumed to be ~-finite. A look at the table shows t h a t in most cases we get the natural answers, t h a t is, we get a ~ r m a t i v e answers to P r o b l e m 2 in the Introduction. There are only the following exceptions: (a) The different containments of co as a c o m p l e m e n t e d subspace, and (b) C o m p l e m e n t e d copies of gl in Loo(#, X). In b o t h cases we see t h a t vector valued function spaces contain complemented copies of Co or ~1 more often t h a n expected. Notice once more t h a t case (b) provides the only answer which depends exclusively on the local structure of the B a n a c h space X.

7. Some Related Open Problems

To finish our work we would like to mention a few lines of research and open problems which are closely related with the results of this monograph. I. If we remember Problem 1 posed in the Introduction, we see that we have a complete solution to it with only one exception: For L ~ (#, X) we know the answer for a-finite spaces, but for arbitrary positive measure spaces we do not. Then we have the following P r o b l e m 7.1. Assume (f2, &, #) is a non a-finite measure space, when does L ~ ( p , X ) have a complemented copy of co!, when does it have a complemented copy of ~,1 "~ It would be particularly interesting to answer these questions for the space L ~ ( p , X ) = eo~ (C, X). Leung and R~biger [90] already posed this problem for complemented copies of co in the sum in the sense of goo of an arbitrary family {Xi : i E F} of Banach spaces. See Section 5.3, and in general Chapter 5. II. In order to consider analogous problems to the ones studied here, a natural thing is to add to our list of spaces (Co, el and too) the fr spaces (1 < r < oe), that is, we can ask: P r o b l e m 7.2. When does C(K, X) or Lp(#, X) have copies or complemented copies of ~r (1 < r < oo)? Tile general problem seems to be very difficult because no good characterization of spaces containing ~,. (1 < r < oc) is known. However, some partial answers have already been given (see [11], [28], [71], [107], [108], [119], [120]). Let us mention at least that Lp(>, X) can contain copies of gr, while neither Lv(#) nor X contain a copy of g, (this is an old known result, see [107], [108] or [94, Remark to Lemma 1.f.8]). And in the same way we can consider LI([0, 1]):

108

7. Some Related Open Problems

P r o b l e m 7.3. If L p ( p , X ) (1 < t) < oc) contains a copy of L1([0,1]) must X contain a copy of LI([0, 1]) ? or, in general, when does L p ( # , X ) have a copy of Cl([0, 1]) These questions were posed to the authors by J. Diestel and they can also be found in [117, Question 10]. The answer to the first one is yes if X is a dual space [117], but we do not know a complete answer even for p = 2. W h a t about complemented copies of L1 ([0, 1]) ? In this case the question is natural for L ~ ( p , X) and C(K, X), too. We have just shown in the Notes and Remarks of Chapter 5 a complete characterization for Lo~(#,X). But concerning the other spaces we do not know anything. 7.4. When does C(K,X) or Lp(p,X) (1 < p < oc) contain a complemented copy of LI([0, 1]) ?

Problem

For C(K,X) this has already been asked by E. and P. Saab in [117, Question 12]. An anonymous referee pointed out to us that Hagler and Stegall paper [67] might help to give a solution to this problem. Actually, it was this remark t h a t led us to formulate the solution tbr Loo(p, X). I I I . All problems considered here can also be posed in more general situations, for instance in locally convex spaces and, in particular in Frdchet spaces. This last case is probably the most important. Of course, some work in this direction has already been done. For instance, in connection with J. Hoffmann-Jorgensen and Kwapiefi's work see [56]. I V . Theorems studied in this monograph have already inspired the search of copies of sequence spaces in several different situations. Let us mention some of them: (a) Tensor products (see [117]). (b) Function spaces more general than Lp(#,X) spaces, like Orlicz vector valued function spaces L~(X) (see, for instance [6] or [7]), or even the more general E(X) spaces, where E is an order continuous Banach lattice which has weak unit as considered in [127] (see for instance [53] or [97]). (c) Vector measure spaces or operator spaces (as in [46] , [47] or [121]). (d) Vector-valued Hardy spaces (as in [44]). Nevertheless, we believe t h a t in these directions many things remain to be done. V . It is well known that vector-valued function spaces may be viewed as operator spaces. For example, the space I~(X, C(K)) of all compact operators from X into C(K) is isometrically isomorphic to C(K, X*) [51, VI.7.1], and Loo(p, X) is very useful in the representation of operators from L I ( # ) into

