Elliptic theory and noncommutative geometry

Operator Theory: Advances and Applications Vol. 183 Editor: I. Gohberg Editorial Office: School of Mathematical Science...

Author: Vladimir E. Nazaykinskiy | A. Yu. Savin | B. Yu. Sternin

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Operator Theory: Advances and Applications Vol. 183 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Böttcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. W. Helton (La Jolla) M. A. Kaashoek (Amsterdam)

H. G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) H. Widom (Santa Cruz)

Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze Universität Potsdam, Germany

Jerome A. Goldstein The University of Memphis, TN, USA

Sergio Albeverio Universität Bonn, Germany

Nobuyuki Tose Keio University, Yokohama, Japan

Michael Demuth Technische Universität Clausthal, Germany

Elliptic Theory and Noncommutative Geometry Nonlocal Elliptic Operators Vladimir E. Nazaikinskii Anton Yu. Savin Boris Yu. Sternin

Birkhäuser Basel · Boston · Berlin

A P D E

Advances in Partial Differential Equations

Authors: Vladimir E. Nazaikinskii Institute for Problems in Mechanics Russian Academy of Sciences Prosp. Vernadskogo 101-1 119526 Moscow Russia e-mail: [email protected]

Anton Yu. Savin Boris Yu. Sternin Independent University of Moscow Bolshoy Vlasyevskiy Pereulok 11 119002 Moscow Russia e-mail: [email protected] [email protected]

2000 Mathematical Subject Classification: 19K, 35J, 39B, 46L, 58J

Library of Congress Control Number: 2008924711

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-8774-7 Birkhäuser Verlag AG, Basel - Boston - Berlin 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 other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany ISBN 978-3-7643-8774-7

e-ISBN 978-3-7643-8775-4

987654321

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Contents Preface

xi

Introduction

1

I Analysis of Nonlocal Elliptic Operators

3

1 Nonlocal Functions and Bundles 1.1 Group Algebras and Crossed Products . . . . . 1.1.1 Group Algebras . . . . . . . . . . . . . . 1.1.2 C ∗ -Crossed Products . . . . . . . . . . . 1.1.3 Isomorphism Theorem . . . . . . . . . . 1.1.4 Smooth Crossed Products . . . . . . . . Motivation . . . . . . . . . . . . . . . . Suﬃcient Condition for Locality . . . . Groups of Polynomial Growth . . . . . . Tempered Actions . . . . . . . . . . . . Schweitzer theorem . . . . . . . . . . . . 1.2 Nonlocal Functions . . . . . . . . . . . . . . . . 1.2.1 Assumptions about the Group Γ . . . . 1.2.2 Continuous Nonlocal Functions . . . . . Nonlocal Functions as Operators in L2 . Changes of Variables and Substitutions 1.2.3 Smooth Nonlocal Functions . . . . . . . 1.3 Nonlocal Bundles . . . . . . . . . . . . . . . . . 1.4 Case of Noncompact Spaces . . . . . . . . . . . 1.4.1 Nonlocal Functions . . . . . . . . . . . . 1.4.2 Special Noncompact Manifolds . . . . .

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5 5 5 7 11 13 13 14 15 16 17 18 18 18 19 19 20 22 22 23 24

2 Nonlocal Elliptic Operators 2.1 Scalar Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Symbol Algebras . . . . . . . . . . . . . . . . . . . . . . . .

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vi

Contents 2.1.2

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Quantization of Symbols and Operators of Quantization of Continuous Symbols . . . Quantization of Smooth Symbols . . . . . 2.1.3 Operators of Arbitrary Order . . . . . . . Operators in Spaces of Sections of Bundles . . . 2.2.1 Operators of Order Zero . . . . . . . . . . 2.2.2 Operators of Arbitrary Order . . . . . . . 2.2.3 Operators on Noncompact Manifolds . . . Symbols as Nonlocal Functions on T ∗ M . . . . .

3 Elliptic Operators over C ∗ -Algebras 3.1 Hilbert Modules and Λ-Index . . . . . . . . . . . 3.1.1 Hilbert Modules . . . . . . . . . . . . . . 3.1.2 Operators in Hilbert Modules . . . . . . . 3.1.3 Fredholm Property and Index . . . . . . . 3.2 Pseudodiﬀerential Operators over Λ . . . . . . . 3.2.1 Λ-bundles and Section Spaces . . . . . . . Λ-bundles . . . . . . . . . . . . . . . . . . Space of Smooth Sections . . . . . . . . . Space L2 . . . . . . . . . . . . . . . . . . Sobolev Spaces . . . . . . . . . . . . . . . 3.2.2 Symbols and Pseudodiﬀerential Operators 3.2.3 Ellipticity and the Λ-Fredholm property . 3.3 Nonlocal Pseudodiﬀerential Operators over Λ . . 3.3.1 Operators in Trivial Λ-Bundles . . . . . . 3.3.2 Operators in General Λ-Bundles . . . . . 3.3.3 Symbols in Trivial Bundles . . . . . . . . 3.3.4 Symbols in Nontrivial Bundles . . . . . . 3.3.5 Ellipticity and the Λ-Fredholm Property .

Order Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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27 28 29 30 32 32 34 35 36

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37 38 38 41 42 44 44 44 45 46 46 47 50 51 51 52 52 54 54

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II Homotopy Invariants of Nonlocal Elliptic Operators

57

4 Homotopy Classiﬁcation 4.1 Ell-Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diﬀerence Construction . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Isomorphism of Ell- and K-Groups . . . . . . . . . . . . . . . . . .

59 59 60 61

5 Analytic Invariants 5.1 Fredholm Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 C ∗ (Γ)-Index and Its Connection with the Fredholm Index . . . . .

63 63 63

Contents 6 Bott 6.1 6.2 6.3 6.4 6.5

Periodicity Preliminary Remarks . . . . . . . . . . . . . Exterior Product of Operators . . . . . . . Euler Operator and the Bott Element . . . Bott Mapping and the Periodicity Theorem Proof of the Periodicity Theorem . . . . . .

vii

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67 67 68 69 70 71

7 Direct Image and Index Formulas in K-Theory 7.1 Direct Image Mapping in K-Theory for Embeddings . . . . . . . 7.1.1 Exterior Products . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Normal Bundle . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Deﬁnition of the Direct Image Mapping . . . . . . . . . . 7.1.4 Λ-Index Is Preserved by the Direct Image Mapping . . . . 7.2 Index Formulas in K-Theory . . . . . . . . . . . . . . . . . . . . 7.2.1 Direct Image in K-Theory for the Projection into a Point 7.2.2 Index Formulas . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Proof of Theorem 7.7 . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Exterior Products of Operators . . . . . . . . . . . . . . . 7.3.2 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Completion of the Proof . . . . . . . . . . . . . . . . . . .

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75 75 75 76 77 78 79 79 79 79 80 82 86

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89 89 89

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90 92 94 95 95 96 96 96 97

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99 99 101 101 102 102 103 105

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8 Chern Character 8.1 Diﬀerential Forms and Graded Traces . . . . . . . . . . . 8.1.1 Noncommutative Diﬀerential Forms . . . . . . . . 8.1.2 Graded Trace on the Algebra of Noncommutative Diﬀerential Forms . . . . . . . . . . . . . . . . . . 8.2 Chern Character of Projections . . . . . . . . . . . . . . . 8.3 Chern Character of Symbols . . . . . . . . . . . . . . . . . 8.4 How to Compute the Chern Character . . . . . . . . . . . 8.4.1 Computation in Terms of the Symbol . . . . . . . 8.4.2 Computation in Terms of Connections . . . . . . . Graded Trace . . . . . . . . . . . . . . . . . . . . . Chern Character of Projections . . . . . . . . . . . Chern Character of Symbols . . . . . . . . . . . . 9 Cohomological Index Formula 9.1 Todd Class . . . . . . . . . . . . . . . . . . . 9.2 Index Theorem . . . . . . . . . . . . . . . . . 9.2.1 Topological Index . . . . . . . . . . . 9.2.2 Chern Character on the Group K0 (Λ) 9.2.3 Statement of the Theorem . . . . . . . 9.3 Vanishing Theorem . . . . . . . . . . . . . . . 9.4 Proof of the Index Theorem . . . . . . . . . .

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viii

Contents 9.5

Proofs of Auxiliary Statements . . . . . . . . . . . 9.5.1 Multiplicativity of the Chern Character . . Case 1. Compact Manifold . . . . . . . . . Case 2. Noncompact Manifold . . . . . . . . 9.5.2 Chern Character of the Symbol of the Euler

10 Cohomological Formula for the Λ-Index 10.1 Noncommutative Diﬀerential Forms . . . 10.2 Graded Traces over Λ∞ . . . . . . . . . . 10.2.1 Cohomology of Groups . . . . . . . 10.2.2 Construction of Traces . . . . . . . 10.2.3 Examples . . . . . . . . . . . . . . Traces of Degree 0 . . . . . . . . . Trace of Degree 1 for the Group Z Trace of Degree 2 for the Group Z2 10.3 Graded Traces over C ∞ (X) Γ . . . . . . 10.4 Chern Character and the Index Formula . 10.4.1 Chern Character . . . . . . . . . . 10.4.2 Index Formula . . . . . . . . . . . 10.4.3 Proofs of Auxiliary Statements . .

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108 108 108 110 114

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115 116 117 118 119 119 119 120 120 120 122 122 123 123

11 Index of Nonlocal Operators over C ∗ -Algebras 11.1 Classiﬁcation of Nonlocal Elliptic Operators . . . . . . . . . 11.1.1 Ell-Group . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Diﬀerence Construction . . . . . . . . . . . . . . . . 11.1.3 Isomorphism of the Groups Ell and K . . . . . . . . 11.1.4 K¨ unneth Formula and Classiﬁcation Modulo Torsion 11.2 Chern Character and the Index Theorem . . . . . . . . . . 11.2.1 Chern Character . . . . . . . . . . . . . . . . . . . . 11.2.2 Index Theorem . . . . . . . . . . . . . . . . . . . . . Case 1. Product of Elements of Even K-Groups . . . Case 2. Product of Elements of Odd K-Groups . . .

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127 128 128 128 128 129 130 130 131 131 132

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III Examples 12 Index Formula on the Noncommutative Torus 12.1 Operators on the Noncommutative Torus 12.2 Index Computation . . . . . . . . . . . . . 12.2.1 Reduction to the Two-Dimensional 12.2.2 Index of Operators on the 2-Torus 12.3 Special Cases . . . . . . . . . . . . . . . .

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139 139 140 140 141 142

Contents

ix

13 An Application of Higher Traces 13.1 Index with Values in Odd K-Groups . . . . . . . . . . . . 13.2 Odd Index Formula . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Suspension . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Index Theorem . . . . . . . . . . . . . . . . . . . . Step 1. Reduction to the Cosphere Bundle and the Simons Character . . . . . . . . . . . . . Step 2. Desuspension and Integration over S1 . . . 13.3 Example of Λ-Index Computation . . . . . . . . . . . . .

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145 145 146 147 147 148 148 149

14 Index Formula for a Finite Group Γ 153 14.1 Trajectory Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 14.2 Index Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

IV Appendices

157

A C ∗ -Algebras A.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Deﬁnitions and Examples . . . . . . . . . . . . . . . . . . . A.1.2 Unital Algebras and Units . . . . . . . . . . . . . . . . . . . A.1.3 Homomorphisms, Ideals, Quotient Algebras, and Extensions A.1.4 Commutative C ∗ -Algebras . . . . . . . . . . . . . . . . . . . A.1.5 Spectrum and Functional Calculus . . . . . . . . . . . . . . A.1.6 Local C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . A.1.7 Positive Elements . . . . . . . . . . . . . . . . . . . . . . . . A.1.8 Projections in C ∗ -Algebras . . . . . . . . . . . . . . . . . . A.2 Representations of C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . A.2.1 Basic Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Existence of Representations . . . . . . . . . . . . . . . . . A.2.3 Representations of Ideals and Quotient Algebras . . . . . . A.2.4 Primitive Ideals . . . . . . . . . . . . . . . . . . . . . . . . . A.2.5 Algebras of Type I . . . . . . . . . . . . . . . . . . . . . . . A.3 Tensor Products and Nuclear Algebras . . . . . . . . . . . . . . . . A.3.1 Minimal and Maximal Tensor Products . . . . . . . . . . . A.3.2 Nuclear Algebras . . . . . . . . . . . . . . . . . . . . . . . . A.3.3 Primitive Ideals in the Tensor Product . . . . . . . . . . . .

161 161 161 162 163 165 165 167 167 168 169 169 172 173 174 175 176 176 177 178

B K-Theory of Operator Algebras B.1 Covariant K-Theory . . . . . B.1.1 Topological K-Theory B.1.2 Group K0 (A) . . . . . B.1.3 Group K1 (A) . . . . . B.1.4 Bott Periodicity . . .

179 179 179 182 184 186

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Contents B.1.5 Long Exact Sequence in K-Theory . . . . . . . . Homomorphism ∂1 . . . . . . . . . . . . . . . . . Homomorphism ∂0 . . . . . . . . . . . . . . . . . B.1.6 Stability of K-Groups . . . . . . . . . . . . . . . B.1.7 K-groups of Local C ∗ -Algebras . . . . . . . . . . B.2 K-Homology . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 K-Homology of a Topological Space . . . . . . . Even Groups . . . . . . . . . . . . . . . . . . . . Odd Groups . . . . . . . . . . . . . . . . . . . . . B.2.2 K-Homology of Operator Algebras: Deﬁnitions . Fredholm Modules . . . . . . . . . . . . . . . . . Dual Algebras . . . . . . . . . . . . . . . . . . . Extensions of C ∗ -Algebras . . . . . . . . . . . . . Equivalence of Diﬀerent Deﬁnitions . . . . . . . . B.2.3 Suspension and Bott Periodicity . . . . . . . . . B.2.4 Long Exact Sequence in K-Homology . . . . . . B.2.5 Stability . . . . . . . . . . . . . . . . . . . . . . . B.2.6 Duality between K-Homology and K-Theory of Algebras . . . . . . . . . . . . . . . . . . . . . . . Even Groups . . . . . . . . . . . . . . . . . . . . Odd Groups . . . . . . . . . . . . . . . . . . . . .

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187 187 188 188 189 189 189 191 191 192 192 193 195 196 197 198 199

C Cyclic Homology and Cohomology C.1 Cyclic Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1.1 Enveloping Diﬀerential Algebra . . . . . . . . . . . . . . . . C.1.2 Graded Traces and Cyclic Functionals . . . . . . . . . . . . C.1.3 Cochains, Cyclic Cochains, and the Hochschild Diﬀerential C.1.4 Cyclic Cohomology and Hochschild Cohomology . . . . . . C.1.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1.6 Cup Product in Cyclic Cohomology . . . . . . . . . . . . . C.1.7 Periodicity in Cyclic Cohomology . . . . . . . . . . . . . . . C.1.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Cyclic Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.1 Deﬁnition of Cyclic Homology . . . . . . . . . . . . . . . . . C.2.2 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2.3 Pairing between Cyclic Homology and Cohomology . . . . . C.2.4 Morita Invariance . . . . . . . . . . . . . . . . . . . . . . . C.2.5 Chern Characters in Homology . . . . . . . . . . . . . . . .

201 201 201 203 204 204 205 206 207 209 209 209 210 211 211 211

Concise Bibliographical Remarks

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Bibliography

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Index

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199 199 199

Preface Noncommutative geometry, which can rightfully claim the role of a philosophy in mathematical studies, undertakes to replace good old notions of classical geometry (such as manifolds, vector bundles, metrics, diﬀerentiable structures, etc.) by their abstract operator-algebraic analogs and then to study the latter by methods of the theory of operator algebras. At ﬁrst sight, this pursuit of maximum possible generality harbors the danger of completely forgetting the classical beginnings, so that not only the answers but also the questions would defy stating in traditional terms. Noncommutative geometry itself would become not only a method but also the main subject of investigation according to the capacious but not too practical formula: “Know thyself.” Fortunately, this is not completely true (or even is completely untrue) in reality: there are numerous problems that are quite classical in their statement (or at least admit an equivalent classical statement) but can be solved only in the framework of noncommutative geometry. One of such problems is the subject of the present book. The classical elliptic theory developed in the well-known work of Atiyah and Singer on the index problem relates an analytic invariant of an elliptic pseudodiﬀerential operator on a smooth compact manifold, namely, its index, to topological invariants of the manifold itself. The index problem for nonlocal (and hence nonpseudodiﬀerential) elliptic operators is much more complicated and requires the use of substantially more powerful methods than those used in the classical case. It should be noted that although elliptic theory (more precisely, its analytic branch) for nonlocal operators has been studied suﬃciently well, meaningful results concerning the index problem for nonlocal elliptic operators have until recently been rather spare. It is only very recently that several important results (some of which are due to the authors) have appeared, suggesting that the solution of this problem is eventually within reach. Therefore, there is a need to gather together the facts already known on the topic. That is why the present book has been written. Methods of K-theory of operator algebras and noncommutative geometry are used here to solve the index problem for nonlocal elliptic operators associated with a countable group of diﬀeomorphisms of a manifold. Furthermore, to make the presentation self-contained and hence the book understandable for readers with standard university education in mathematics,

xii

Preface

we have decided to include the Appendix, which contains some material used in noncommutative elliptic theory, namely, C ∗ -algebras and their K-theory as well as basics of the theory of cyclic homology and cohomology. The main results contained in the book were obtained during the authors’ stay as guests researchers at Institut f¨ ur Analysis, Leibniz Universit¨ at Hannover (Hannover, Germany). The authors thank Director of the Institute, Professor E. Schrohe and Leibniz Universit¨ at Hannover for kind hospitality. We also wish to express our gratitude to Professors A. B. Antonevich and A. V. Lebedev of Minsk Univeristy, Byelorussia, for useful discussions. We are grateful to Professor B.-W. Schulze of Potsdam University, who suggested publishing our recent results on the noncommutative theory of diﬀerential equations with Birkh¨ auser. Finally, we especially wish to thank Professor G. V. Rozenblyum of G¨oteborg University, Sweden, who was the ﬁrst to acquaint us with the beautiful world of noncommutative geometry. The book was supported by the Russian Foundation for Basic Research (grant No. 06-01-00098) and by the Deutsche Forschungsgemeinschaft (project 436 RUS 113/849/0-1“K-theory and noncommutative geometry of stratiﬁed manifolds”). The second author gratefully acknowledges the support from the Deligne 2004 Balzan prize in mathematics.

Hannover–Moscow, December 2007

Introduction Diﬀerential equations containing values of unknown functions and their derivatives at diﬀerent points of a manifold are called nonlocal diﬀerential equations. The simplest equation of this type has the form D1 u(x) + D2 u(g(x)) = f (x),

x ∈ Ω,

where D1 and D2 are some diﬀerential operators, u is the unknown function, and g : Ω → Ω is a self-mapping of the domain where the equation is considered. We shall consider only equations in which the mapping g is invertible. Such equations arise in numerous physical and mathematical problems, in particular, in problems related to noncommutative geometry. We present only some of them: 1. Elliptic theory on the noncommutative torus and the quantum Hall eﬀect. Diﬀerential operators on the noncommutative torus were studied by Connes in [24, 27], who, in particular, obtained an index formula for such operators. The coeﬃcients of these operators contain shift operators generated by irrational rotations. 2. More general nonlocal operators related to deformations of function algebras on toric manifolds, in particular, to quantum spheres obtained by noncommutative isospectral deformations. (See Connes–Landi [29], Connes–DuboisViolette [28], Landi–van Suijlekom [50], etc.) 3. Nonlocal boundary value problems1 (Carleman [23], Antonevich [3], Bitsadze [21], Dezin [34], Skubachevskii [71], etc.). These examples naturally justify interest in general nonlocal elliptic operators, i.e., in diﬀerential or pseudodiﬀerential operators whose coeﬃcients include not only operators of multiplication by functions but also shift operators induced by a discrete group Γ of diﬀeomorphisms of the manifold. Finiteness theorems for such operators were obtained by Antonevich and Lebedev (e.g., see [1,2,4] and references therein). The present book deals mainly with the topological 1 Note,

however, that we do not consider nonlocal boundary value problems in this book.

2

Introduction

(or, if you like, noncommutative-geometric) aspects of the theory. Namely, for general nonlocal operators we obtain a cohomological index formula. Let us explain the main results of the book in more detail. We consider diﬀerential operators whose coeﬃcients contain shift operators corresponding to the action of a discrete group Γ on a smooth closed manifold. Under the assumption that the group is of polynomial growth and the action is embedded in an action of a compact Lie group of diﬀeomorphisms, we show that to a nonlocal elliptic operator one can assign a Fredholm operator in Hilbert modules over the group C ∗ -algebra C ∗ (Γ). The latter operator has a well-deﬁned index that is an element of the K-group of this algebra: indC ∗ (Γ) D ∈ K0 (C ∗ (Γ)).

(0.1)

The Fredholm index of the original operator can be obtained as the image of the index (0.1) under the mapping induced by the trivial representation C ∗ (Γ) → C. We present formulas that allow us to calculate the index (0.1) in terms of the symbol of the operator. First, we derive an index formula in K-theory. To this end, we establish the stable homotopy classiﬁcation of nonlocal elliptic operators, construct the direct image mapping for nonlocal elliptic symbols under an embedding of manifolds, and generalize the Bott periodicity theorem to the case of inﬁnite discrete groups. Then, in Chaps. 9 and 10, we obtain cohomological formulas for the coupling of the index (0.1) with cyclic cocycles over a smooth local subalgebra in C ∗ (Γ). The simplest of these formulas (Chap. 9) leads to formulas for the Fredholm index. Cohomological formulas are given in terms of the Chern character determined here for the symbol and the Todd class modiﬁed in the spirit of [15]. Finally, we construct formulas for the Λ-index for elliptic nonlocal operators acting in Hilbert modules over a C ∗ -algebra Λ in a suﬃciently wide class of algebras. The result is obtained by combining the methods developed in the present book and the classical approach in [57] (where index formulas were obtained for local elliptic operators over C ∗ -algebras).

Part I

Analysis of Nonlocal Elliptic Operators

Chapter 1

Nonlocal Functions and Bundles 1.1 Group Algebras of Discrete Groups and Crossed Products An important notion in the theory of nonlocal operators is the notion of the crossed product of a C ∗ -algebra by a discrete group. Such products (where for the algebra we consider an appropriate algebra of functions) play the role of algebras of symbols in this theory. In this section, we present necessary deﬁnitions and main facts of interest to us in the theory of crossed products.1

1.1.1 Group Algebras Let Γ be a countable discrete group. Consider the space F (Γ) of ﬁnite (i.e., nonzero at ﬁnitely many points) functions f : Γ → C. This space is a normed (but noncomplete) ∗-algebra with respect to the multiplication, involution, and norm given by the formulas f1 (k)f2 (h) (group convolution), (1.1) (f1 f2 )(g) = k,h∈Γ kh=g

f ∗ (g) = f (g −1 ), f 1 = |f (g)|.

(1.2) (1.3)

g∈Γ

Deﬁnition 1.1. The enveloping C ∗ -algebra of the algebra F (Γ), i.e., its completion with respect to the norm f = sup π(f ) , π

1 The

C ∗ -crossed

notion of product is deﬁned not only for discrete but also for arbitrary locally compact topological groups; e.g., see [27]. But we are interested only in the discrete case.

6

Chapter 1. Nonlocal Functions and Bundles

where the supremum is taken over all ∗-representations π : F (Γ) −→ BH that are continuous in the norm · 1 (here BH is the C ∗ -algebra of bounded operators in a Hilbert space H), is called the group C ∗ -algebra of the group Γ and is denoted by C ∗ (Γ). Remark 1.2. The group C ∗ -algebra has the following universality property: any unitary representation Γ → BH can be continued to a unitary representation C ∗ (Γ) → BH. Of course, the continuation is unique. In particular, consider the regular representation C ∗ (Γ) −→ Bl2(Γ), f −→ f (g)Lg , g∈Γ

where Lg is the left shift operator on the group Γ: [Lg ϕ](h) = ϕ(g −1 h). The image of the algebra C ∗ (G) under the regular representation is a C ∗ -subalgebra in Bl2 (Γ). This image is called the reduced group C ∗ -algebra of the group Γ ∗ and is denoted by Cred (Γ). ∗ (Γ) are diﬀerent. But there is an In general, the algebras C ∗ (Γ) and Cred important class of discrete groups for which they coincide. Deﬁnition 1.3. A discrete group Γ is said to be amenable if there exists a ﬁniteadditive left-invariant probability measure on Γ. Free groups with ≥ 2 generators can serve as examples of nonamenable groups. The following amenability criterion (the Følner condition) holds. A discrete group Γ is amenable if and only if, for any ﬁnite set K ⊂ Γ and any ε > 0, there exists a nonempty ﬁnite set U ⊂ Γ such that |sU U | < ε, |U |

∀s ∈ K.

Here is the symmetric diﬀerence of sets, and |V | is the cardinality of a set V . Examples of amenable groups include ﬁnite, Abelian, and solvable groups. Any subgroup of an amenable group is amenable. A class of amenable groups important for our studies will be introduced in Sec. 1.1.3. ∗ Proposition 1.4. If Γ is a discrete amenable group, then C ∗ (Γ) = Cred (Γ). More∗ ∗ over, in this case the group C -algebra C (Γ) is nuclear.

1.1. Group Algebras and Crossed Products

7

Both statements are speciﬁc cases of corresponding, more general, statements for crossed products (see the next section). Example 1.5. Let Γ = Z. The mapping f (n)einϕ f → n∈Z

takes elements of the algebra F (Γ) to trigonometric polynomials on the circle S1 with coordinate ϕ mod 2π. The multiplication in F (Γ) becomes the multiplication of functions, and the involution becomes the complex conjugation. Further, under the same transformation (the discrete Fourier transform), the space l2 (Γ) becomes the space L2 (S1 ) on which the above trigonometric polynomials act under the regular representation by multiplications. The norm of such a multiplication operator is the maximum absolute value of the polynomial under study; therefore (by the Weierstrass theorem about the approximation of continuous functions by ∗ (Z) C(S1 ), and hence, by the preceding trigonometric polynomials), we have Cred ∗ 1 proposition, C (Z) = C(S ). Similarly, we have C ∗ (Zn ) = C(Tn ).

1.1.2 C ∗ -Crossed Products The construction of a C ∗ -group algebra has the following natural generalization. Suppose that a discrete group Γ acts on a C ∗ -algebra A by unital ∗-automorphisms τ (g), τ (g)(ab) = τ (g)(a)τ (g)(b),

(τ (g)a)∗ = τ (g)(a∗ ),

τ (g)1 = 1,

In this case, one says that a C ∗ -dynamical system is given, A, Γ, τ : Γ → Aut(A) .

g ∈ Γ. (1.4)

(1.5)

In what follows, instead of τ (g)(a) we simply write g(a) if this does not lead to a misunderstanding. Deﬁnition 1.6. The C ∗ -crossed product A Γ ≡ A τ Γ is deﬁned to be the completion of the ∗-algebra of ﬁnite A-valued functions on Γ with the multiplication [h−1 (f1 (k))]f2 (h) (1.6) (f1 f2 )(g) = k,h∈Γ kh=g

and the involution f ∗ (g) = g −1 [f (g −1 )]

(1.7)

(the bar stands for complex conjugation) with respect to the norm f = sup U (g)π(f (g)) , (π,U ) g∈Γ

BH

(1.8)

8

Chapter 1. Nonlocal Functions and Bundles

where the supremum is taken over all possible covariant representations of the C ∗ -dynamical system (1.5), i.e., over pairs (π, U ) consisting of a representation π of the C ∗ -algebra A on a Hilbert space H and a unitary representation U of the group Γ in H with the property π(g(a)) = U (g)π(a)U (g −1 ).

(1.9)

Remark 1.7. The crossed product A Γ has the following universality property: any covariant representation (π, U ) of the C ∗ -dynamical system (1.5) on a Hilbert space H can be continued to a unitary representation A Γ → BH. Of course, this continuation is unique. The reduced C ∗ -crossed product A red Γ is deﬁned to be the image of the algebra A Γ under its regular representation in the Banach space l2 (Γ, A). of ﬁnitely supported A-valued The space l2 (Γ, A) is the completion of the space functions on Γ with respect to the norm f = g∈Γ f ∗ (g)f (g). The regular representation corresponds to the covariant representation [π(a)v](g) = g −1 (a)v(g),

[U (g)v](h) = v(g −1 h),

a ∈ A, g ∈ Γ, v ∈ l2 (Γ, A)

of the C ∗ -dynamical system (A, Γ, τ ) on the Banach space l2 (Γ, A). For more details about spaces of l2 (Γ, A)-type, which are called Hilbert modules, see Chap. 3. The group C ∗ -algebra and the reduced group C ∗ -algebra of the group Γ are obtained from these deﬁnitions for A = C. The following statement holds; it contains Proposition 1.4 as a special case. Proposition 1.8 (see [27], p. 172, and [69], p. 1766). 1. If Γ is a discrete amenable group and A is a C ∗ -algebra on which Γ acts by automorphisms, then A Γ = A red Γ. 2. If, in addition, the C ∗ -algebra A is nuclear, then the crossed product A Γ is also nuclear. We point out two other obvious properties of C ∗ -crossed products. 1. If the action of the group Γ on the C ∗ -algebra A is trivial, then A Γ = A ⊗min C ∗ (Γ). (Here ⊗min is the minimal tensor product of C ∗ -algebras; see Appendix A.) 2. If J ⊂ A is an ideal invariant under the action of the group Γ, then the following exact sequence of C ∗ -algebras holds: 0 −→ J Γ −−−−→ A Γ −−−−→ (A/J) Γ −→ 0, where the action of the group on the quotient algebra is deﬁned in a natural way.

1.1. Group Algebras and Crossed Products

9

Example 1.9 (Noncommutative torus). Let θ be a real number. We consider the action of the group Z on the unit circle S1 by rotations by angles that are integer multiples of 2π/θ. The corresponding action in the space of functions has the form (Tk f )(ϕ) = f (ϕ − 2πk/θ),

k ∈ Z.

This action determines a C ∗ -dynamical system, and the corresponding crossed product C(S1 ) Z is called the noncommutative torus.2 Let us say some words about the name of this C ∗ -algebra. In the crossed product C(S1 ) Z realized as a subalgebra in BL2 (S1 ), consider the elements (U f )(ϕ) = eiϕ f (ϕ), (V f )(ϕ) = f (ϕ − 2π/θ). These two elements are unitary and satisfy the relation U V = e2πi/θ V U. Clearly, the crossed product C(S1 ) Z is obtained as the closure of the algebra of polynomials in U and V . On the other hand, the algebra of functions on the torus T2 with coordinates (ϕ, ψ) contains the dense subalgebra consisting of polynomials in the elements U = eiϕ and V = eiψ ; thus, The crossed product C(S1 ) Z can be interpreted as a noncommutative deformation of the algebra of functions on the torus T2 . We can also consider the representation of the noncommutative torus in the space L2 (R) on the straight line, where the action of the operators U and V is determined by the formulas (U f )(x) = e−2πix/θ f (x),

(V f )(x) = f (x + 1).

In the noncommutative torus, there is a remarkable projection p = pθ called the Rieﬀel projection, which is deﬁned as follows. Let θ = 1/n, n ∈ N. To deﬁne the Rieﬀel projection, we choose a small number ε > 0 such that the intervals [0, ε] and [1, 1 + ε] are disjoint as sets on the circle R/θZ. Then the projection p = p∗ is deﬁned by the formula (1.10) p = V −1 g + f + gV, where f is a smooth function such that f ≡ 1 on the interval [ε, 1], f ≡ 0 for t > 1 + ε, and f (t) + f (1 + t) = 1 for t ∈ (0, ε) and g = f − f 2 for t ∈ [0, ε] and g = 0 outside this interval (see Fig. 1.1). It is easy to verify that the element p thus deﬁned is indeed a projection. 2 Sometimes,

this algebra is also called the algebra of rotations.

10

Chapter 1. Nonlocal Functions and Bundles

1

f (x) g(x)

0

ε

1

1+ε

θ

x

Figure 1.1: Functions f and g deﬁning Rieﬀel’s projection pθ = V −1 g + f + gV x

−∞

−2θ

−θ

0

θ

2θ

+∞

Figure 1.2: Compactiﬁcation C of the cylinder C = R × S1 Example 1.10 (Noncommutative cylinder). Let C = R × S1 be an inﬁnite cylinder whose base is the circle S1 and the coordinates are denoted by x ∈ R and ϕ mod 2π ∈ S1 . By A0 we denote the algebra of bounded continuous functions on C stabilizing as x → ±∞ to (necessarily continuous) functions of ϕ. This algebra is isomorphic to the algebra A0 = C(C) of continuous functions on the compactiﬁcation C [0, 1] × S1 of the cylinder obtained by adding two circles corresponding to x = ±∞ (see Fig. 1.2). The algebra A0 acts (by pointwise multiplication) in the space L2 (C; dx dϕ). The set of elements with ﬁnite Fourier series ak (x)eikϕ , (1.11) a= k∈Z

where the coeﬃcients ak (x) are continuous (or even smooth) functions on R stabilizing at ±∞, is dense in the algebra A0 . Thus, the algebra A0 is generated

1.1. Group Algebras and Crossed Products

11

by smooth stabilizing functions of the self-adjoint operator x and by the unitary element eiϕ . We deﬁne the noncommutative cylinder as the one-parameter deformation Aθ , θ ∈ R, of the algebra A0 under which the pair of commuting operators x and eiϕ in (1.11) is replaced by the operators u = x,

v = vθ = Tθ eiϕ ,

(1.12)

where Tθ is the operator of shift by θ in the variable x: (Tθ f )(x, ϕ) = f (x − θ, ϕ). These operators satisfy the commutation relation uv = v(u + θ).

(1.13)

(Thus, for θ = 0 we obtain exactly the algebra A0 .) The noncommutative cylinder is deﬁned to be the subalgebra Aθ ⊂ BL2(C) obtained as the closure of the set of ﬁnite sums ak (x)vθk . (1.14) a= k∈Z

The composition of elements in this algebra is determined by the formula

ak (x)vθk bl (x)vθl = ak (x)bl (x − kθ) vθs . (1.15) k∈Z

l∈Z

s∈Z k+l=s

By construction, the noncommutative cylinder is generated by functions in C(C) and by the operators vθ containing the shift Tθ ; therefore, we have the embedding Aθ ⊂ C(C) Z, where, on the right-hand side, the action of the group Z on the compactiﬁcation of the cylinder C is induced by the shift (x, ϕ) → (x + θ, ϕ).

1.1.3 Isomorphism Theorem The key role in the proof of the ﬁniteness theorem for nonlocal operators is played by the isomorphism theorem [1]. Let a C ∗ -algebra A be realized as a subalgebra in the algebra BH of operators in a Hilbert space H, and let a unitary representation T of a group Γ in H be given such that T (g)aT (g −1) ∈ A, a ∈ A, g ∈ Γ. We denote the least C ∗ -subalgebra in BH containing A and all the operators T (g), g ∈ Γ, by C ∗ (A, T ).

12

Chapter 1. Nonlocal Functions and Bundles

The unitary representation T determines an action τ of the group Γ on A by the formula g ∈ Γ, a ∈ A, τ (g)(a) = T (g)aT (g −1), and hence it determines the dynamical system (A, Γ, τ ). Obviously, the algebra C ∗ (A, T ) is the image of the C ∗ -crossed product A Γ under the homomorphism π : A Γ −→ BH, generated by the covariant representation ( π , T ) of the above dynamical system, where π

is the embedding of the algebra A in BH. In general, one and the same dynamical system (A, Γ, τ ) can be associated with various algebras C ∗ (A, T ). The isomorphism theorem implies suﬃcient conditions under which any two such algebras C ∗ (A, T ) and C ∗ (A, T ) are naturally isomorphic. (The isomorphism is identical on the algebra A and takes T (g) to T (g) for all g ∈ Γ.) If the fact that A Γ, just as any C ∗ -algebra, can be realized as a subalgebra in BH is taken into account, then the statement about the natural isomorphism (in terms of which this theorem is stated in [1]) is equivalent to the statement that π is a monomorphism. We formulate this theorem precisely in this form, which is more convenient for us. To state the condition on the action τ , we note that this action generates the action τ : Γ −→ Homeo(Prim A) of the group Γ by homeomorphisms on the set Prim A of primitive ideals of the algebra A equipped with the Jacobson topology. (The deﬁnition of the Jacobson topology is given in Appendix A.) Indeed, let J ∈ Prim A be a primitive ideal in A. Then τ (g)(J) = τ (g)J ≡ {τ (g)a : a ∈ J}. The ideal τ (g)(J) is primitive. Indeed, if J = ker α, where α is an irreducible representation of the algebra A, then τ (g)(J) = ker(α ◦ τ (g)−1 ), and obviously, the representation α ◦ τ (g)−1 is also irreducible, so that τ (g)(J) ∈ Prim A. One can show that the mapping τ (g) is a homeomorphism in the Jacobson topology. Let (Prim A)g be the set of ﬁxed points of the mapping τ (g) : Prim A −→ Prim A.

1.1. Group Algebras and Crossed Products

13

Deﬁnition 1.11. We say that the action τ of the group Γ on A is topologically free if, for every ﬁnite set of nonunit elements g1 , . . . , gk ∈ Γ \ {e} of the group Γ, the union k (Prim A)gj j=1

of the sets of ﬁxed points of the homeomorphisms τ gk has empty interior (in the Jacobson topology). Theorem 1.12 (Isomorphism theorem [1]). If Γ is a discrete amenable group and its action on the C ∗ -algebra A is topologically free, then any C ∗ -homomorphism of the crossed product A Γ that is exact on the subalgebra A is exact.

1.1.4 Smooth Crossed Products Motivation In the theory of nonlocal pseudodiﬀerential operators, it is desirable to work not only with continuous but also with smooth symbols. This is especially important in studying homotopy invariants of nonlocal elliptic operators and in constructing index formulas. In this case, it is essential that the inverse of a smooth elliptic symbol is again a smooth symbol. For this to be the case, it is required that the algebra of smooth symbols be a local subalgebra in the C ∗ -algebra of continuous symbols. (The deﬁnition of a local subalgebra is discussed in Appendix A.) The set of smooth nonlocal symbols will be deﬁned as the crossed product of the algebra of smooth functions by the group Γ. Thus, the speciﬁc problem whose solution we need consists in the following. 1. We know that the algebra C ∞ (X) of smooth functions on a compact manifold X is a local subalgebra in the C ∗ -algebra of continuous functions. 2. How to deﬁne the algebra C ∞ (X) Γ, i.e., the smooth crossed product? 3. How to show that this algebra is local in C(X) Γ? It turns out that it is relevant to give answers to these questions in a more general context not necessarily starting from commutative C ∗ -algebras.3 The corresponding theory was developed by Schweitzer [67, 68]. We readily note that a tempting simple scheme like “if A∞ ⊂ A is an invariant local subalgebra, then, under an appropriate deﬁnition of the crossed product, the subalgebra A∞ Γ ⊂ A Γ is also local” does not work and one has to impose additional conditions on both the algebra A∞ and the group Γ; however, these conditions are satisﬁed in the applications that are of interest to us. Now let us study the theory developed in [67, 68] and, in particular, present conditions ensuring that given smooth crossed products of subalgebras are local. 3 However,

C ∗ -algebras.

we need noncommutative algebras only when we deal with elliptic operators over

14

Chapter 1. Nonlocal Functions and Bundles

Note that Γ was assumed to be a Lie group in the cited works. We are interested only in the very special (from this standpoint) case of a discrete group Γ; in this case, the theory is substantially simpler. The presentation in this section is given precisely for this case. Let A be a C ∗ -algebra, and let A∞ be a dense ∗-subalgebra. For simplicity, we assume that both A and A∞ are unital (with the same unity), although the results (with the corresponding standard modiﬁcations) also hold in the general case. Further, we assume that A∞ is a Fr´echet algebra (i.e., its underlying linear space is a Fr´echet space, and the multiplication is jointly continuous with respect to the corresponding topology) with a system of seminorms { · m }, m ∈ N, and the seminorm · 0 coincides with the restriction of the norm in A to A∞ . (In particular, the embedding A∞ ⊂ A is continuous.) Suﬃcient Condition for Locality Deﬁnition 1.13. The algebra A∞ is said to be strongly spectrally invariant in A if there exists a constant C > 0 such that, for any m, n, and elements a1 , . . . , an of the algebra A∞ , the inequalities a1 · · · an m ≤ Dm C n a1 k1 · · · an kn (1.16) k1 +···+kn ≤pm

hold with constants Dm and pm depending only on m (but independent of n and of the elements a1 , . . . , an ). Remark 1.14. Inequality (1.16) generalizes the properties of seminorms in the algebra of inﬁnitely diﬀerentiable functions if the maximum of the absolute value of the mth derivative is denoted by · m . Namely, as the number n of factors increases, the number of factors with nonzero indices kj on the right-hand side in inequality (1.16) does not increase. For the algebra of inﬁnitely diﬀerentiable functions, this readily follows from the Leibniz rule. The importance of condition (1.16) is illustrated by the following statement, which is obtained if we combine [68, Theorem 1.17] and [67, Lemma 1.2]. Proposition 1.15. If the algebra A∞ is strongly spectrally invariant in A, then it is a local subalgebra in A. Thus, the strong spectral invariancy condition is stronger than the locality condition; in fact, this stronger condition will be needed to establish the locality of the smooth crossed product. But we note that the latter is yet to be deﬁned. To this end, we should impose a certain condition on the group Γ.