7. Some Related Open Problems

109

X [39, Chapter III]. For this reason the results given here have inspired and have found applications in the study of several operator spaces (see [12], [13], [43], [48], and see also [61] and [24] in connection with [48]). Among these applications are some complementation results more or less related to the famous open problem of the uncomplementability of the space K2(X, Y), of compact operators from X to Y, in the space s Y ) of all operators from X to Y (see [48]). This should not be surprising because it was already shown in [78, Theorem 6] that such a problem is related to the presence of copies of co or goo in operator spaces. VI. We have so far been mainly interested in isomorphic copies of spaces. Another natural point of view is to consider isometric copies. Some results in this line have been recently obtained by Koldobsky [81] (see also [65], I103], [108] and [66]). V I I . Instead of asking about subspaces we could also ask about quotients. That is, P r o b l e m 7.5. Wizen do the spaces considered here have quotients isomorphic to any of th,e classical sequence spaces? In fact Dfaz [29] has already got an interesting answer to the preceding question. He has shown that: "For" 1 <_p <_ oc, if the measure Ix is not purely atomic and X* contains a copy of ~1, then Lp(Ix, X ) has a quotient isomorphic to e0 ". This result has allowed him to get a striking characterization of Grothendieck Lp(Ix, X) spaces. The study of quotients was continued by Dfaz with Schliichternann in [34]. The result we have .just mentioned recalls us that many properties considered in Banach space theory are connected with the presence of copies or quotients of certain spaces. For instance, property (V*) of Petczynski is connected with the existence of complemented/ql-sequences [7], and Grothendieck property with the lack of" complemented copies of Co [26]. Of course these connections haw', already been exploited ([7], [25], [26], [29], [59], [104]), tout it seems natural to think that they could be used further. V I I I . Let us finally consider the tbllowing problem: It is well known that ~ and Loo([0, 1]) are isomorphic [93, p. 111]. However, let us suppose that X has a copy but not a complemented copy of co. For instance X may be ~'.~ or (qoo/co (a well studied space, see for example [49]), or more generally X may be any Grothendieck C ( K ) space or a quotient of it. It, follows from Theorem 5.1.4 that Loo([0, 1],X) does contain a complemented copy of co, but f,~(X) does ',tot. Hence L~o([0, 1],X) and g ~ ( X ) are not isomorphic. Then the tbllowing natural question arises: P r o b l e m 7 . 6 . / n which conditions aT'e Loo ([0, 1], X ) and goo(X) isomorphic ? We do not know the answer even for separable spaces X.

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Index

(~-2 | (&), 12 (r~), 42

102

C(K, X), 8 E(X), 108 G-space, 74 Lq~. (#, X*), 26 L1 ([0, +o0)), 61 Lp(~,X), 8 L~(X), 108 X DY, 8 X DY, 8 (c) [A], 13 [N], 13

Lp(#), 8 /3N, 68, 74 &, 35 5~m, 10 gp(X), 32

& ( M ) , 13

K(X, C(K)), 108

c ( x , Y), 109 s (A, x*), 19 s s

s

(#, X*), 26 18

18

P(N), 11 m} 8 (s.s.) sequence, 37 B(K), 34 C(K, X)*, 35

C(K), 8 i~(~,X)*, 17

cabv(Z), 8 cabv(Z, X ), 8 co-sequence, 9 gp-sequence, 9 &-sequence, 9 rcabv(K, X*), 35 ( E eX~)p, 39, 102

&~(X), 32 &o(F), 101

Bessaga-Petczyfiski theorem 9

&o(F,X), 101, 107

Caratheodory theorem, 40 compact space - extremally disconnected (or Stonean), 73 scattered (or dispersed), 65 complemented copy, 8 complemented sequence, 9 conditional expectation operator, 29 contains/~'s uniformly, 94 contains g~'s uniformly complemented, 95 contains ~p'S uniformly, 94 contains gp'S uniformly complemented, 95 contraction principle, 44 copy, 8 - complemented, 8 counting measure, 32

II. I1~, 8 II. I1r1628 II ~ il, 8

p, 18 C (for measure spaces), 27 ba(N), 84 c0(X), 96 coo, 44

ca(N), 84

cabv(t3(K), X*), 35 cabv~(Z, X*) , 19 d(X, Y), 94 rn,~, 35

rcabv(K), 66 w*-measurable, 18 fi, 19 ~ ( X , Y), 109

118

Index

density function, 17 dispersed (or scattered) compact space, 65 Dunford-Pettis, 34 extremally disconnected, 73, 74, 84 Fr6chet spaces, 108 function - w*-measurable, 18 Grothendieck property (or space), 68, 74, 109 Hardy spaces, 108 injective, 30 isometric copy, 15, 109 isometric theory of Lp(#) spaces, 32, 40 lifting, 17, 18 linear, 23 Local Reflexivity Principle, 95 locally complemented subspace, 103

-

martingale difference sequence, 63 Maurey-Pisier Great Theorem, 96 measure - w*-countably additive, 35 w*-regular, 35 regular, 35 strongly additive, 14 measure algebra, 40 non-trivial weak-Cauchy sequence, 37 operation (A), see Souslin operation, 30 Orlicz vector valued function spaces, 108 Phillips Theorem, 13 probability in Banach spaces, 61, 62 property (V*) of Petczynski, 109 purely atomic, 28, 32 Rademacher functions, 42 Radon-Nikod3~m property, 18 Radon-Nikod3~m type theorem, 19, 22 random variables, 61 independent, 61 symmetric, 61 Riesz-Singer theorem, 35 Rosenthal's Disjointification lemma, 11

scattered (or dispersed) compact space, 65 seminormalized, 9 semivariation, 8 separable measure space, 27 sequence (s.s.), 37 co-sequence, 9 - gv-sequence, 9 ~?l-sequence, 9 complemented, 9 - non-trivial weak-Cauchy, 37 seminormalized, 9 strongly summing, 37 Souslin operation, 5, 30, 31, 39 Stone-Cech compactification of N, 68 Stonean, 73 strong compactness in Lp(#, X), 63 strongly summing, 37 subsequence splitting lemma, 54, 62 UMD spaces, 63 unconditional basic sequence (in Lp(#)), 63 unconditional basis, 63 unconditionally converging operator, 12 uniform integrability, 27, 33, 34, 40 uniformly integrable, 34, 52, 54 56, 59-61, 77, 78 variation, 8 w.u.C., 9 weak compactness in Lv(#, X), 63 weak convergence in LB(#, X), 63 weakly compact operator, 12 weakly unconditionally Cauchy, 9 weakly unconditionally convergent (or weakly unconditionally Cauchy) series, 9