1.1. Group Algebras and Crossed Products

15

Groups of Polynomial Growth In [55], Milnor introduced the following class of groups. Deﬁnition 1.16. Let Γ be a discrete group. This group is said to be of polynomial growth if 1. The group Γ is ﬁnitely generated. 2. For a given (and thus for an arbitrary) set {g1±1 , . . . , gs±1 } of generators, the growth function of the group Γ, i.e., the number of elements whose length does not exceed r (where the length |g| of an element g is understood as the minimum length of words representing this element in the above generators) increases as r → ∞ not faster than some power of the number r. Gromov [38] showed that a discrete group is a group of polynomial growth if and only if it contains a nilpotent subgroup of ﬁnite index. Moreover, a discrete group of polynomial growth is always amenable (which allows us to use the isomorphism theorem in the subsequent sections). Now we present several examples of groups of polynomial growth. Example 1.17. It is clear that any ﬁnitely generated Abelian group is of polynomial growth. Example 1.18. Consider the group Γ generated by rotations of the sphere by an irrational angle about the axis OZ (see Fig. 1.3) and by reﬂections in the plane XOZ. This group is an extension 0 −→ Z −→ Γ −→ Z2 −→ 0 of the (Abelian) group of rotations by reﬂections and hence is of polynomial growth. (By the way, this can readily be veriﬁed by a straightforward argument.) Example 1.19. Consider the group Γ of upper triangular matrices ⎛ ⎞ 1 a b ⎝0 1 c ⎠ , 0 0 1 where a, b, c are integers. This group is called the discrete Heisenberg group. Let us show that it is nilpotent. One can readily show that this group is generated by the matrices ⎛ ⎛ ⎛ ⎞ ⎞ ⎞ 1 1 0 1 0 0 1 0 1 e1 = ⎝0 1 0⎠ ; e2 = ⎝0 1 1⎠ ; e3 = ⎝0 1 0 ⎠ . 0 0 1 0 0 1 0 0 1 Since [e1 , e2 ] = e3 , [e1 , e3 ] = 1, and [e2 , e3 ] = 1, the group is indeed nilpotent. Hence it follows from the Gromov theorem that it is a group of polynomial growth.

16

Chapter 1. Nonlocal Functions and Bundles

Z

Z

Y

Y

X

X

0

θ

Figure 1.3: Rotation by θ and reﬂection in the plane XOZ Tempered Actions Now we assume that a discrete group Γ of polynomial growth acts on the Fr´echet algebra A∞ by automorphisms τ (g), g ∈ Γ. Deﬁnition 1.20. The action τ is said to be tempered if, for any positive integer m, there exists a positive integer number k and a polynomial P (x) such that τ (g)(a)m ≤ P (|g|) ak ,

g ∈ Γ,

a ∈ A.

Now we can deﬁne the smooth crossed product. Deﬁnition 1.21. Let a tempered action of a discrete group Γ of polynomial growth on a Fr´echet algebra A∞ be given. The smooth crossed product A∞ Γ is deﬁned to be the space of functions f : Γ → A∞ for which the seminorms f d,m = sup|g|d f (g)m ,

d, m ∈ N,

(1.17)

g∈Γ

are ﬁnite. This space is equipped with the multiplication (1.6) and the involution (1.7). Proposition 1.22. The multiplication in A∞ Γ is well deﬁned (i.e., the corresponding series converge and determine an element of A∞ Γ). In this case, A∞ Γ is a Fr´echet ∗-algebra with respect to the seminorms (1.17). Proof is based on a careful estimate of norms for the series.

1.1. Group Algebras and Crossed Products

17

Schweitzer theorem Now we can ﬁnally state the main theorem of this section. Theorem 1.23. Let a C ∗ -dynamical system (A, Γ, τ ), where Γ is a discrete group of polynomial growth, be given, and let A∞ ⊂ A be a dense strongly spectral invariant Fr´echet ∗-subalgebra in A. Further, we assume that the subalgebra A∞ is invariant under the action τ and the restriction of this action to A∞ is tempered. Then the smooth crossed product A∞ Γ is a local subalgebra of the C ∗ -crossed product A Γ. Proof. This statement is a special case of Corollary 7.16 in [68] with Lemma 1.2 in [67] taken into account. Remark 1.24. We note that the length of an element in the group Γ satisﬁes the obvious inequality |gh| ≤ |g| + |h| and, in particular, is a subpolynomial gauge on Γ in the sense of [68]. (In the deﬁnition of a subpolynomial gauge, an arbitrary polynomial of |g| and |h| is allowed on the right-hand side in the last inequality.) This condition is used in the proofs of the statements given in [68]. Summarizing the above, we can say that, to verify that the smooth crossed product A∞ Γ is well deﬁned and is a local subalgebra, one should verify whether the action of the discrete group Γ of polynomial growth on the algebra A∞ is tempered and the Fr´echet ∗-algebra A∞ itself is dense in A and strongly spectrally invariant. Example 1.25. Let A = A∞ = C. Then, by deﬁnition, the smooth crossed product A∞ Γ is the set of rapidly decaying functions f : Γ −→ C; i.e., for any k ∈ N there exists a constant C(k) such that |f (g)| ≤ C(k)(1 + |g|)−k

(1.18)

for all g ∈ Γ. In this case, the multiplication of functions is determined by the convolution (f1 ∗ f2 )(k) = f1 (g)f2 (g −1 k). g∈Γ

If Γ = Z, then the (discrete) Fourier transform can be applied. Then the convolution becomes the usual (pointwise) product of functions, and the functions satisfying the condition (1.18) become inﬁnitely diﬀerentiable functions on the circle. This explains the term smooth crossed product.

18

Chapter 1. Nonlocal Functions and Bundles

1.2 Nonlocal Functions If a group Γ acts on a manifold (or, more generally, a locally compact Hausdorﬀ space) X, then this action generates an action of the group Γ on algebras of functions on X. In the deﬁnition of the class of nonlocal operators studied in this book, the corresponding crossed products of these algebras by the group Γ play the same role as the usual function algebras do in the deﬁnition of classical pseudodiﬀerential operators. With some abuse of terminology, the elements of these crossed products will be called nonlocal functions.

1.2.1 Assumptions about the Group Γ Now we state several conditions on the group Γ and its action, which will be assumed to be satisﬁed in the entire book. We assume that Γ is a countable discrete group of polynomial growth. From now on, we assume that Γ is a dense subgroup of some compact Lie group Γ of homeomorphisms of the space X. In the case where X is a manifold, we assume that Γ is a compact Lie group of diﬀeomorphisms of the manifold X.

1.2.2 Continuous Nonlocal Functions Let the group Γ act on a compact Hausdorﬀ topological space X. This action induces the action τ : Γ → Aut(C(X)) of the group Γ on the algebra C(X) by automorphisms (1.19) (τ (g)a)(x) = a(g −1 (x)). Thus, the C ∗ -crossed product C(X) Γ is deﬁned, which we brieﬂy denote by C(X)Γ in what follows. The space of primitive ideals of the commutative algebra C(X) is the space X of its maximal ideals, so that the isomorphism theorem 1.12 in this case acquires the following form. Theorem 1.26. If the action of the group Γ on X is topologically free (i.e., for any ﬁnite set (g1 , . . . , gk ) ⊂ Γ of elements gj = e the interior of the union of the sets ﬁx gj of their ﬁxed points is empty), then each C ∗ -homomorphism of the crossed product C(X)Γ faithful on the subalgebra C(X) is faithful. By way of example, consider an invertibility criterion for an element a ∈ C(X)Γ , which follows from this theorem. For each x ∈ X, there is natural representation πx of the algebra C(X)Γ on the Hilbert space l2 (Γ) of square integrable functions on the group Γ. On the dense subset in C(X)Γ consisting of ﬁnite functions ϕ : Γ → C(X), it is given by the formula πx (ϕ) = Φ, where Φ is the matrix with entries Φgh = ϕg−1 h (h−1 x),

g, h ∈ Γ.

This representation is called the trajectory representation, because it uses only the values of the functions ϕg on the Γ-orbit of the point x.

1.2. Nonlocal Functions

19

Proposition 1.27 ( [1]). Suppose that the action of the group Γ on X is topologically free. Then an element a ∈ C(X)Γ is invertible if and only if the operator πx (a) is invertible for all x ∈ X. Proof. It suﬃces to apply Theorem 1.26 to the representation x∈X πx and, in addition, note that the element πx (a) continuously (with respect to the norm) depends on the point x. Nonlocal Functions as Operators in L2 In what follows, we are primarily interested in the following speciﬁc realization of the algebra C(X)Γ . We assume that the space X is equipped with a ﬁnite Γinvariant measure μ and the measure of any nonempty open subset in X is positive. Consider the representation of the algebra C(X) by the operators of multiplication by functions a ∈ C(X) in the space L2 (X, μ) and the representation g → T (g) of the group Γ by the unitary operators T (g)u(x) = u(g −1 (x))

(1.20)

in the same space. This pair determines a covariant representation of the C ∗ dynamical system (C(X), Γ, τ ) and hence a representation of the algebra C(X)Γ by operators on the Hilbert space L2 (X, μ). Theorem 1.26 implies the following statement. Proposition 1.28. If the action of the group Γ on X is topologically free, then the above representation is faithful. Changes of Variables and Substitutions Let Y be also a compact Hausdorﬀ Γ-space, and let f : X → Y be a continuous Γ-mapping (a mapping intertwining the actions of the group Γ). Then the induced homomorphism (1.21) f ∗ : C(Y )Γ −→ C(X)Γ is well deﬁned; on the dense subset in C(Y )Γ , consisting of ﬁnitely supported functions ϕ : Γ → C(Y ), it is given by the formula (f ∗ ϕ)g (x) = ϕg (f (x)).

(1.22)

Notation 1.29. Sometimes, we less formally write ϕ(y) instead of ϕ and ϕ(f (x)) instead of f ∗ ϕ. i

In particular, if K ⊂ X is a Γ-invariant closed subset, then we obtain the restriction homomorphism i∗ : C(X)Γ −→ C(K)Γ . (1.23) In what follows, i∗ ϕ is also denoted by ϕK . We say that ϕ is invertible on K if ϕK is invertible.

20

Chapter 1. Nonlocal Functions and Bundles

Proposition 1.30. Suppose that the space X is represented as the union X = α Kα of closed Γ-invariant subsets Kα ⊂ X and the group Γ acts topologically free both on X and on each Kα . Then the following statement holds: an element a ∈ C(X)Γ is invertible if and only if all its restrictions a|Kα are invertible. Proof. By Proposition 1.27, the invertibility of an element a is equivalent to the invertibility of all operators πx (a), x ∈ X, and the invertibility of an element

x (a|Kα ), x ∈ Kα , where a|Kα is equivalent to the invertibility of the operators π π

x is the trajectory representation of the algebra C(K)Γ . It remains to note that πx (a) = π

x (a|Kα ).

1.2.3 Smooth Nonlocal Functions After the above deﬁnitions, we consider the most interesting case in which the space X on which nonlocal functions are deﬁned is a smooth compact manifold. Recall that in this case we additionally assume that Γ is a dense subgroup of some compact Lie group Γ of diﬀeomorphisms of the manifold X. Then, without loss of generality, we can assume (this will be used later) that X is a Riemannian manifold with a metric invariant under the diﬀeomorphisms in the group Γ (such a metric can be constructed by averaging an arbitrary Riemannian metric over the action of the group Γ) and with the invariant volume form dμ associated with the metric. For simplicity, from now on we assume (although this is not always essential) that X is connected. Proposition 1.31. If X is connected, then the action of the group Γ on X is topologically free. Proof. 1. The set of ﬁxed points of an isometry is a ﬁnite union of smooth submanifolds. Indeed, let us take a ﬁxed point x ∈ Xg . The diﬀerential dg : Tx X −→ Tx X at this point is an orthogonal transformation. If Tx X = L ⊕ L⊥ is the decomposition of the tangent space at a point x into the eigenspace of the diﬀerential corresponding to the eigenvalue 1 and its orthogonal complement, then the set Xg coincides near the point x with the image of the subspace L under the exponential mapping (see Fig. 1.4) expx : Tx X −→ X. For more detail, e.g., see [9].

1.2. Nonlocal Functions

21

L⊥ x

L

Tx X

Xg X

Figure 1.4: Fixed point submanifold Xg = expx (L) 2. Since the manifold X is connected, the set Xg either coincides with the entire manifold (which is possible only for the identity mapping) or has empty interior. Therefore, it follows from Proposition 1.28 that the representation of the algebra C(X)Γ by operators in L2 (X, μ) described in Sec. 1.2.2 is faithful. On a smooth manifold, along with continuous functions, one can consider smooth nonlocal functions. The latter are constructed as follows. The group Γ acts on the Fr´echet algebra C ∞ (X) by the automorphisms τ (g), g ∈ Γ, by formula (1.19). This action is bounded uniformly in g ∈ Γ in the topology of the Fr´echet space (because it is the restriction of the action of the compact Lie group Γ) and hence is tempered, so that, taking into account the fact that Γ is a group of polynomial growth, we obtain a well-posed deﬁnition of the smooth crossed product C ∞ (X) Γ, which, in what follows, is denoted by C ∞ (X)Γ (see Sec. 1.1.4). Recall that, by Deﬁnition 1.21, C ∞ (X)Γ is the ∗-algebra of C ∞ (X)-valued rapidly decaying functions f = f (g), g ∈ Γ, on the group Γ, where the multiplication is given by formula (1.6) and the involution is given by formula (1.7). Here the rapid decay condition means that, for any of the seminorms · α determining the topology on the Fr´echet space C ∞ (X), the estimates f (g)α ≤ C(1 + |g|)−N

(1.24)

hold for all N ∈ Z+ , where, of course, the constant C depends on f , α, and N . One can readily see that C ∞ (X)Γ is a ∗-subalgebra of the algebra C(X)Γ . Namely, under the representation described in Sec. 1.2.2, any element f ∈ C ∞ (X)Γ

22

Chapter 1. Nonlocal Functions and Bundles

is represented by the series

T (g)f (g),

g∈Γ

which converges in the operator norm in BL2(X, dμ). Proposition 1.32. The algebra C ∞ (X)Γ is a dense local subalgebra of the algebra C(X)Γ . Proof. This statement follows from Theorem 1.23.

1.3 Nonlocal Bundles Nonlocal pseudodiﬀerential operators deﬁned in the subsequent chapter will act in spaces of sections of bundles on the manifold X. First, we should analyze the problem of estimating what bundles are most appropriate to be considered here. Consider the classical situation. Classical (pseudo)diﬀerential operators are operators whose coeﬃcients belong to the algebra of functions on the manifold X. Such operators are considered in spaces of sections of vector bundles on X; by the Serre–Swan theorem, these spaces are precisely the spaces that can be represented as the images of projections over the above-mentioned algebra of functions. The operators studied in the present book are, in fact, (pseudo)diﬀerential operators whose coeﬃcients belong to an algebra wider than the algebra of functions on M , namely, the algebra C ∞ (X)Γ . Therefore, from the standpoint of noncommutative geometry, it is more natural to consider nonlocal operators not in the spaces of sections of usual vector bundles on X but in the wider class of spaces of sections of “nonlocal bundles,” i.e., of spaces determined by (orthogonal ) projections over the algebra C ∞ (X)Γ . In what follows, such bundles are simply called bundles, and bundles in the usual sense will be called classical bundles if a misunderstanding is possible otherwise. Of course, in the case of nonclassical bundles, not the bundles themselves are deﬁned but only their spaces of sections (i.e., the ranges of the corresponding projections realized by the standard representation described in Sec. 1.2.2 as projections in the spaces of vector functions on X). We note that, because of the locality of the subalgebra C ∞ (X)Γ , the supply of bundles does not increase if we consider projections over the algebra C(X)Γ , which we shall often do.

1.4 Case of Noncompact Spaces We also have to deal with nonlocal functions and bundles on noncompact spaces (in particular, on spaces of the form Rn or T ∗ M ). In the present section, we generalize

1.4. Case of Noncompact Spaces

23

the above notions to the case of noncompact spaces X. This generalization is quite natural in form. (In fact, we deﬁne a nonlocal function on a noncompact space X in terms of its restrictions to an arbitrary Γ-invariant compact subset in X.) We believe that this intuitive remark is quite suﬃcient for understanding the subsequent material, and hence this section can be omitted on the ﬁrst reading if the reader is not interested in technical details.

1.4.1 Nonlocal Functions Let X be a locally compact Hausdorﬀ Γ-space. We assume that in X there exists an exhaustive system {Kα }α∈I of Γ-invariant compact subsets, where I is an at most countable directed set and iαβ : Kα ⊂ Kβ

for α ≺ β.

(1.25)

The embeddings (1.25) induce the restriction homomorphisms i∗αβ : C(Kβ )Γ −→ C(Kα )Γ .

(1.26)

We denote the projective limit of the system of algebras {C(Kα )Γ }α∈I with the homomorphisms (1.26) by C(X)Γ . This limit is a Fr´echet algebra with respect to the system of seminorms uα = u|Kα C(Kα )Γ , and neither the algebra C(X)Γ itself nor the topology in this algebra depend on the choice of the system of subsets {Kα }. The induced mappings (1.21) are also well deﬁned in the noncompact case. In the algebra C(X)Γ , we distinguish the subalgebra of elements that are zero outside some compact subset Kα (diﬀerent for diﬀerent elements). Its closure with respect to the norm u = sup uα α

∗

is a C -subalgebra in C(X)Γ , which will be denoted by C0 (X)Γ . Just as in the compact case, we can deﬁne a representation of the algebra C(X)Γ in the space L2loc (X, μ), where μ is a Γ-invariant (not necessarily ﬁnite) measure on X. This representation is faithful if the measure of any nonempty open set is nonzero. If there is a smooth structure on X, then, following the above argument, we can deﬁne the set C ∞ (X)Γ of smooth nonlocal functions. In the case of noncompact X, the space C ∞ (X)Γ is a ∗-subalgebra of the algebra C(X)Γ , and any element f ∈ C ∞ (X)Γ can be represented by the series T (g)f (g) g∈Γ

24

Chapter 1. Nonlocal Functions and Bundles

converging in each of the norms in BL2 (Kα , dμ), where {Kα } is an exhaustive system of Γ-invariant compact subsets of X. Proposition 1.33. The algebra C ∞ (X)Γ is a dense local subalgebra of the algebra C(X)Γ . Proof. For the case in which the manifold X is compact, this statement has already been proved (Proposition 1.32). The general case can be reduced to this case if we note that it is possible to pass to the projective limit, because 1. A homomorphism of C ∗ -algebras does not increase the spectrum of an element. 2. The image of an analytic function of an element of the algebra under such a homomorphism coincides with the function of the image of the element itself.

1.4.2 Special Noncompact Manifolds In what follows, we consider an important special case of noncompact spaces, namely, the total spaces of vector bundles on compact manifolds (and subsets of these). Let X be a compact Riemannian manifold on which the group Γ acts by isometries, and let π : E → X be a ﬁnite-dimensional real vector Γ-bundle with ﬁber F . A point in the ﬁber will be denoted by ξ. In the ﬁbers, we introduce an inner product so that the ﬁberwise mappings determined by the elements of the group Γ are isometries. (Using a Γ-invariant connection on E, we can also equip the total space E with a Γ-invariant metric that coincides with the above inner product on vertical subspaces and with the pullback of the metric from X on the horizontal subspaces; we shall use this fact below.) Now we can deﬁne the algebra C(E)Γ ; in this case, the role of Kα is played by the Γ-invariant sets BR = {|ξ| ≤ R} ⊂ E. We denote the corresponding seminorms by uR = u|BR . Here we can consider the cotangent bundle T ∗ X of the Riemannian Γ-manifold X as an important example. We identify the cotangent bundle of the manifold X with the tangent bundle by using the Riemannian metric in the standard way. Then the group Γ acts on T ∗ X T X by the diﬀerential ∗ X, dgx : Tx∗ X −→ Tg(x)

x ∈ X,

which preserves the inner product determined in the ﬁbers by the metric.

1.4. Case of Noncompact Spaces

25

Now let the manifold X = E itself be the total space of a ﬁnite-dimensional real vector Γ-bundle π : E → M over a smooth manifold M with ﬁber F . There is a natural exact sequence 0 −→ π ∗ (T ∗ M ) −→ T ∗ E −→ T ∗ F −→ 0

(1.27)

of Γ-bundles (where T ∗ F is the bundle over E of the cotangent spaces to the ﬁbers of F ). We use the Γ-invariant inner product in the ﬁbers of the bundle T ∗ E to construct the orthogonal decomposition T ∗ E = π ∗ (T ∗ M ) ⊕ T ∗ F

(1.28)

from the sequence (1.27); without loss of generality we can assume that the metric in the ﬁbers of T ∗ E is the direct sum of the metrics naturally deﬁned on the summands. Let x be coordinates on M , let ξ be linear orthogonal coordinates in the ﬁber F (i.e., the coordinates with respect to some local orthogonal reference frame in the bundle E), and let (px , pξ ) be the canonical dual coordinates in the ﬁbers of the bundle T ∗ E. Further, let p and η be the coordinates dual to x and ξ in the ﬁbers of the bundles T ∗ M and T ∗ F , respectively. Straightforward calculations prove the following statement. Lemma 1.34. The passage from the coordinates (px , pξ ) to the coordinates (p, η) determined by the decomposition (1.28) has the form px = p,

pξ = η + A(x, ξ)p,

(1.29)

where the matrix function A(x, ξ) (linear in ξ) depends on the choice of the coordinate system. Obviously, using the decomposition (1.28), we can treat T ∗ E as the bundle TM ⊕ E ⊕ E

(1.30)

over M with ﬁber Tx M ⊕ F ⊕ F and with coordinates (p, η, ξ) in the ﬁbers. This decomposition (just as the decomposition (1.28)) is convenient, because the group Γ acts in the ﬁbers by linear transformations separately in each of the three variables p, η, and ξ. It follows that the orthogonal projections onto the direct summands in the decompositions (1.28) and (1.30) are Γ-mappings.

Chapter 2

Nonlocal Elliptic Operators 2.1 Scalar Operators Just as in the case of usual local elliptic (pseudo)diﬀerential operators, the theory of nonlocal elliptic operators is based on the scalar case, i.e., the case of operators acting in spaces of functions. Indeed, a majority of typical facts and situations of the theory (including the issues related to the ﬁniteness theorem) fully manifest themselves already in this case. After the theory has been developed for scalar operators, the generalization of the deﬁnitions and theorems to the case of operators acting in spaces of sections of arbitrary bundles on the manifold M is a nearly (though not 100%) standard exercise. Thus, we start from scalar operators.

2.1.1 Symbol Algebras The action of the Lie group Γ on the cotangent bundle T ∗ M preserves the norm in the ﬁbers, so that its restriction to the cosphere bundle S ∗ M treated as the bundle of unit spheres in T ∗ M is well deﬁned. Therefore, one can consider the algebra C ∞ (S ∗ M )Γ , whose elements play the role of smooth symbols in what follows, and also the algebra C(S ∗ M )Γ of continuous symbols. If M is noncompact, then we consider only symbols that are independent of the covariables outside some compact set (i.e., are obtained there by lifting some elements of C(M )Γ to S ∗ M by means of the projection π : S ∗ M → M ). Such symbols are said to be multiplicatively compactly supported.

2.1.2 Quantization of Symbols and Operators of Order Zero We still consider the case of a compact manifold M . Our problem is to quantize the symbols constructed, i.e., to construct the corresponding operators on the manifold M (modulo compact operators).

28

Chapter 2. Nonlocal Elliptic Operators

Quantization of Continuous Symbols Since our symbols are associated with the space S ∗ M , we can, just as in the classical theory, construct operators of any prescribed order in the Sobolev scale on the manifold M for the same symbol. First, we explain how symbols give rise to operators of order zero acting in the space L2 (M ). From the standpoint of the theory of C ∗ -algebras, quantization is nothing else but a homomorphism π : C(S ∗ M )Γ −→ QL2 (M ) = BL2 (M )/KL2 (M ) of the symbol algebra C(S ∗ M )Γ into the Calkin algebra of the space L2 (M ). Any such a homomorphism (just as any homomorphism of the C ∗ -crossed product) is determined by its restriction π|C(S ∗ M) : C(S ∗ M ) −→ QL2 (M ) and by a unitary representation (consistent with it in the sense of formula (1.9)) U : Γ −→ QL2 (M ) of the group Γ. For π|C(S ∗ M) we take the standard pseudodiﬀerential quantization of the algebra C(S ∗ M ) (taking each continuous symbol a ∈ C(S ∗ M ) to the class of pseudodiﬀerential operator a of order zero with the same symbol in the Calkin algebra), and for U we take the representation induced in the Calkin algebra by the unitary representation T of the group Γ by shifts: [T (g)u](x) = u(g −1 x),

u ∈ L2 (M ),

g ∈ Γ.

The consistency of these mappings follows from the formula of change of variables in pseudodiﬀerential operators. Thus, the quantization π is well deﬁned. Let ΨΓ (M ) be the set of operators A ∈ BL2(M ) whose classes in the Calkin algebra lie in the image of the homomorphism π. Such operators are called nonlocal pseudodiﬀerential operators on M with symbols in the algebra C(S ∗ M )Γ . Since the standard pseudodiﬀerential quantization π|C(S ∗ M) of the algebra of continuous functions on M is injective (a noncompact operator is always assigned a nonzero symbol), we can apply the isomorphism theorem and obtain the following result. Proposition 2.1 (ψDO calculus and the ﬁniteness theorem [1]). The quantization π realizes an isomorphism π : C(S ∗ M )Γ −→ ΨΓ (M )/KL2 (M ).

(2.1)

In other words, the principal symbol σ(A) = π −1 A mod KL2 (M ) ∈ C(S ∗ M )Γ of a pseudodiﬀerential operator A ∈ ΨΓ (M ) is well deﬁned. It is zero if and only if the operator itself is compact, and it is invertible if and only if the operator itself is Fredholm.

2.1. Scalar Operators

29

Quantization of Smooth Symbols As we see, the quantization on the entire algebra C(S ∗ M )Γ of continuous symbols is speciﬁed in a rather implicit way. But if we restrict ourselves to the dense subalgebra of smooth symbols, then we can already write explicit formulas. Namely, let a ∈ C ∞ (S ∗ M )Γ . For each g ∈ Γ, to the symbol a(g) ∈ C ∞ (S ∗ M ) we assign a of order zero in L2 (M ) so that the operators a(g) pseudodiﬀerential operator a(g) 1 rapidly converge to zero as |g| → ∞ in the standard Fr´echet topology on the set of pseudodiﬀerential operators of order zero. This can be done by using a ﬁxed covering of the manifold M by coordinate neighborhoods, a ﬁxed partition of unity subordinate to this covering, and the standard Kohn–Nirenberg formulas [49] for pseudodiﬀerential quantization in local coordinates. In this case, the operators rapidly converge to zero with respect to the operator norm in L2 (M ), and a(g) we can set a= T (g)a(g). g∈Γ

Accordingly, if a ∈ C ∞ (S ∗ M )Γ is an elliptic (i.e., invertible) symbol, then the inverse symbol a−1 also lies in C ∞ (S ∗ M )Γ (because the algebra of smooth symbols is local); i.e., a−1 = T (g)b(g). g∈Γ

Then we obtain the almost inverse operator R for a by quantizing the symbol a−1 : R=

T (g)b(g).

g∈Γ

We note that here we do not study the problem concerning the conditions under which the symbol a is invertible or try to compute the inverse symbol. In contrast to the traditional case (where the symbols of scalar operators are simply functions), here the symbol is an element of a noncommutative algebra, and studying its invertibility can be a very diﬃcult problem. We refer the reader interested in these problems to the corresponding literature (see the bibliographical remarks at the end of the book). Remark 2.2. The above constructions can be transferred with obvious changes to the case of a noncompact manifold M . The main change is that only multiplicatively compactly supported symbols are considered and it is required that outside a compact set the pseudodiﬀerential operator coincides with the operator of multiplication by the nonlocal function on M whose lift to S ∗ M coincides with the symbol. 1 This

topology can be obtained from the C ∞ -topologies on the spaces of complete symbols in local coordinates and in the space of operators with smooth kernels; see [36].

30

Chapter 2. Nonlocal Elliptic Operators

2.1.3 Operators of Arbitrary Order We use the idea underlying the construction just presented and deﬁne algebras of nonlocal pseudodiﬀerential operators of arbitrary order on a compact manifold.2 A nonlocal pseudodiﬀerential operator of order m is deﬁned to be an operator of the form T (g)D(g), (2.2) D= g∈Γ

where the D(g) are the classical pseudodiﬀerential operators of order m on the manifold M rapidly converging to zero as |g| → ∞ in the standard Fr´echet topology on the set of pseudodiﬀerential operators of order m [36]. One can readily show that such a sum converges in the operator norm and determines a well-deﬁned operator D : H s (M ) −→ H s−m (M ) (2.3) for all s ∈ R. Here the key point is the fact that the operators T (g) are unitary in all Sobolev spaces H s (M ) if the latter are deﬁned by using the Beltrami–Laplace operator on M corresponding to the invariant metric; this operator commutes with all T (g). We denote the set of nonlocal pseudodiﬀerential operators of order m by Ψm Γ (M ) and deﬁne the symbol mapping ∞ ∗ σm : Ψm Γ (M ) −→ C (S M )Γ

by the formula σm (D)(g) = σm (D(g)) (where D is the operator (2.2)). We omit the index m if this does not lead to a misunderstanding. The following proposition, which completely describes the calculus of nonlocal pseudodiﬀerential operators with smooth symbols on M , is a simple consequence of the above-introduced deﬁnitions and the polynomial growth condition satisﬁed by the group Γ. Proposition 2.3. The symbol mapping σm is well deﬁned, i.e., is independent of the representation of the operator D in the form of the sum (2.2). The operator m−1 (M ) if and only if σm (D) = 0. The multiplication D ∈ Ψm Γ (M ) belongs to ΨΓ of operators induces a mapping m+l l (M ), Ψm Γ (M ) × ΨΓ (M ) −→ ΨΓ

and moreover,

σm+l (DD ) = σm (D)σl (D ).

Finally, we can state the ﬁniteness theorem. Theorem 2.4 (ﬁniteness theorem). An operator D ∈ Ψm Γ (M ) is Fredholm in the spaces (2.3) if and only if it is elliptic (i.e., its symbol σ(D) is invertible in the algebra C(S ∗ M )Γ ). 2 On

noncompact manifolds, we consider only pseudodiﬀerential operators of order zero.

2.1. Scalar Operators

31

Proof of Proposition 2.3 and Theorem 2.4. It suﬃces to reduce the proof to the case of operators of order zero in L2 (M ) by multiplying the operator D on the right and on the left by appropriate powers of the Laplace–Beltrami operator. Since the latter commutes with the operators T (g) and has unit symbol on the cosphere bundle, the symbol of the operator D does not change under such a procedure (usually referred to as order reduction). Remark 2.5. 1. Since the subalgebra C ∞ (S ∗ M )Γ is local in the algebra C(S ∗ M )Γ , the inverse of an invertible symbol σ ∈ C ∞ (S ∗ M )Γ always itself lies in C ∞ (S ∗ M )Γ . 2. Closing (for a ﬁxed s) the space Ψm Γ (M ) with respect to the operator norm in the spaces (2.3), we obtain nonlocal pseudodiﬀerential operators in the above spaces with continuous symbols σ(D) ∈ C(S ∗ M )Γ . In particular, the closure of the algebra Ψ0Γ (M ) with respect to the operator norm in the space L2 (M ) gives the algebra ΨΓ (M ). Example 2.6. On the unit circle S1 with coordinate ϕ, consider the operator d d + αTθ , D= dϕ dϕ where (Tθ f )(ϕ) = f (ϕ − θ) is the shift operator and α is a complex parameter. The symbol of this operator is equal to σ(D) = iξ + αTθ : L2 (S ∗ S1 ) −→ L2 (S ∗ S1 ). The invertibility condition for this symbol can be obtained using information about the spectrum of the shift operator Tθ . We see that our symbol is invertible for |α| = 1. (The inverse operator is given by the Neumann series.) But if |α| = 1, then the invertibility condition is stated in terms of the angle θ. If θ/2π is irrational, then the symbol can never be invertible, but if θ/2π is rational, then the operator is invertible provided that α does not lie in the corresponding set of roots of unity. In this example, verifying the ellipticity conditions amounts to studying the spectrum of a functional operator. In several cases, the ellipticity condition can be veriﬁed by using an argument of algebraic character. Let us consider the corresponding example. Example 2.7 (Connes operators on the noncommutative torus). Consider the representation of the noncommutative torus in the space L2 (R) (see Example 1.9). We denote the coordinate on the straight line by x and consider the ﬁrst-order nonlocal operator d (2pθ − 1), D = x+ dx acting in the Schwartz space, where pθ is the Rieﬀel projection (1.10). The symbol of this operator is equal to σ(D)(x, ξ) = x + iξ(2pθ − 1) : L2 (R) −→ L2 (R);

(2.4)

32

Chapter 2. Nonlocal Elliptic Operators

it is considered as a function of the pair (x, ξ) ranging in operators with shifts. The ellipticity condition, i.e., the invertibility condition for the symbol is satisﬁed for x2 + ξ 2 = 0, since |σ(D)(x, ξ)|2 = x2 + ξ 2 . (This relation follows from (2.4) and relation p2θ = pθ .) Note that in this case the deﬁnition of symbol is somewhat diﬀerent from what was introduced above. The term x is included in the symbol because we deal with operators on the noncompact space R and accordingly consider symbols that are homogeneous not in the momentum variable ξ alone but rather in the pair (x, ξ). We do not develop the theory of such symbols and operators in full generality, because we do not need it elsewhere. The interested reader can ﬁnd all relevant deﬁnitions (for the local case) in the monograph [70].

2.2 Operators in Spaces of Sections of Bundles Now we can deﬁne pseudodiﬀerential operators of general form. We start from the case in which the manifold M is compact.

2.2.1 Operators of Order Zero First, we consider the case of operators of order zero in the spaces L2 . Here it is natural to pass to the closure with respect to the norm and consider operators with continuous symbols in the spaces determined by projections over the algebra C(M )Γ .3 Now we can in a rather standard way deﬁne nonlocal operators in the spaces of sections of noncommutative bundles. Note that the algebra C(M )Γ is naturally embedded in ΨΓ (M ) and is in this case realized by operators in the space L2 (M ), so that the group Γ acts by the shift operators T (g) and the subalgebra C(M ) ⊂ C(M )Γ acts by multiplication operators. We use this embedding to associate elements of the algebra C(M )Γ with their symbols treated as nonlocal pseudodiﬀerential operators. Obviously, in this case the diagram

C(M )Γ

n6 nnn n n n

ΨΓ (M ) σ

QQQ QQQ Q( C(S ∗ M )Γ

3 Note that since C ∞ (M ) is a dense local subalgebra of the C ∗ -algebra C(M ) , their KΓ Γ theories coincide, so that the passage to the closure with respect to the norm does not result in new classes of stable homotopy equivalence of projections.

2.2. Operators in Spaces of Sections of Bundles

33

(where the diagonal arrows are natural embedding) commutes, so that the symbols of elements of the algebra C(M )Γ can be identiﬁed with these elements themselves, which shall be done later. Consider two Hermitian projections P1 , P2 ∈ MatN (C(M )Γ ) over the algebra C(M )Γ . They act in the space (L2 (M ))N and determine Hilbert subspaces there, i.e., their ranges Im P1 and Im P2 . We also assume that D ∈ MatN (ΨΓ (M )) is a matrix nonlocal pseudodiﬀerential operator on M intertwining the projections P1 and P2 , i.e., satisfying the condition (2.5) P2 D = DP1 . Then its restriction to the subspace Im P1 determines the well-deﬁned operator (2.6) D = DIm P1 : Im P1 −→ Im P2 . The operator (2.6) is uniquely determined by the triple (D, P1 , P2 ), and we sometimes write D = (D, P1 , P2 ). This operator does not change if we replace D by P2 DP1 , so that, in what follows, without loss of generality, we assume that D satisﬁes the relation D = P2 DP1 . Then the operator D is uniquely deﬁned, and its symbol satisﬁes the relation σ(D) = P2 σ(D)P1 . Deﬁnition 2.8. The operator (2.6) is called a nonlocal pseudodiﬀerential operator in the spaces determined by the projections P1 and P2 . The operator σ(D) ≡ (σ(D), P1 , P2 ) = σ(D)Im P1 : Im P1 −→ Im P2 will be called its symbol. Deﬁnition 2.9. The operator D is said to be elliptic if its symbol σ(D) is invertible. Just as in the classical theory, ellipticity serves as a criterion for the Fredholm property. Theorem 2.10 (Finiteness theorem). The operator D is Fredholm if and only if it is elliptic. How to verify whether an operator is elliptic? The following statement answers this question. Proposition 2.11. An operator D = (D, P1 , P2 ) is elliptic if and only if the symbol σ(D) realizes an equivalence of the projections P1 and P2 in MatN (C(S ∗ M )Γ ), i.e., if there exists a symbol r ∈ MatN (C(S ∗ M )Γ ) such that rσ(D) = P1 ,

σ(D)r = P2 .

Proof of Theorem 2.10 and Proposition 2.11. The operator D = (D, P1 , P2 )

34

Chapter 2. Nonlocal Elliptic Operators

is Fredholm (or elliptic) if and only if so are the operators D∗ D = (D∗ D, P1 , P1 )

DD∗ = (DD∗ , P2 , P2 ).

and

Moreover, the symbol σ(D) realizes an equivalence of the projections P1 and P2 if and only if σ(D∗ D) realizes a self-equivalence of the projection P1 and σ(DD∗ ) realizes a self-equivalence of the projection P2 . Therefore, it suﬃces to verify both of these statements for operators D such that P1 = P2 = P . In this case, the Fredholm property of the operator D is equivalent to the Fredholm property of the matrix operator D + 1 − P . The latter operator is Fredholm if and only if its symbol σ(D) + 1 − P is invertible. This readily implies the desired statements. It follows from the proof that if the projections P1 and P2 and the symbol σ(D) lie in a matrix algebra over the subalgebra of smooth symbols, then the symbol r also has the same property. Indeed, since the algebra C ∞ (S ∗ M )Γ is local, it follows that so are the matrix algebras over it [67].

2.2.2 Operators of Arbitrary Order From the formal standpoint, the construction of nonlocal pseudodiﬀerential operators of arbitrary order m in the Sobolev spaces of sections of nonlocal bundles on M is not very diﬀerent from the above case of operators of order zero in the spaces of L2 -sections. The main distinction is that the projections P1 and P2 should be taken to be smooth, i.e., they should be taken in the algebra MatN (C ∞ (M )Γ ), and the operators D should be taken in the space MatN (Ψm Γ (M )). Such projections are bounded and, modulo compact operators, self-adjoint in any Sobolev space (H s (M ))N . Now the entire argument in the preceding subsection can be carried out with obvious modiﬁcations. The space of nonlocal pseudodiﬀerential operators of order m (respectively, pseudodiﬀerential operators of order zero with continuous symbols) acting in the spaces determined by the projections P1 and P2 will be denoted by Ψm Γ (P1 , P2 ) (respectively, ΨΓ (P1 , P2 )). Nonlocal pseudodiﬀerential operators in spaces of sections of classical bundles are a special case of such operators. The following two constructions allow one to obtain examples of nonlocal elliptic operators. Example 2.12. Let D : C ∞ (M, Cm ) → C ∞ (M, Cm ) be a usual elliptic operator on M , and let P be a projection operator in the algebra Matn (C ∞ (M ) Γ). We assume that the operator D is equivariant with respect to the action of the group Γ, i.e., DT (g) = T (g)D

for any g ∈ Γ.

2.2. Operators in Spaces of Sections of Bundles

35

Then the nonlocal operator (P ⊗1)(1⊗D) : Im(P ⊗1) −→ Im(P ⊗1),

where Im(P ⊗1) ⊂ L2 (M, Cnm ), (2.7)

is elliptic. Indeed, the operators 1 ⊗ D and P ⊗ 1 commute modulo compact operators. Therefore, the symbol of the almost inverse operator is (P ⊗ 1)(1 ⊗ σ(D)−1 ). Example 2.13. Conversely, let P : C ∞ (M, Cm ) −→ C ∞ (M, Cm ) be a pseudodiﬀerential projection equivariant with respect to the action of the group Γ, and let U be an invertible element of the algebra Matn (C ∞ (M ) Γ). Then the nonlocal operator (U ⊗ 1)(1 ⊗ P ) + 1 ⊗ (1 − P ) : L2 (M, Cnm ) −→ L2 (M, Cnm )

(2.8)

is elliptic.

2.2.3 Operators on Noncompact Manifolds Now let E be a noncompact manifold (the total space of a real vector bundle over a compact manifold). Just as in the scalar case, we consider only operators of order zero in L2loc -type spaces. Let P1 and P2 be two projections over C(E)Γ . From now on, we follow the notation introduced in Sec. 1.4.2. We use the same letters to denote the pullbacks of P1 and P2 to S ∗ E ≡ S(T ∗ E) under the natural projection S ∗ E → E. Further, let a be an element of the matrix algebra over C(S ∗ E)Γ intertwining these two projections and, for large |ξ|, independent of the covariables p, η (i.e., coinciding outside the balls BR with the pullback of some element b intertwining the projections P1 and P2 over the algebra C(E \ BR )Γ ). In this case, we say that a = (a, P1 , P2 ) is a multiplicatively compactly supported symbol, or a symbol that is multiplicatively trivial at inﬁnity. For such a symbol, using an obvious construction, we obtain the nonlocal pseudodiﬀerential operator a : L2loc (E, Im P1 ) → L2loc (E, Im P2 ),

(2.9)

coinciding with the operator b for |ξ| > R and having a as its symbol, a = σ( a). Further, if the symbol a is elliptic (i.e., if a realizes the equivalence between a is Fredholm both in the spaces the projections P1 and P2 ), then the operator (2.9), and in their restrictions to the balls BR for all R > R; moreover, its index, the kernel, and the cokernel in all these pairs of spaces are the same.

36

Chapter 2. Nonlocal Elliptic Operators

2.3 Symbols as Nonlocal Functions on T ∗ M In the preceding, symbols were deﬁned as nonlocal functions on the cosphere bundle S ∗ M . In what follows, it is convenient to assume that the symbol is deﬁned on the entire space T ∗ M . To this end, we continue it by homogeneity of degree zero (this operation is well deﬁned, since the action of the group R+ in the ﬁbers of the bundle T ∗ E commutes with the action of the group Γ), and in a neighborhood of the zero section, we arbitrarily smooth it to obtain a continuous function. (In the case of a noncompact manifold M , smoothing must be performed only over the compact domain in M where the symbol is not the pullback of a bundle homomorphism and hence there is a discontinuity on the zero section.) Such symbols, deﬁned on the entire space T ∗ M and multiplicatively trivial at inﬁnity in the case of a noncompact manifold M , are called homogeneous elliptic symbols (although they are not homogeneous in a neighborhood of the zero section).

Chapter 3

Elliptic Operators over C ∗-Algebras Just as in the usual elliptic theory, in the theory of nonlocal elliptic operators we need not restrict ourselves to individual operators but can also consider families of nonlocal elliptic operators. The index of such a family is already not a number but an element of the K-group of the parameter space (or, which is the same, the K-group of the algebra of continuous functions on the parameter space.) In the classical elliptic theory, there is a long- and well-known construction that permits studying both the problem on the index of a single operator and the problem on the index of families, as well as several other problems (for more detail, see the literature cited below) in the framework of a uniﬁed approach related to studying the elliptic operators over an arbitrary C ∗ -algebra Λ. (The problem on the index of a single operator is obtained for Λ = C, and the problem on the index of families is obtained for Λ = C0 (X), where X is the parameter space.) In the present chapter, we generalize the theory of elliptic operators over C ∗ -algebras to the case of nonlocal elliptic operators. Such a generalization is necessitated not only by applications but also by intrinsic needs of the theory of nonlocal elliptic operators. In Part 2, we study homotopy invariants of nonlocal elliptic operators. The most important invariant is described in terms of the index of operators over the C ∗ -algebra Λ = C ∗ (Γ). In the ﬁrst two sections of the present chapter, we recall some preliminaries concerning operators over C ∗ -algebras, in particular, elliptic pseudodiﬀerential operators. Our presentation mainly follows the papers [53, 56, 57, 72], where the reader can ﬁnd a detailed presentation of these results. In the third section, we introduce and study the nonlocal elliptic operators.

Chapter 3. Elliptic Operators over C ∗ -Algebras

38

3.1 Hilbert Modules and Λ-Index 3.1.1 Hilbert Modules Let Λ be a C ∗ -algebra. Deﬁnition 3.1. A pre-Hilbert Λ-module is deﬁned to be a right Λ-module M equipped with a Λ-valued inner product, i.e., a sesquilinear form ( · , · ) : M × M −→ Λ

(3.1)

with the following properties: 1. (x, y) = (y, x)∗ , x, y ∈ M . 2. (x, x) ≥ 0, x ∈ M ; (x, x) = 0 only for the zero element x. 3. (x, yλ) = (x, y)λ, x, y ∈ M , λ ∈ Λ. Example 3.2. An arbitrary right ideal J ⊂ Λ is a pre-Hilbert Λ-module with respect to the inner product (x, y) = x∗ y. xn , . . . ), Example 3.3. Consider the linear space M of sequences x = (x1 , . . . , where xj ∈ Jj are elements of given right ideals Jj ∈ Λ and the sum j xj converges. This space can be transformed into a pre-Hilbert Λ-module if one introduces the inner product by the formula x∗j yj . (x, y) = j

(Obviously, the series in the right-hand side converges.) Example 3.4. (This example demonstrates the passage from a degenerate form to a nondegenerate.) Let K be a right Λ-module equipped with a sesquilinear form (3.1) that satisﬁes all conditions 1–3 except for the nondegeneracy condition in 2. Consider the set N = {x ∈ K : (x, x) = 0}. This set is a right submodule in K. Indeed, an element x ∈ K lies in N if and only if (y, x) = 0 for any y ∈ K. To prove this statement, consider the positive element a(t) = (y − tμx, y − tμx),

t ∈ R,

0 = μ ∈ C.

Since (x, x) = 0, we have a(t) = (y, y) − t[μ(y, x) + μ(x, y)]. Let ϕ be a positive functional on Λ. Then 0 ≤ ϕ(a(t)) = ϕ((y, y)) − tϕ [μ(y, x) + μ(x, y)]

3.1. Hilbert Modules and Λ-Index for all t ∈ R. It follows that

39

ϕ [μ(y, x) + μ(x, y)] = 0,

and since the functional ϕ and the number μ are arbitrary, (x, y) = (y, x) = 0. Now it is obvious that N is a right submodule: if x ∈ N , then (y, xλ) − (y, x)λ = 0 for any y ∈ K, λ ∈ Λ. The quotient module K/N is equipped with the structure of a pre-Hilbert module with respect to the inner product inherited from K. In a pre-Hilbert module, one can introduce a norm by the formula (3.2) x = (x, x). (One can verify that such a function is indeed a norm.) It has the following properties: 1. (boundedness) xλ ≤ x λ. 2. (The Cauchy–Schwarz inequality) (x, y) ≤ x y. Deﬁnition 3.5. A pre-Hilbert module that is a Banach space with respect to the norm (3.2) is called a Hilbert module. Obviously, the completion of a pre-Hilbert module with respect to the norm (3.2) is a Hilbert module. Example 3.6. Let Mj , j = 1, . . . , n, be Hilbert Λ-modules. Their ﬁnite direct sum M=

n

Mj ,

j=1

equipped with the inner product (x, y) =

n (xj , yj ), j=1

is a Hilbert Λ-module. Example 3.7. The countable direct sum M=

∞

Mj

j=1

consisting of elements x = (x1 , . . . , xn , . . . ) for which the series (x, x) =

∞ (xj , xj ) j=1

Chapter 3. Elliptic Operators over C ∗ -Algebras

40

converges in Λ is a Hilbert Λ-module with the inner product determined by the series ∞ (xj , yj ). (x, y) = j=1

(One can readily show that this space is complete.) Example 3.8. Let H be a Hilbert A-module, and let T be a compact Hausdorﬀ space. Then the space C(T, H) of continuous H-valued functions on T is a Hilbert Λ-module for the C ∗ -algebra A ≡ C(T, A) Λ = C(T ) ⊗ is the tensor product of C ∗ -algebras) with respect to the inner product (here ⊗ (x, y)(t) = (x(t), y(t)),

t ∈ T.

If, in Example 3.7, all modules Mj coincide with some projective module P , then the countable direct sum is denoted by l2 (P ). If P = Λ, then the countable direct sum is denoted by HΛ and is called the standard Hilbert module. If M is a Hilbert Λ-module and N ⊂ M is a closed submodule, then one can deﬁne the orthogonal complement N ⊥ = {x ∈ M : (x, y) = 0 for all y ∈ N }. In general, the relation N ⊕ N ⊥ = M does not hold. For example, the orthogonal complement of the submodule C0 ((0, 1)) in Λ = C([0, 1]) is trivial. A submodule N ⊂ M is said to be (topologically) complemented if it has a complement, i.e., if there exists a closed submodule L ⊂ M whose direct sum with N coincides with the ambient module M . It can happen that a submodule N is topologically complemented but the sum of N with the orthogonal complement N ⊥ does not coincide with the entire module. For example, in the C([0, 1])-module M = C([0, 1]) ⊕ C0 ((0, 1)), consider the submodule N = diag{C0 ((0, 1)), C0 ((0, 1))} = {(f, f ) : f ∈ C0 ((0, 1))}. Its orthogonal complement has the form N ⊥ = diag C0 ((0, 1)), −C0 ((0, 1)) = {(g, −g) : g ∈ C0 ((0, 1))}, and

N ⊕ N ⊥ = C0 ((0, 1)) ⊕ C0 ((0, 1)) = M.

But the submodule C([0, 1]) ⊕ 0 is a complement of the submodule N . A Hilbert Λ-module M is said to be countably generated if is contains a countable set of elements whose Λ-linear combinations are dense in M .

3.1. Hilbert Modules and Λ-Index

41

Theorem 3.9 (Kasparov stabilization theorem). Let M be a countably generated Hilbert Λ-module. Then there exists an isomorphism M ⊕ H Λ = HΛ of Hilbert modules (where ⊕ is the orthogonal direct sum).

3.1.2 Operators in Hilbert Modules Let M1 and M2 be Hilbert Λ-modules. Deﬁnition 3.10. A mapping T : M1 → M2 such that for some mapping T ∗ : M2 → M1 (called the adjoint operator of T ), the relation (u, T v) = (T ∗ u, v)

holds for all u ∈ M2 and v ∈ M1

is called an operator from M1 into M2 . An operator from M1 into M2 is automatically Λ-linear (i.e., is a homomorphism of Λ-modules) and bounded (i.e., satisﬁes the relation T u ≤ C u for all u ∈ M1 ). The space of such operators is denoted by HomΛ (M1 , M2 ). For M1 = M2 = M , this space of operators is a C ∗ -algebra, which we denote by End∗Λ (M ). If T : M1 → M2 is an operator in the sense of Deﬁnition 3.10, then we say that it admits an adjoint, or is adjointable. The subspace K(M1 , M2 ) ⊂ HomΛ (M1 , M2 ) of compact operators is determined as the norm closure of the space generated by operators of rank 1, i.e., by operators of the form θx,y , θx,y z = x(y, z), Obviously,

(θx,y )∗ = θy,x ,

x ∈ M2 ,

y, z ∈ M1 .

θx,y θu,v = θx(y,u),v = θx,v(u,y) ,

and if S ∈ HomΛ (L, M1 ), then θx,y S = θx,S ∗ y . For M1 = M2 = M , the space K(M ) = K(M, M ) of compact operators is a C ∗ -ideal in End∗Λ (M ). Proposition 3.11. For a unital algebra Λ, the compactness of an operator T in the standard module HΛ is equivalent to the condition1 lim T Ln = 0, n→∞

where Ln ⊂ HΛ is the submodule consisting of elements whose ﬁrst n components are zero. 1 Warning.

For nonunital algebras, this is not true.

Chapter 3. Elliptic Operators over C ∗ -Algebras

42

This statement also holds for operators in modules of the form l2 (P ), where P is a ﬁnitely generated projective Λ-module. Proposition 3.12. The following natural isometric isomorphisms hold: K(Λ) Λ,

K(Λn ) Matn (Λ),

Λ, K(HΛ ) K ⊗

where K is the ideal of (usual ) compact operators in l2 . If a submodule of a Hilbert Λ-module has a complement (not necessarily orthogonal), then the projection onto this submodule along the additional submodule is deﬁned. If the submodule does not have any orthogonal complement, then the projection may be nonadjointable (i.e., is not an operator in the Hilbert Λ-module). Theorem 3.13. Let M and N be Hilbert Λ-modules, and let T ∈ Hom∗Λ (M, N ); moreover, suppose that the range of the operator T is closed. Then the kernel and the range of the operator T are orthogonally complemented submodules. Corollary 3.14. The range of an adjointable idempotent is orthogonally complemented. Corollary 3.15. The range of an ajointable embedding F : M → N is orthogonally complemented. The following proposition also holds. Proposition 3.16. A ﬁnitely generated Hilbert submodule of a Hilbert module over a unital algebra is orthogonally complemented.

3.1.3 Fredholm Property and Index Let Λ be a unital C ∗ -algebra. Now we deﬁne Fredholm operators in Hilbert Λ-modules. (Such operators are also called Λ-Fredholm if they can be confused with the usual Fredholm operators in the context.) We start from operators in standard Λ-modules. Deﬁnition 3.17. An operator F ∈ End∗Λ (HΛ ) is said to be Λ-Fredholm if there exist decompositions HΛ = M 0 ⊕ M 1 ,

HΛ = N0 ⊕ N1

of the space HΛ into direct sums of closed modules such that 1. The modules M1 and N1 are ﬁnitely generated. 2. With respect to these expansions, the operator can be represented by a diagonal matrix

F0 0 : M0 ⊕ M1 −→ N0 ⊕ N1 , F = 0 F1 where the operator F1 : M0 → N0 is invertible.

3.1. Hilbert Modules and Λ-Index

43

It turns out that if the assumptions of this deﬁnition are satisﬁed, then the modules M1 and N1 are automatically projective and thus can be described as the ranges of projections in the free ﬁnitely generated module ΛN for some N . Hence they generate the classes [M1 ] and [N1 ] in the group K0 (Λ). Deﬁnition 3.18. The Λ-index of a Λ-Fredholm operator F is deﬁned to be the element indΛ F = [M1 ] − [N1 ] ∈ K0 (Λ). Proposition 3.19. The index of a Λ-Fredholm operator is well deﬁned, i.e., is independent of the choice of decompositions of the module HΛ into direct sums contained in its deﬁnition. Moreover, it is homotopy invariant in the class of ΛFredholm operators, does not vary if a Λ-compact perturbation is added to the operator, and satisﬁes the so-called logarithmic property: if F1 and F2 are Λ-Fredholm operators, then the product F1 F2 is Λ-Fredholm, and indΛ F1 F2 = indΛ F1 + indΛ F2 . Now we pass to the general case. Let M and N be countably generated Hilbert Λ-modules. Deﬁnition 3.20. An operator F : M → N is said to be Λ-Fredholm if its stabilization (3.3) S(F ) = F ⊕ 1 : HΛ ≡ M ⊕ HΛ −→ N ⊕ HΛ = HΛ treated as an operator in the standard module HΛ is Fredholm. The index of an operator F is deﬁned to be the index of its stabilization: indΛ F = indΛ S(F ). It turns out that all standard properties of the index (homotopy invariance, invariance with respect to compact perturbations, and the logarithmic property) established in Proposition 3.19 for operators in the standard module can be transferred without changes to the case of operators in arbitrarily countably generated Hilbert modules. This can be done by using the following theorem. Theorem 3.21. The following conditions are equivalent: 1. The operator F : M → N is Fredholm in the sense of Deﬁnition 3.20. 2. The operator F has left and right inverse operators modulo Λ-compact operators. 3. There exist decompositions M = M0 ⊕ M1 , N = N0 ⊕ N1 into direct sums of closed submodules, where the modules M1 and N1 are ﬁnitely generated, such that the operator F has a block-diagonal form with respect to these decompositions and the restriction F M0 : M0 −→ N0 is an isomorphism.

Chapter 3. Elliptic Operators over C ∗ -Algebras

44

Further, the index of the operator F satisﬁes the formula ind F = [M1 ] − [N1 ].

3.2 Pseudodiﬀerential Operators over Λ An important example of Λ-Fredholm operators is obtained if we consider elliptic pseudodiﬀerential operators over the C ∗ -algebra Λ. Such operators act in the Sobolev spaces of sections of Λ-bundles on a smooth compact manifold M . By analogy with usual pseudodiﬀerential operators, they have symbols that are Λ-homomorphisms of the pullbacks of these bundles to the cotangent bundle T ∗ M \ {0} without the zero section (or to the cosphere bundle) and ellipticity (i.e., the invertibility of the symbol) guarantees that the operator under study is Λ-Fredholm in appropriate Sobolev spaces. Now we brieﬂy present these results, which are essentially parallel to the corresponding results for usual pseudodiﬀerential operators.

3.2.1 Λ-bundles and Section Spaces Λ-bundles Let M be a smooth compact manifold, let Λ be a unital C ∗ -algebra, and let R be a ﬁnitely generated projective right Hilbert Λ-module. A locally trivial Λ-bundle E with ﬁber R over M is deﬁned to be a locally trivial bundle π : E → M with ﬁber R such that its transition functions, which smoothly depend on a point of the base, are automorphisms of the Λ-module R. More precisely, if U, V ⊂ M are two neighborhoods over which the bundle E can be trivialized, ϕU : E|U U × R,

ϕV : E|V V × R,

(3.4)

then the transition mapping ϕV U : ϕV ◦ ϕ−1 U : (U ∩ V ) × R −→ (U ∩ V ) × R has the form ϕV U (x, y) = (x, W (x)y), where W (x), x ∈ U ∩ V , is a smooth function ranging in the set Aut∗Λ (R) ⊂ End∗Λ (R) of automorphisms of the module R. Without loss of generality, we can and will assume that all the mappings W (x) are unitary (i.e., satisfy the condition W ∗ (x) = W (x)−1 ). A smooth mapping τ : E1 → E2 commuting with the projection and ﬁberwise Λ-linear is called

3.2. Pseudodiﬀerential Operators over Λ

45

a homomorphism of such bundles π1 : E1 → M1 and π2 : E2 → M2 over a smooth mapping f : M1 → M2 . More precisely, the diagram τ

E1 −−−−→ ⏐ ⏐ π1

E2 ⏐ ⏐π 2

M1 −−−−→ M2 f

should be commutative, so that τ “splits” into a family of mappings τx : E1x → E2f (x) , and

τx ∈ Hom∗Λ (R1 , R2 ).

Most frequently, we deal with the case in which M1 = M2 = M and f is the identity mapping of the manifold M . Proposition 3.22. 1. Let P : M → MatN (Λ) be a smooth projection-valued function on a (connected ) manifold M . Then the set EP = {(x, y) ∈ M × ΛN : y ∈ Im P (x)} ⊂ M × ΛN

(3.5)

has the natural structure of a locally trivial Λ-bundle with ﬁber R = Im P (x0 ) over the smooth compact manifold M . (Here x0 ∈ M is an arbitrary point.) 2. Let E be a locally trivial Λ-bundle with ﬁber R over a smooth compact manifold M . Then for some N there exists a smooth projection-valued function P : M → MatN (Λ) such that the bundle E is unitarily isomorphic to the bundle EP . Space of Smooth Sections Smooth sections of a locally trivial Λ-bundle π : E → M are determined in a standard way as smooth mappings s : M → E, so that π ◦ s = id. Of course, a smooth mapping is deﬁned to be a mapping that, in local trivializations of the bundle E, is speciﬁed by smooth functions U x −→ s(x) ∈ R. One can readily see that the smoothness condition is independent of the choice of the trivialization. Obviously, the isomorphism E EP takes the space C ∞ (M, E) of smooth sections of the bundle E to a subspace of the space C ∞ (M, ΛN ) of smooth ΛN -valued functions on M . This subspace consists of functions u(x) such that P (x)u(x) = u(x) for all x ∈ M . This subspace is the range Im P = P C ∞ (M, ΛN ) of the matrix projection P ∈ MatN (C ∞ (M, Λ)) determined by the family P (x).

Chapter 3. Elliptic Operators over C ∗ -Algebras

46 Space L2

Now for Λ-bundles we introduce analogs of spaces of “generalized” sections, i.e., of the L2 space and Sobolev spaces. We start from the space L2 . On the manifold M , we ﬁx a Riemannian metric and let dμ denote the measure on M corresponding to this metric. On the space C ∞ (M, E), we introduce the structure of a pre-Hilbert Λ-module by setting (u, v) = (u(x), v(x)) dμ(x). (3.6) M

Here the inner product (u(x), v(x)) of values of sections at a point x ∈ M is determined via a local trivialization (where u(x) and v(x) can be treated as elements of the module R), but is independent of it because the transition functions are unitary. The corresponding Hilbert module obtained from this module by completion with respect to the norm u = (u, u) is denoted by L2 (M, E) and is called the space of L2 -sections of the bundle E. Obviously, the isomorphism E EP induces a unitary isomorphism of the space L2 (M, E) onto the subspace P L2 (M, ΛN ). Sobolev Spaces Now we deﬁne Sobolev spaces of sections of the bundle E. We start from the case in which E is a trivial bundle over M with ﬁber ΛN . Let Δ be the positive Beltrami– Laplace operator corresponding to the Riemann metric on M . For any2 s ∈ Z, the operator (1 + Δ)s is well deﬁned and continuous in the Fr´echet space C ∞ (M ) and hence in the space C ∞ (M, ΛN ) = C ∞ (M ) ⊗ ΛN (the tensor product of Fr´echet spaces), where, with respect to the factor ΛN , it acts as the unit operator. For s ∈ Z, we deﬁne a Λ-valued inner product on the Λ-module C ∞ (M, ΛN ) by setting (3.7) (u, v)s = (u, (1 + Δ)s v). The completion of the space C ∞ (M, ΛN ) with respect to the corresponding norm us = (u, u)s is a Hilbert module (to which the inner product (3.7) can naturally be continued), which we denote by H s (M, ΛN ) and call the Sobolev space of order s of sections of the bundle ΛN . In particular, H 0 (M, ΛN ) = L2 (M, ΛN ). Now let E be an arbitrary locally trivial Λ-bundle over the compact manifold M . We choose some isomorphism E EP , which exists by Proposition 3.22. 2 In

fact, for any real s.

3.2. Pseudodiﬀerential Operators over Λ

47

Proposition 3.23. The operator P can be extended by continuity to a continuous adjointable operator in any space H s (M, ΛN ): P ∈ End∗Λ (H s (M, ΛN )). By this proposition, the range P H s (M, ΛN ) ⊂ H s (M, ΛN ) is a Hilbert Λ-submodule in H s (M, ΛN ). Therefore, we can introduce the following deﬁnition. Deﬁnition 3.24. The Sobolev space H s (M, E) is deﬁned by the relation def

H s (M, E) = P H s (M, ΛN ). Proposition 3.25. Up to the passage to an equivalent inner product, the Hilbert Λ-module H s (M, E) is independent of the choice of the projection P determining the isomorphic bundle EP and of the speciﬁc choice of the isomorphism E EP . The identity mapping of the Λ-module C ∞ (M, E) can be continued to Λcompact (and, in particular, adjointable) embeddings H s (M, E) ⊂ H l (M, E), The following relation holds:

s > l,

s, l ∈ Z.

H s (M, E) = C ∞ (M, E).

s∈Z

Remark 3.26. The spaces H s (M, E) can also be deﬁned diﬀerently by “gluing together,” with the use of a partition of unity, the spaces H s (U, R) of R-valued functions on neighborhoods U ⊂ M where the bundle E can be trivialized (see [57]). The deﬁnition thus obtained is equivalent to the above deﬁnition.

3.2.2 Symbols and Pseudodiﬀerential Operators Now we can deﬁne pseudodiﬀerential operators in the space of sections of a Λbundle over the manifold M . Let E and F be locally trivial Λ-bundles over M with ﬁbers R1 and R2 , respectively. First, we deﬁne pseudodiﬀerential operators locally. Let f (x, ξ) : R1 −→ R2 be a smooth operator-valued function ranging in Hom∗Λ (R1 , R2 ), and let this function be compactly supported in x and, for some integer m, satisfy the estimates α+β ∂ f (x, ξ) m−|β| ∂xα ∂ξ β ≤ Cαβ (1 + |ξ|)

Chapter 3. Elliptic Operators over C ∗ -Algebras

48

for all x and ξ, where α and β are multiindices. We deﬁne the operator f acting on smooth compactly supported R1 -valued functions of x (and taking them to compactly supported R2 -valued functions) by the standard formula 1

∂ 1 2 f u ≡ f x, −i u(ξ) dξ, eixξ f (x, ξ) u= ∂x (2π)n/2

where u

(ξ) is the Fourier transform of the function u(x). One can readily verify that this operator can be extended by continuity to an operator f: H s (Rn , R1 ) −→ H s−m (Rn , R2 ),

s ∈ Z.

Now we are in a position to give the global deﬁnition. Let f : π ∗ E −→ π ∗ F be a Λ-homomorphism of the pullbacks of the bundles E and F to the cotangent bundle of M without the zero section. Here π : T ∗ M \ {0} −→ M is the natural projection. We assume that f is positive homogeneous of degree m ∈ Z, i.e., f (x, tξ) = tm f (x, ξ),

t ∈ R+ ,

(x, ξ) ∈ T ∗ M \ {0}.

Now we deﬁne the operator f: C ∞ (M, E) −→ C ∞ (M, F ) using a partition of unity {e2α } by the formula fu = e α f (eα u),

(3.8)

(3.9)

α

where the separate summands e α f are calculated in the corresponding trivializations (3.4) of the bundles E and F and, of course, the operator-valued function eα (x)f (x, ξ) : R1 −→ R2 is arbitrarily smoothed in a neighborhood of the set {ξ = 0}. Proposition 3.27. The operator (3.9) can be continued to an adjointable continuous operator of order m in the Sobolev spaces of sections of the bundles E and F , i.e., to a continuous operator f: H s (M, E) −→ H s−m (M, F )

(3.10)

for any s ∈ Z. Modulo operators of order m − 1 (and hence operators Λ-compact in the spaces H s (M, E) → H s−m (M, F )), this operator is independent of the arbitrariness in the construction related to the choice of trivializations, the partition of unity, and the smoothing of the symbol in a neighborhood of the zero section {ξ = 0} in T ∗ M .

3.2. Pseudodiﬀerential Operators over Λ

49

Proof. We prove only the statement concerning the property of being adjointable (since this statement is not contained in [57]). It suﬃces to verify the statement for each individual summand in (3.9), i.e., for operators in Rn . In this case, the adjoint of the operator 1

∂ 2 f x, −i : H s (Rn , R1 ) −→ H s−m (Rn , R2 ) ∂x

is determined by the explicit formula 2

∂ 1 f x, −i (1 + Δ)s−m : H s−m (Rn , R2 ) −→ H s (Rn , R1 ) ∂x

−s ∗

(1 + Δ)

where f ∗ (x, ξ) is the adjoint symbol. The proof of the desired statement is complete. Deﬁnition 3.28. The operator (3.9) is called a Λ-pseudodiﬀerential operator (or a pseudodiﬀerential operator over Λ) with symbol f . In what follows, we shall write f = σ(f). Remark 3.29. For the symbol we can also take the restriction of the homomorphism f to the bundle S ∗ M of unit spheres in T ∗ M . But in this case we should explicitly indicate the order m of the symbol in question. For Λ-pseudodiﬀerential operators, we have the following composition theorem, which reproduces the standard composition theorem for pseudodiﬀerential operators word for word. Theorem 3.30 (Calculus of Λ-pseudodiﬀerential operators). Let D1 : C ∞ (M, E1 ) −→ C ∞ (M, E2 ),

D2 : C ∞ (M, E2 ) −→ C ∞ (M, E3 )

be two Λ-pseudodiﬀerential operators of orders m1 and m2 , respectively. Then the composition D2 D1 : C ∞ (M, E1 ) −→ C ∞ (M, E3 ) is a Λ-pseudodiﬀerential operator of order m1 +m2 , and its symbol can be calculated by the formula σ(D2 D1 ) = σ(D2 )σ(D1 ). (3.11) With Proposition 3.27 taken into account, we obtain the following statement. Corollary 3.31. If σ1 : π ∗ E1 −→ π ∗ E2 ,

σ2 : π ∗ E2 −→ π ∗ E3

are symbols of orders m1 and m2 , respectively, then the operator σ 2 σ 1 is 2 σ1 − σ of order m1 + m2 − 1 and is Λ-compact in the spaces H s (M, E1 ) → H s−m1 −m2 (M, E3 ).

Chapter 3. Elliptic Operators over C ∗ -Algebras

50

3.2.3 Ellipticity and the Λ-Fredholm property Theorem 3.30 and the equivalence of statements 1 and 2 in Theorem 3.21 imply the following criterion for the Λ-Fredholm property of the Λ-pseudodiﬀerential operator. Theorem 3.32 (Finiteness theorem). Let D : H s (M, E) −→ H s−m (M, F )

(3.12)

be a Λ-pseudodiﬀerential operator of order m. If its symbol σ(D) : π ∗ E −→ π ∗ F, where π : S ∗ M → M is a natural projection, is everywhere invertible, then the operator (3.12) is Λ-Fredholm. Thus, it is natural to introduce the following standard deﬁnition. Deﬁnition 3.33. A Λ-pseudodiﬀerential operator on a manifold M is said to be elliptic if its symbol is invertible everywhere on S ∗ M . For elliptic Λ-pseudodiﬀerential operators, one has the index formula indΛ D = chΛ ([σ(D)]) Td M, [M ] ∈ K0 (Λ) ⊗ Q. Here we do not dwell on the deﬁnition of the Chern Λ-character of the symbol of the operator D and refer the reader to the original papers cited at the beginning of this chapter, because in Chap. 11 we present a more general deﬁnition modeling the above deﬁnition and applicable in the case of nonlocal elliptic operators over C ∗ -algebras. Example 3.34. Let us show how the theory of families of elliptic operators can be treated as a speciﬁc case of the theory of operators over C ∗ -algebras. Let D(x) : C ∞ (M ) −→ C ∞ (M ),

x ∈ X,

be a continuous family of elliptic operators parametrized by points of some compact set X. For simplicity, we assume that the family consists of operators of order zero. Then we can treat the space C(X, L2 (M )) as a Hilbert module over the algebra C(X), deﬁning the module structure as (u · f )(x, m) = u(x, m)f (x),

u ∈ C(X, L2 (M )),

and the inner product as (u, v)(x) =

u(x)v(x)dm. M

f ∈ C(X),

3.3. Nonlocal Pseudodiﬀerential Operators over Λ

51

Clearly, the family {D(x)} determines an elliptic operator D : C(X, L2 (M )) −→ C(X, L2 (M ))

(3.13)

in Hilbert C(X)-modules. Finally, the index of the family ind{Dx } ∈ K 0 (X) (treated as the virtual diﬀerence of the kernel and the cokernel) coincides with the index of the operator (3.13) if we use the natural isomorphism K 0 (X) K0 (C(X)).

3.3 Nonlocal Pseudodiﬀerential Operators over Λ The calculus of nonlocal pseudodiﬀerential operators described in Chap. 2 is generated by the usual (classical) pseudodiﬀerential operators and by the shift operators T (g), g ∈ Γ. We generalize this construction to the case in which, instead of usual pseudodiﬀerential operators, one deals with the above-described class of pseudodiﬀerential operators over some unital C ∗ -algebra Λ. All our constructions are parallel to the constructions presented earlier in Chap. 2; hence our description is concise, and we mainly draw the reader’s attention to distinctions. Further, we restrict ourselves to the case of operators of order zero acting in spaces of L2 -sections of Λ-bundles. The simplest method for passing to operators of an arbitrary order in Sobolev spaces of sections of Λ-bundles is to multiply the operators to be considered on the right and on the left by appropriate powers of the Beltrami–Laplace operator. Just as in Chap. 2, we assume that Γ is a discrete group of polynomial growth and a dense subgroup of a compact Lie group Γ of diﬀeomorphisms of the manifold M.

3.3.1 Operators in Trivial Λ-Bundles We start from nonlocal pseudodiﬀerential operators in trivial Λ-bundles. These are operators of the form D= T (g)D(g) : L2 (M, ΛN ) −→ L2 (M, ΛN ), (3.14) g∈Γ

where

D(g) : L2 (M, ΛN ) −→ L2 (M, ΛN ) is a pseudodiﬀerential operator of order zero in the spaces of sections of trivial Λ-bundles and T (g), just as above, is the shift operator [T (g)u](x) = u(g −1 (x)),

x ∈ M.

The shift operators are unitary in the spaces L2 (M, ΛN ). We assume that the operators D(g) rapidly decay as |g| → ∞ in the natural Fr´echet topology on the set of pseudodiﬀerential operators of order zero. Under this condition, their norms also rapidly decay, so that the sum (3.14) is well deﬁned.

52

Chapter 3. Elliptic Operators over C ∗ -Algebras

3.3.2 Operators in General Λ-Bundles A natural class of bundles in which nonlocal Λ-pseudodiﬀerential operators act consists of nonlocal Λ-bundles. Their section spaces are the ranges of nonlocal selfadjoint projections in trivial Λ-bundles, i.e, of projections P ∈ C ∞ (M, MatN (Λ))Γ in the Hilbert Λ-modules L2 (M, ΛN ). A nonlocal Λ-pseudodiﬀerential operator D is a triple D = (D, P1 , P2 ), where P1 , P2 are projections of the form described above and D is a Λ-pseudodiﬀerential operator in trivial Λ-bundles such that D = P2 DP1 .

(3.15)

Such a triple is identiﬁed with the operator D Im P : Im P1 −→ Im P2 . 1

(3.16)

Consider the operator (3.16). The spaces Im P1 and Im P2 are Hilbert Λ-modules; therefore, it is natural to ask whether the operator D has an adjoint. Proposition 3.35. Nonlocal Λ-pseudodiﬀerential operators are adjointable. Proof. To prove this proposition, we use the representation (3.15). The adjoint operator, if it exists, must have the form D∗ = (D∗ , P2 , P1 ), so that the problem is reduced to the adjointability of operators acting in the spaces of L2 -sections of bundles with free ﬁber. In this case, we can use the representation (3.14). The limit of adjointable operators is adjointable as well, the operators T (g) are ajointable, (T (g)∗ = T (g −1 )), and hence it suﬃces to verify the adjointability of a Λ-pseudodiﬀerential operator D(g). But this has already been done earlier. The space of zero-order nonlocal Λ-pseudodiﬀerential operators in L2 -sections of bundles determined by projections P1 and P2 will be denoted by ΨΓ,Λ (M ; P1 , P2 ), and its closure will be denoted by ΨΓ,Λ (M ; P1 , P2 ). (We sometimes omit the arguments-projections.)

3.3.3 Symbols in Trivial Bundles Now we discuss the notion of symbol for nonlocal Λ-pseudodiﬀerential operators. First, consider operators acting in the space of L2 -sections of a trivial bundle with ﬁber ΛN . To such an operator of the form (3.14), we assign the symbol σ(D) ∈ C(S ∗ M, MatN (Λ)) Γ,

σ(D)(g) = σ(D(g)).

(3.17)

(Obviously, the element thus deﬁned belongs to the crossed product, because the coeﬃcients rapidly decrease with respect to the norm.) A trivial veriﬁcation shows

3.3. Nonlocal Pseudodiﬀerential Operators over Λ

53

that the product of symbols of the form (3.17) corresponds to the product of operators written in the form (3.14). Remark 3.36. Note that it has not been veriﬁed yet that the symbol of an operator is well deﬁned (i.e., is independent of exactly how the operator is represented in the form (3.14)). This will be done below, simultaneously with the proof of the ﬁniteness theorem. In what follows, we denote the C ∗ -crossed product C(S ∗ M, MatN (Λ)) Γ by C(S ∗ M, MatN (Λ))Γ . The set of symbols of the form (3.17), where σ(D)(g) ∈ C ∞ (S ∗ M, MatN (Λ)) rapidly decays in the topology of the Fr´echet space C ∞ (S ∗ M, MatN (Λ)) as |g| → ∞, is denoted by C ∞ (S ∗ M, MatN (Λ))Γ . This is a dense involutive subalgebra in C(S ∗ M, MatN (Λ))Γ . We also note that it is the smooth crossed product of the Fr´echet algebra C ∞ (S ∗ M, MatN (Λ)) by the group Γ in the sense of Sec. 1.1.4. Proposition 3.37. The following relation holds: MatN (Λ) C(S ∗ M, MatN (Λ))Γ = C(S ∗ M )Γ ⊗

(3.18)

is understood as the tensor product of C ∗ -algebras). (where ⊗ Proof. 1. The group Γ is amenable and locally compact. Therefore, the C ∗ -algebra C(S ∗ M )Γ is nuclear by Proposition 1.8, so that the tensor product in (3.18) is uniquely determined. 2. The relation MatN (Λ) C(S ∗ M, MatN (Λ)) = C(S ∗ M ) ⊗ holds; thus, we should prove that MatN (Λ)]Γ = C(S ∗ M )Γ ⊗ MatN (Λ). [C(S ∗ M ) ⊗

(3.19)

3. Since the algebra C(S ∗ M ) is nuclear, we have the relation MatN (Λ)) = Prim(C(S ∗ M )) × Prim(MatN (Λ)) Prim(C(S ∗ M ) ⊗ for the spaces of primitive ideals of the corresponding algebras (see Proposition A.55). Since the group Γ acts topologically free on Prim(C(S ∗ M )) and trivially on Prim(MatN (Λ)), it follows that this group also acts topologically free MatN (Λ)). Now, to prove formula (3.19), it suﬃces to note on Prim(C(S ∗ M ) ⊗ that MatN (Λ) is embedded as a C ∗ -subalgebra in the 1. The C ∗ -algebra C(S ∗ M ) ⊗ ∗ MatN (Λ). (The mapping is induced by the embedding algebra C(S M )Γ ⊗ C(S ∗ M ) ⊂ C(S ∗ M )Γ

Chapter 3. Elliptic Operators over C ∗ -Algebras

54

of nuclear algebras; the fact that, as a result, we obtain an embedding of tensor products, follows from the continuity property for the spatial norm [69, Sec. 2, p. 1763].) 2. Both algebras in (3.19) are obtained by closure from the ﬁnitely supported functions MatN (Λ). Γ g → a(g) ∈ C(S ∗ M ) ⊗ It remains to apply the isomorphism Theorem 1.12.

The relation C ∞ (S ∗ M, MatN (Λ))Γ = C ∞ (S ∗ M )Γ ⊗ MatN (Λ)

(3.20)

(where ⊗ is the tensor product of Fr´echet spaces) is obvious by deﬁnition. Combining this with relation (3.18) and using Proposition 1.32, we see that the algebra C ∞ (S ∗ M, MatN (Λ))Γ is a dense local subalgebra in C(S ∗ M, MatN (Λ))Γ .

3.3.4 Symbols in Nontrivial Bundles The symbol of the operator D = (D, P1 , P2 ) acting in section spaces of nontrivial bundles is deﬁned (with obvious identiﬁcations) as the triple σ(D) = (σ(D), P1 , P2 ). In this case, it follows from (3.15) that σ(D) = P2 σ(D)P1 . The symbol σ(D) is said to be elliptic on S ∗ M if σ(D) realizes an equivalence of the projections P1 and P2 on S ∗ M , i.e., if there is an element R ∈ C(S ∗ M, MatN (Λ))Γ such that Rσ(D) = P1 , σ(D)R = P2 .

3.3.5 Ellipticity and the Λ-Fredholm Property Now one can readily prove that the symbol is well deﬁned and thus obtain the following theorem. Theorem 3.38 (Finiteness theorem). A nonlocal Λ-pseudodiﬀerential operator D is Λ-Fredholm if and only if its symbol is elliptic on S ∗ M . Proof. 1. The scalar case. Consider the quotient algebras Σ = ΨΛ (M )/K(L2 (M, Λ)),

ΣΓ = ΨΓ,Λ (M )/K(L2 (M, Λ))

with respect to Λ-compact operators. Then we have

3.3. Nonlocal Pseudodiﬀerential Operators over Λ

55

1. The embedding Σ ⊂ ΣΓ . (The set of classes corresponding to local Λ-pseudodifferential operators can be embedded in the set of classes corresponding to nonlocal Λ-pseudodiﬀerential operators.) 2. The isomorphism Σ C(S ∗ M, Λ). (This follows from the preceding results on pseudodiﬀerential Λ-operators.) It remains to apply the isomorphism Theorem 1.12 to the homomorphism Σ Γ −→ ΣΓ (the action of the group Γ on Prim Σ is topologically free by item 3 in the proof of Proposition 3.37) to prove that ΣΓ C(S ∗ M, Λ)Γ and the isomorphism is given by the symbol mapping (which is thus well deﬁned). Further, this implies the statement of the ﬁniteness theorem in a standard manner. 2. The case of operators in section spaces of bundles can be considered by analogy with the proofs of Theorem 2.10 and Proposition 2.11.

Part II

Homotopy Invariants of Nonlocal Elliptic Operators

Chapter 4

Homotopy Classiﬁcation In this second part of the book, we wish to obtain index formulas for nonlocal elliptic operators. By analogy with the classical situation of elliptic operators on closed manifolds considered by Atiyah and Singer, the study of this problem naturally starts from establishing the homotopy classiﬁcation of elliptic operators. It is well known that in the classical case this classiﬁcation is given by the K-group with compact supports of the cotangent bundle. Our situation diﬀers from the classical situation in that the algebra of symbols is bigger (and already noncommutative): it is obtained as the crossed product of the classical algebra of symbols by a discrete group Γ. Hence it is no miracle that the homotopy classiﬁcation in this case is similar to the classical one. Namely, it is given by the K-group of the crossed product of the algebra of functions on the cotangent bundle by the group Γ. In this chapter, we give a detailed account of this result.

4.1 Ell-Group Let M be a Γ-manifold, either compact or the total space of a real vector Γ-bundle over a compact base Y . Denote by EllΓ (M ) the group of stable homotopy classes of nonlocal elliptic pseudodiﬀerential operators of order zero1 in sections of nonlocal bundles over M . (If M is noncompact, we consider only pseudodiﬀerential operators multiplicatively trivial at inﬁnity.) This group is deﬁned in the standard way, just as for the classical pseudodiﬀerential operators. Let us brieﬂy describe the deﬁnition of this group. A nonlocal elliptic zero-order operator D on M is said to be trivial if it is induced by an isomorphism of nonlocal bundles on M . (In particular, its symbol σ(D) does not depend on the covariables, i.e., it is a pullback from M to S ∗ M by the canonical projection S ∗ M → M .) Two 1 For

the purpose of homotopy classiﬁcation, it suﬃces to restrict ourselves to zero-order operators, since order reduction is available.

60

Chapter 4. Homotopy Classiﬁcation

elliptic operators D1 and D2 are said to be stably homotopic if for some trivial operators D , D the direct sums D1 = D1 ⊕ D

and D2 = D2 ⊕ D

are homotopic in the class of elliptic operators. (More precisely, the spaces of sections in which the operators D1 and D2 are deﬁned should be identiﬁed by some vector bundle isomorphisms.) The set EllΓ (M ) of classes of nonlocal elliptic operators on M modulo this equivalence relation is an Abelian group with the operation induced by the direct sum of elliptic operators. Indeed, this is obviously an Abelian semigroup. In addition, for each [D] ∈ EllΓ (M ) there is an opposite element; it is given by the equivalence class of an almost inverse nonlocal elliptic operator. As in the classical case, it is possible to describe these classes in terms of elements of some K-group. We give this description in Sec. 4.3. But before that let us deﬁne the diﬀerence construction.

4.2 Diﬀerence Construction It follows from the deﬁnitions that the stable classiﬁcation of nonlocal bundles over a compact space X (i.e., projections in matrix algebras over C(X)Γ ) is given by the group K0 (C(X)Γ ). For a noncompact space, by analogy with the classical situation, it is useful to introduce the K-group with compact supports, i.e., the group of stable equivalence classes of virtual bundles vanishing at inﬁnity. Let us give the formal deﬁnition. Let E be a noncompact total space of a vector Γ-bundle. Further, let P1 , P2 ∈ MatN (C(E)Γ ) be two projections over the algebra C(E)Γ ; let a ∈ MatN (C(E)Γ ) be a homomorphism of nonlocal bundles deﬁned by the projections, i.e., an element such that P2 aP1 = a. We denote this homomorphism by a = (a, P1 , P2 ).

(4.1)

An element a is called an isomorphism of bundles (or said to be invertible, for i

short) outside a Γ-invariant set K ⊂ E if, for any Γ-invariant compact set K1 ⊂ E disjoint with K, the restriction a|K1 deﬁnes an equivalence of the projections i∗ (P1 ) and i∗ (P2 ) over C(K1 )Γ . An element invertible at inﬁnity is an element (4.1) invertible outside some compact Γ-invariant set. A homotopy of elements invertible at inﬁnity is a continuous family of such elements (invertible for all parameter values outside a given Γ-invariant compact set K). A trivial element is an element invertible everywhere. Proposition 4.1. The group of stable homotopy classes of elements (4.1) (i.e., homotopy classes modulo addition of trivial elements) invertible at inﬁnity coincides with the K-group K0 (C0 (E)Γ ).

4.3. Isomorphism of Ell- and K-Groups

61

The proof is standard.

The class [a] ∈ K0 (C0 (E)Γ ) associated with an element a invertible at inﬁnity is called the diﬀerence construction of a. Below we mainly use this representation of elements of K-groups in terms of elements invertible at inﬁnity. Remark 4.2. If E were compact, the element [a] associated with the triple a = (a, P1 , P2 ) would be just [P1 ] − [P2 ] ∈ K0 (C(E)Γ ). This explains why this construction is called the “diﬀerence construction.” The following technical proposition proves useful if we wish to show that two elements are homotopic. Proposition 4.3. Two elements invertible at inﬁnity are homotopic if and only if for some R they are invertible outside the ball BR ⊂ E and homotopic in the class of elements invertible on the sphere ∂BR . Proof. The “only if” claim is obvious. Let us prove the converse. Consider an element invertible on ∂Bt for all t ≥ R. Then it is invertible outside the open ball BR \ ∂BR . (This follows from Proposition 1.30.) Using this fact, it is easy to write out the desired homotopy. First, on each sphere ∂Bt , for t ≥ R we make a radial homotopy to the values on the sphere ∂BR , while for t ≤ R we leave the original values. Then we make a homotopy on the ball BR (for t ≥ R the homotopy corresponds to the values on the sphere ∂BR ), and ﬁnally return to the new values on ∂Bt . Remark 4.4. In the proof, we have substantially used the special form of the noncompact manifold in question.

4.3 Isomorphism of Ell- and K-Groups Theorem 4.5 (Homotopy classiﬁcation). The mapping χ : EllΓ (M ) −→ K0 (C0 (T ∗ M )Γ ) that takes an elliptic operator D to the diﬀerence construction [σ(D)] of its symbol is an isomorphism. Proof. This theorem follows from Proposition 4.6 below.

σ ] ∈ K0 (C0 (T ∗ M )Γ ) contains eleProposition 4.6. Each stable homotopy class [σ ments invertible at inﬁnity that are homogeneous elliptic symbols.2 If two homogeneous elliptic symbols deﬁne the same element in the K-group, then they are stably homotopic in the class of homogeneous elliptic symbols. 2 Recall

that our deﬁnition of homogeneous elliptic symbols in Sec. 2.3 requires multiplicative triviality at inﬁnity when M is noncompact.

62

Chapter 4. Homotopy Classiﬁcation

Proof. 1. To start with, note that any projection over C(T ∗ M )Γ is homotopic to a projection that is the pullback of a projection over C(M )Γ . Indeed, consider the family of mappings pt : T ∗ M −→ T ∗ M, t ∈ [0, 1], deﬁned by ﬁberwise multiplication by t. This family commutes with the action of Γ and thus deﬁnes a continuous family of endomorphisms p∗t of C(T ∗ M )Γ . For t=1, we obtain the identity endomorphism, while for t = 0 any element in the range is a pullback from C(M )Γ . It remains to apply p∗t to the projections to obtain the desired homotopy. Finally, a homotopy of projections can be lifted to a homotopy of the triple (4.1) invertible at inﬁnity. 2. Thus, without loss of generality, we can assume that the projections P1 and P2 in the triple σ = (σ, P1 , P2 ) are pullbacks from the base M . Now the claim for compact M follows from Proposition 4.3. 3. Now consider the second case, where M is the total space of a real vector bundle over Y . On T ∗ M , we consider the coordinates (x, ξ, p, η) introduced in Sec. 1.4.2. Let σ be an element invertible outside the ball

R = {ξ 2 + p2 + η 2 ≤ R2 } ⊂ T ∗ M. B Consider the continuous family σ t = (σt , P1 , P2 ) of elements invertible at inﬁnity deﬁned by the formula σt (x, ξ, p, η) = σ x, ξ, λ(t, ξ)p, λ(t, ξ)η , (here we use Notation 1.29), where ⎧ ⎪ ⎨t, λ(t, ξ) = ⎪ ⎩t + (1 − t)

|ξ| ≥ R, − , p2 + η 2 }

R2

max{1,

ξ2

|ξ| ≤ R.

For t = 1, the symbol σt is equal to the original symbol, and for t = 0 it gives the desired multiplicatively trivial elliptic symbol

R2 − ξ 2 R2 − ξ 2 σ0 (x, ξ, p, η) = σ x, ξ, p, η . (4.2) max{1, p2 + η 2 } max{1, p2 + η 2 } This construction enables one to connect homogeneous elliptic symbols homotopic in the class of elements invertible at inﬁnity by a homotopy of homogeneous elliptic symbols. This completes the proof of the proposition. The homotopy classiﬁcation theorem enables us to identify stable homotopy invariants of nonlocal elliptic operators (and symbols) and functionals on the Kgroups K0 (C0 (T ∗ M )Γ ). We freely use this identiﬁcation in subsequent chapters.

Chapter 5

Analytic Invariants We start the study of homotopy invariants of nonlocal elliptic operators by constructing some analytic homotopy invariants.

5.1 Fredholm Index The most standard (and obvious) analytic invariant of nonlocal elliptic operators is the Fredholm index, i.e., the diﬀerence of dimensions of the kernel and cokernel of the Fredholm operator. The classiﬁcation theorem enables us to deﬁne the Fredholm index as a homomorphism inda : K0 (C0 (T ∗ M )Γ ) −→ Z σ ] ∈ K0 (C0 (T ∗ M )Γ ) and consider a homogeneous elliptic symbol as follows. Take [σ σ ]. We deﬁne σ ∈ [σ σ ] = ind σ , inda [σ

where σ is a nonlocal elliptic operator with symbol σ . It follows from the above that the homomorphism is well deﬁned.

5.2 C ∗(Γ)-Index and Its Connection with the Fredholm Index Nonlocal elliptic operators possess a very natural analytic homotopy invariant, which is more general than the Fredholm index. Let σ = (σ, P1 , P2 ) be an elliptic symbol, where σ = {σ(g)}g∈Γ ∈ MatN (C ∞ (T ∗ M )Γ ) is a homogeneous symbol and P1 = {P1 (g)},

P2 = {P2 (g)} ∈ MatN (C ∞ (M )Γ )

64

Chapter 5. Analytic Invariants

are projections. We now use this symbol to deﬁne an elliptic operator over the group C ∗ algebra Λ = C ∗ (Γ).1 Let L(g) be the operator of left translation by g in the free Λ-module ΛN . We deﬁne P j (g) = Pj (g) ⊗ L(g),

j = 1, 2,

σ

(g) = σ(g) ⊗ L(g).

(5.1)

Then σ

= P 2 σ

P 1 . Hence we obtain a well-deﬁned elliptic symbol σ = ( σ , P 1 , P 2 ) over the C ∗ -algebra Λ, and some operator

Im P : Im P 1 −→ Im P 2 σ 1

(5.2)

with this symbol over Λ, which acts in the ranges of the projections P j : L2 (M, ΛN ) −→ L2 (M, ΛN ) considered as right Λ-modules. Proposition 5.1. The operator (5.2) is Λ-Fredholm. Proof. Indeed, the mapping σ → σ induces a unital ∗-homomorphism of the corresponding symbol algebras. Thus, the operator (5.2) is elliptic, and by the ﬁniteness theorem for nonlocal Λ-elliptic operators, it is Λ-Fredholm. Set

∈ K0 (Λ), σ ] = indΛ σ indΛ [σ

(5.3)

where indΛ stands for the Λ-index of a Λ-Fredholm operator. Thus, we have a well-deﬁned group homomorphism indΛ : K0 (C0 (T ∗ M )Γ ) −→ K0 (Λ), which we call the Λ-Fredholm index mapping or Λ-index for short. By abuse of language, in what follows, the notation indΛ D for a nonlocal elliptic operator D will stand for the Λ-index indΛ σ(D) of its symbol. It turns out that there is a very simple relation between the Fredholm index and the Λ-Fredholm index. To explain it, consider the homomorphism α : Λ −→ C of C ∗ -algebras generated by the trivial representation Γ → {1} of the group Γ. Let α∗ : K0 (Λ) −→ K0 (C) = Z be the induced homomorphism of K-groups. 1 We

shall follow the convention Λ = C ∗ (Γ) from now until Chap. 11; then Λ will denote an arbitrary C ∗ -algebra.

5.2. C ∗ (Γ)-Index and Its Connection with the Fredholm Index

65

Proposition 5.2. Let σ be a nonlocal elliptic symbol. Then σ ] = α∗ indΛ [σ σ ]. inda [σ Proof. The tensor product · → · ⊗α C of a right Λ-module by C over the homomorphism α takes L2 (M, ΛN ) to L2 (M, CN ), the ranges of the projections P 1 and

to σ , P 2 to the ranges of the projections P1 and P2 , respectively, the operator σ and the kernel and cokernel of the ﬁrst operator to the kernel and cokernel of the second operator. This implies the desired statement. Let us recall the notion of (interior ) tensor product of Hilbert modules (e.g., see [22]) appearing in this proof. Let M1 be a Hilbert A-module, and let M2 be a Hilbert B-module, where A and B are C ∗ -algebras. Let us also suppose that we are given a homomorphism ρ : A −→ End∗B (M2 ). Then we deﬁne a B-valued inner product on the algebraic tensor product M1 ⊗ M2 by the formula (m1 ⊗ m2 , m1 ⊗ m2 ) = (m2 , ρ((m1 , m1 ))m2 ). Passing to the quotient of M1 ⊗ M2 by the null space of this inner product and completing the quotient as in Example 3.4, we obtain a Hilbert B-module, which we denote by M1 ⊗ρ M2 and call the (interior) tensor product of the modules M1 and M2 over the homomorphism ρ. Note that, by construction, elements m1 a ⊗ m2

and m1 ⊗ ρ(a)m2 ,

where m1,2 ∈ M1,2 , and a ∈ A, deﬁne the same class in M1 ⊗ρ M2 . In what follows, we focus our attention on the computation of the Λ-index. The formulas for the Fredholm index will be obtained as corollaries of these computations using the proposition just proved. Remark 5.3. A priori, it is not clear whether the Λ-index gives new homotopy invariants in comparison with the usual Fredholm index. (For example, it might happen that the Λ-index is always in the range of the homomorphism κ : Z −→ K0 (Λ) induced by the embedding C ⊂ Λ, z → z1.) In Chap. 6 (Corollary 6.6), we shall show that new invariants do appear. Moreover, the range of the Λ-index is the entire group K0 (Λ).

Chapter 6

Bott Periodicity 6.1 Preliminary Remarks The Bott periodicity theorem in K-theory of operator algebras gives the isomor C0 (R2 )) and (more generally) K0 (A) and phism of K-groups K0 (A) and K0 (A ⊗ 2n K0 (A ⊗ C0 (R )). In the special case where A = C(X), this isomorphism turns into the isomorphism of K-groups1 K0 (C(X)) K0 (C0 (X × R2n )).

(6.1)

Of course, a similar isomorphism can also be written out if C(X) is replaced by the crossed product C(X)Γ : K0 (C(X)Γ ) K0 (C0 (X × R2n )Γ ).

(6.2)

Here, however, the action of the group Γ on X × R2n is the product of its original action on X and the trivial action on R2n . From this point of view, the isomorphism (6.2) cannot be considered as a full-ﬂedged analog of the isomorphism (6.1). In the present chapter, we consider the case in which the action of the group Γ on R2n is nontrivial. More precisely, we assume that an orthogonal action of Γ on Rn is given and consider the Γ-module R2n obtained as the direct sum of two copies of the Γ-module Rn . In this situation, we establish a “true” analog of (6.1), admittedly only for the case in which X is a one-point space. (This is suﬃcient for our purposes.) Namely, we shall show that K0 (Λ) K0 (C0 (R2n )Γ ), where Λ = C ∗ (Γ) is the group C ∗ -algebra of Γ. 1 We

assume that X is compact.

(6.3)

68

Chapter 6. Bott Periodicity

The classical Bott isomorphism is given by the exterior product by the Bott element, which is essentially none other than the symbol of the Euler operator d + d∗ acting from even diﬀerential forms to odd diﬀerential forms in Rn . The isomorphism in (6.3) has a similar structure, except that in the Γ-situation some natural changes should be incorporated in the deﬁnition of the exterior product. If Γ is ﬁnite, then the K-groups of algebras in (6.3) can be replaced by the equivariant K-groups of the corresponding spaces, and the isomorphism becomes KΓ0 ({pt}) KΓ0 (R2n ), which is just the Bott periodicity in equivariant topological K-theory. The proof given below follows the scheme of Atiyah’s proof [7]. The most signiﬁcant diﬀerence of our proof is that we use the C ∗ -index instead of the index of elliptic families.

6.2 Exterior Product of Operators The deﬁnition of exterior product of nonlocal operators has certain speciﬁc features. While in “classical” theory any two symbols (or operators) on manifolds X and Y can be cross-multiplied, giving a symbol (operator) on the product X × Y , the exterior product of two nonlocal symbols is not deﬁned. A well-deﬁned product can be constructed only for the case in which one of the symbols is equivariant. Let us give the corresponding deﬁnitions. Let X and Y be manifolds with an action of the group Γ. Let us assume further that in CN we are given a unitary representation ρ : Γ −→ U(N, C)

(6.4)

of the group Γ. Deﬁnition 6.1. An element b ∈ MatN (C(X)) is said to be equivariant (with respect to ρ) if ρ(g)T (g)b = bρ(g)T (g), g ∈ Γ. Equivariant operators on X are deﬁned in a similar way. Let us take a ∈ MatN (C(Y )Γ ); and deﬁne the tensor product a ⊗ b ∈ MatN N (C(Y × X)Γ ) by the formula [a ⊗ b](g) = a(g) ⊗ ρ(g)b.

(6.5)

(We assume that the action of Γ on Y × X is the direct product of its actions on Y and X.) Now let a = (a, P1 , P2 ) and b = (b, Q1 , Q2 ) be symbols over C(Y )Γ and C(X) respectively, where b is equivariant.

6.3. Euler Operator and the Bott Element

69

Deﬁnition 6.2. The exterior product of the symbols a and b is the symbol a#b over C(Y × X)Γ deﬁned by the formula

a ⊗ 1 −1 ⊗ b∗ 0 0 P1 ⊗ Q1 P2 ⊗ Q1 a#b = , , . 0 P2 ⊗ Q2 0 P1 ⊗ Q2 1 ⊗ b a∗ ⊗ 1 We shall abridge this notation and simply write

a −b∗ a#b = , b a∗ omitting the projections and the tensor products by identity operators. One deﬁnes the exterior product in a similar way if the ﬁrst factor is equivariant. More generally, whenever we write an expression of the form a#b, we implicitly assume that one of the factors is equivariant, and depending on which of the factors is equivariant, we apply the corresponding deﬁnition. (If both symbols are equivariant, we can use either of the deﬁnitions; both give the same result.) It is easy to check that the exterior product of two elliptic symbols is elliptic, and consequently, homotopies of the factors in the corresponding classes of elliptic symbols result in homotopies of the exterior product.

6.3 Euler Operator and the Bott Element Let us construct the “Euler operator” in Rn , which we need. This operator will not coincide exactly with the classical Euler operator, since we wish this operator (unlike the classical Euler operator) to have order zero and be multiplicatively trivial at inﬁnity. Recall that the classical Euler operator on a Riemannian manifold M is deﬁned by the formula d + d∗ : C ∞ (M,

#ev

(M )) −→ C ∞ (M,

#odd

(M )).

(6.6)

It takes diﬀerential forms of even degree to diﬀerential forms of odd degree. Here d is the exterior derivative and d∗ is the adjoint with respect to the Riemannian volume form and the inner product on forms deﬁned by the Hodge ∗ operator. If M = Rn , then the symbol of the operator (6.6) is equal to σ(d + d∗ )(x, ξ) = iξ ∧ +(iξ∧)∗ :

#ev

(Cn ) −→

#odd

(Cn )

and is invertible for ξ = 0. Let us modify this operator and obtain a zero-order operator multiplicatively trivial at inﬁnity. To this end, we apply formula (4.2) to the matrix-valued function B(x, ξ) = (x + iξ) ∧ +((x + iξ)∧)∗ :

#ev

(Cn ) −→

#odd

(Cn ),

(6.7)

70

Chapter 6. Bott Periodicity

which is invertible for x2 + ξ 2 = 0. The resulting O(n)-equivariant symbol is denoted by Q(x, ξ). Now consider an O(n)-equivariant zero-order operator : L2 (Rn , #ev (Cn )) −→ L2 (Rn , #odd (Cn )) Q = 0, dim ker Q = 1 and moreover, the with symbol Q(x, ξ) such that dim coker Q kernel of Q consists of O(n)-invariant sections. To make the exposition self-contained, recall the construction of the operator Q (see [47], [13]). Denote by C some O(n)-equivariant operator with symbol Q(x, ξ). It is known that the equivariant index (see [10]) of this operator is equal to the unit element: 0 (pt). indO(n) C = 1 ∈ KO(n) Thus, there exists an equivariant surjection P : ker C −→ ker C ∗ , whose kernel is one-dimensional and O(n)-invariant. (Here we use the following fact: the set of ﬁnite-dimensional representations of a compact Lie group is a semigroup with cancellation property with respect to direct sum, i.e., the relation x ⊕ y = x ⊕ z for representations x, y, z implies that y = z; this property can be proved using decomposition of representations into irreducible representations.) = C + P has the Let us extend P by zero to (ker C)⊥ . Then the operator Q desired properties. thus constructed is called the Euler operator (in Deﬁnition 6.3. The operator Q n R ), while its symbol Q is called the Bott element.

6.4 Bott Mapping and the Periodicity Theorem Suppose we are given an orthogonal action of Γ on the space Rn . Let us deﬁne the Γ-module R2n as the direct sum of two copies of the Γ-module Rn and set A = C0 (R2n )Γ . One has the obvious isomorphism A = C0 (T ∗ Rn )Γ , where Γ acts on the cotangent bundle by diﬀerentials. Deﬁnition 6.4. The mapping β : K0 (Λ) −→ K0 (A) deﬁned by β[a] = [a#Q], where Q is the Bott element, is called the Bott mapping.

(6.8)

6.5. Proof of the Periodicity Theorem

71

Theorem 6.5 (Bott periodicity). The Bott mapping (6.8) is an isomorphism. The inverse mapping (6.9) β −1 = indΛ : K0 (A) −→ K0 (Λ) is equal to the Λ-Fredholm index on the space Rn . Corollary 6.6. An arbitrary element of the group K0 (Λ) is the Λ-index indΛ D of some nonlocal elliptic operator D. Namely, such operators can be constructed in RN as exterior products of projections over Λ∞ by the Euler operator.

6.5 Proof of the Periodicity Theorem 1. Let us prove that indΛ ◦β = 1. To this end, note that if [a] = [P ] ∈ K0 (Λ), where P is a projection over Λ∞ , then the class [a#Q] ∈ K0 (A) can be represented by the elliptic symbol 1N ⊗ Q, P ⊗ 1∧ev (Cn ) , P ⊗ 1∧odd (Cn ) , where we consider the tensor product in the sense of (6.5). Hence the Λ-index indΛ [a#Q] is equal to the Λ-index of the operator : (P ⊗ 1)L2 (Rn , ΛN ⊗ 1⊗Q

#ev

(Cn )) −→ (P ⊗ 1)L2 (Rn , ΛN ⊗

#odd

(Cn )). (6.10)

is trivial, and the kernel is one-dimensional and consists However, the cokernel of Q of Γ-invariant elements. Thus, the cokernel of the operator (6.10) is trivial, and the kernel is = Im P ⊗ ker Q Im P ⊂ ΛN . ker 1 ⊗ Q We obtain indΛ [a#Q] = [P ], as desired. 2. To prove that β ◦ indΛ = 1, let us use an auxiliary construction. Namely, consider the doubled space R2n = Rn × Rn and the auxiliary mapping β : K0 (A) −→ K0 (C0 (R4n )Γ ), [σ] −→ [σ#Q]. We deﬁne the index mapping indA : K0 (C0 (R4n )Γ ) −→ K0 (A) as follows. Let (x, ξ) be variables in R2n , and let (x, ξ, y, η) be variables in R4n . σ ] ∈ K0 (C0 (R4n )Γ ) contains a representative that is Each class [σ

72

Chapter 6. Bott Periodicity

1. Homogeneous of order zero in η for large |η|. 2. Equal for large x2 + ξ 2 + y 2 to the identity homomorphism of trivial bundles. (Such a representative can be constructed by using formula (4.2) and then by making suitable stable homotopies of projections.) Denote a representative with properties 1 and 2 by σ = (σ, P1 , P2 ). Now consider the symbol σ deﬁned using (5.1). This is a well-deﬁned nonlocal ∗ + elliptic symbol on R2n y,η in Hilbert modules over the C -algebra A obtained from + σ ] as the A -index of the corresponding A by adjoining the unit. We deﬁne indA [σ

. A priori this index lies in K0 (A+ ), but one can readily A+ -Fredholm operator σ

to the identity show that the homomorphism A+ → C, whose kernel is A, takes σ operator, whose index is zero, and hence σ ] ∈ K0 (A) ≡ ker K0 (A+ ) −→ K0 (C) . indA [σ By analogy with the case of the mapping β, the following formula is valid: indA ◦β = 1.

(6.11)

Its proof, however, is somewhat more clumsy. We shall not give it here, since it is similar to the proof of Theorem 7.7 in the next chapter. Let us now take an arbitrary class [a] ∈ K0 (A). Then, on the one hand, [a] = indA [a#Q] by (6.11). On the other hand, the element a#Q is homotopic within elliptic symbols to an element unitary equivalent to Q#a. Indeed, the homotopy σt = a(x cos t + y sin t, ξ cos t + η sin t)#Q(y cos t − x sin t, η cos t − ξ sin t) for t ∈ [0, π/2] takes a(x, ξ)#Q(y, η) to a(y, η)#Q(−x, −ξ), and then the 180◦ rotation in the (x, ξ)-plane takes it to the symbol unitary equivalent to Q#a. Lemma 6.7. indA [Q#a] = β indΛ [a]. This lemma will be proved below. By combining these results, we obtain [a] = indA [a#Q] = indA [Q#a] = β indΛ [a], as desired. The proof of the periodicity theorem is complete.

6.5. Proof of the Periodicity Theorem

73

Proof of Lemma 6.7. Consider a homogeneous representative a(y, η) in the class Q is realized as a homogeneous element [a] ∈ K0 (A). Then the Bott element equal to identity isomorphism for large x2 + ξ 2 . The exterior product Q#a can be represented by a homogeneous element equal to the identity isomorphism of trivial bundles for large x2 + ξ 2 :

Q(x, ξ) ⊗ 1 −χ(x, ξ) ⊗ a∗ (y, η) Q#a = , χ(x, ξ) ⊗ a(y, η) Q∗ (x, ξ) ⊗ 1 where χ(x, ξ) is a cutoﬀ function equal to 1 at the origin and 0 for large x2 + ξ 2 . Thus, according to the deﬁnition, $ % ∂ Q(x, ξ) ⊗ 1 −χ(x, ξ) ⊗ a∗ y, −i ∂y indA (Q#a) = indA+ , (6.12) ∂ Q∗ (x, ξ) χ(x, ξ) ⊗ a y, −i ∂y where we recall that the symbol a is constructed using symbol a as follows: if a = {a(g)}, then a = {a(g) ⊗ L(g)}, where L(g) is the operator of left translation on Γ. The operators

∂ ∂ a∗ y, −i A= a y, −i and A∗ = ∂y ∂y are treated here as operators over the C ∗ -algebra Λ. (And after tensor product by χ(x, ξ), we obtain operators over the C ∗ -algebra A.) The operator A is ΛFredholm; hence it is equal to the direct sum A = A1 ⊕ A2 , where the operator A1 is invertible and the operator A2 is acting between certain ﬁnitely-generated projective Λ-modules N1 and N2 . Moreover, indΛ a = [N1 ] − [N2 ] ∈ K0 (Λ). In its turn, the operator on the right-hand side in (6.12) (let us denote this operator by D) is also represented as a direct sum, D = D1 ⊕ D2 . The operator 1 ⊗ A1 is invertible. Indeed, if E(g)T (g) ⊗ L(g), A1 = then 1 ⊗ A1 =

E(g)T (g) ⊗ L(g) ⊗ ρ(g).

Let the inverse of A1 be A−1 1 =

E (g)T (g) ⊗ L(g);

then the operator 1 ⊗ A−1 1 =

E (g)T (g) ⊗ L(g) ⊗ ρ(g)

74

Chapter 6. Bott Periodicity

is the inverse of 1 ⊗ A1 . (This becomes obvious once we note that the coeﬃcients in the decomposition D= D(g)T (g) ⊗ L(g), where the D(g) are pseudodiﬀerential operators, are uniquely determined.) The invertibility of 1 ⊗ A1 implies the invertibility of D1 . Now the operator D2 can be connected by a homotopy within the class of A+ -Fredholm operators to the operator

Q(x, ξ) ⊗ 1N1 0 D2 = , 0 Q∗ (x, ξ) ⊗ 1N2 whose A+ -index is obviously equal to β([N1 ] − [N2 ]) = β(indΛ a). This completes the proof of the lemma.

Chapter 7

Direct Image and Index Formulas in K-Theory 7.1 Direct Image Mapping in K-Theory for Embeddings Let i : M ⊂ X be a Γ-embedding of Γ-manifolds. In this section, we construct the direct image mapping i! : K0 (C0 (T ∗ M )Γ ) −→ K0 (C0 (T ∗ X)Γ )

(7.1)

and prove that it preserves the Λ-index. The mapping (7.1) takes the symbol of a nonlocal elliptic operator on M to the symbol of a nonlocal elliptic operator on X. As in the classical theory of Atiyah and Singer, the new symbol is represented in a neighborhood of the submanifold M in X as the exterior product of the original operator by the Euler operator in the normal direction, and outside this neighborhood the new symbol is multiplicatively trivial. Note that the main complication compared with the construction of the Bott isomorphism, which was given in the previous chapter, is due to the fact that a tubular neighborhood of M in X is not in general the Cartesian product of M by Rcodim M but is a locally trivial bundle over M . Let us now proceed to a detailed description of the corresponding construction.

7.1.1 Exterior Products Let X be a manifold with an action of the group Γ, and let a = (a, P1 , P2 ) and b = (b, Q1 , Q2 ) be symbols over C(X)Γ and C(X), respectively, such that the latter symbol is equivariant.

76

Chapter 7. Direct Image and Index Formulas in K-Theory

Deﬁnition 7.1. The exterior product of the symbols a and b is the symbol

a −b∗ a#b = b a∗

a ⊗ 1 −1 ⊗ b∗ P1 ⊗ Q1 P2 ⊗ Q1 0 0 = , , . 1 ⊗ b a∗ ⊗ 1 0 P2 ⊗ Q2 0 P1 ⊗ Q2 Remark 7.2. This formula resembles that in Deﬁnition 6.2. However, there is a substantial diﬀerence between the two formulas: the factors in Deﬁnition 6.2 are symbols on two diﬀerent manifolds, and the product a#b is a symbol on the product of these manifolds, whereas now both factors and the product are symbols on the same manifold. The following two statements are useful when proving ellipticity of exterior products. Proposition 7.3. Let Y ⊂ X be a closed Γ-invariant subset on which the action of Γ is topologically free. If Y can be represented as a union of closed Γ-invariant sets on each of which either a or b is invertible and the action of Γ is topologically free, then the symbol a#b is invertible on Y . Proof. The invertibility of a#b is equivalent to the invertibility of (a#b)∗ (a#b) and (a#b)(a#b)∗ . The oﬀ-diagonal entries of the matrix a#b commute with the diagonal entries. Thus, (a#b)∗ (a#b) and (a#b)(a#b)∗ are diagonal matrices with diagonal entries equal to a∗ a + bb∗ and aa∗ + b∗ b. The remainder of the proof (taking into account Proposition 1.30) is obvious once we note that a sum of two positive elements in a C ∗ -algebra is invertible provided that one of the summands is invertible. Corollary 7.4. If, in addition to the assumptions of Proposition 7.3, the subset Y is represented as a union Y = K ∩ U of closed and open invariant subsets on each of which Γ acts topologically free, if a is homotopic to some element a in the class of elements invertible on K, and if b is homotopic to some element b in the class of equivariant elements invertible on any closed invariant subset in U , then a#b is homotopic to a #b in the class of elements invertible on Y , where the homotopy is just the exterior product of homotopies of the factors.

7.1.2 Normal Bundle Let E = N M be the normal bundle of the embedding i, let F be the ﬁber of E, and let dim F = n. Then F is equipped with an inner product (induced by the metric). Since the action of Γ on both M and X is isometric, E is naturally a Γ-bundle. Further, using the geodesic exponential mapping, one can identify a Γ-invariant tubular neighborhood (see Figure 7.1) U = {x ∈ X | dist(x, M ) < ε}

7.1. Direct Image Mapping in K-Theory for Embeddings

77

X

U F M Figure 7.1: Tubular neighborhood U of the submanifold M ⊂ X

of the zero section in E; this exponential of M in X with a neighborhood U mapping is a Γ-mapping. The homeomorphism

tan(π|ξ|/2ε)

U U −→ E, (x, ξ) −→ x, ξ |ξ| preserves Γ-structures and hence induces an isomorphism of algebras, C0 (T ∗ U )Γ C0 (T ∗ E)Γ .

(7.2)

Further, the embedding U ⊂ X induces the embedding C0 (T ∗ U )Γ ⊂ C0 (T ∗ X)Γ .

(7.3)

We shall deﬁne the mapping (7.1) as the composition of three mappings, j!

K0 (C0 (T ∗ M )Γ ) → K0 (C0 (T ∗ E)Γ ) K0 (C0 (T ∗ U )Γ ) → K0 (C0 (T ∗ X)Γ ). (Here j stands for the embedding of M in E as the zero section.) The second and third mappings in the composition are induced by the algebra homomorphisms (7.2) and (7.3). Clearly, these mappings preserve the analytical index. Thus, we have reduced the problem to that of deﬁning the mapping j! .

7.1.3 Deﬁnition of the Direct Image Mapping Let Q be the Bott element constructed in Sec. 6.3. In terms of this element, we deﬁne a vector bundle homomorphism over T ∗ F by setting Q : πF∗

#ev

(T ∗ F ) −→ πF∗

#odd

(T ∗ F ).

(7.4)

78

Chapter 7. Direct Image and Index Formulas in K-Theory

(Here πF : T ∗ F → F stands for the natural projection.) In the canonical coordinates (ξ, η), where ξ are the coordinates in F with respect to some orthonormal basis and η are the dual covariables, we set this element to be equal to the Bott element Q(ξ, η). This deﬁnition does not depend on the choice of the basis, since an orthogonal mapping in F induces a unitary transformation in the space T ∗ F F ⊕ F viewed as a complex vector space with operator of multiplication by i deﬁned by the symplectic structure. The homomorphism (7.4) is O(n)-equivariant (and hence Γ-equivariant). Using the projection onto the second factor in the orthogonal decomposition T ∗ E = π ∗ (T ∗ M ) ⊕ T ∗ F

(7.5)

(see (1.28)), we lift this homomorphism to T ∗ E. Now consider a class [σ] ∈ K0 (C0 (T ∗ M )Γ ). Using the projection onto the ﬁrst factor in the orthogonal decomposition (7.5), we lift σ to T ∗ E. Deﬁnition 7.5. The direct image of the element [σ] is the element j! ([σ]) = [σ#Q] ∈ K0 (C0 (T ∗ E)Γ ),

(7.6)

where the exterior product is described in Deﬁnition 7.1. Proposition 7.6. The direct image is well deﬁned. Proof. We should verify that the element σ#Q is invertible at inﬁnity in T ∗ E and its stable homotopy class does not depend on the choice of a representative σ in [σ]. Let σ = σ(x, p) be invertible for |p| > R. Consider the closed neighborhood K1 ∪ K2 of inﬁnity in T ∗ E formed by the closed Γ-invariant sets K1 = {|p| ≥ 2R},

K2 = {η 2 + ξ 2 ≥ 1}.

Then σ is invertible on the ﬁrst set and Q is invertible on the second set. Hence σ#Q is invertible on K1 ∪ K2 by Proposition 7.3. Now let σt be a homotopy of elements invertible at inﬁnity on T ∗ M ; then the same reasoning as above shows that σt #Q is a homotopy of elements invertible at inﬁnity in T ∗ E. Finally, if σ is trivial (invertible everywhere), then the same is true for σ#Q; hence stabilization of the ﬁrst factor does not change the class of the exterior product.

7.1.4 Λ-Index Is Preserved by the Direct Image Mapping Now let us state an important theorem, which will be proved in Sec. 7.3. Theorem 7.7. The following diagram commutes: K0 (C0 (T ∗ M )Γ ) OOO OOO OO indΛ OOO '

j!

K0 (Λ).

/ K0 (C0 (T ∗ E)Γ ) o ooo o o oo w oo indΛ o

(7.7)

7.2. Index Formulas in K-Theory

79

7.2 Index Formulas in K-Theory 7.2.1 Direct Image in K-Theory for the Projection into a Point In the previous section, the direct image in K-theory was deﬁned for embeddings. Here we use this result and Bott periodicity to deﬁne direct image for the projection into a one-point space, which will enable us to compute the Λ-index of nonlocal elliptic operators in K-theoretic terms. Let p : M −→ {pt} be the projection of M into a one-point space. We deﬁne the direct image mapping in K-theory, (7.8) p! : K0 (C0 (T ∗ M )Γ ) −→ K0 (Λ), as follows. It is well known that any Γ-manifold M can be embedded in a Γ-space RN with a linear action of Γ for suﬃciently large N [9]. Denote such an embedding by j : M → RN and set p! = β −1 ◦ j! . Since any two Γ-embeddings are homotopic (in the direct sum of the representation spaces), it follows that formula (7.8) gives a well-deﬁned mapping; i.e., it does not depend on the choice of the embedding.

7.2.2 Index Formulas The above results give us the following index theorem for nonlocal elliptic pseudodiﬀerential operators. Theorem 7.8 (on the Λ-Fredholm index). Let D be a nonlocal elliptic operator on a manifold M . Then (7.9) indΛ D = p! [σ(D)]. Corollary 7.9 (on the Fredholm index). Let D be a nonlocal elliptic operator on a manifold M . Then (7.10) inda D = α∗ (p! [σ(D)]), where α : Λ → C is the homomorphism corresponding to the trivial representation of the group Γ.

7.3 Proof of Theorem 7.7 We shall represent the K-theory classes occurring in the theorem by nonlocal Λelliptic operators and prove the equality of the analytic Λ-indices of these operators by an explicit computation.

80

Chapter 7. Direct Image and Index Formulas in K-Theory

7.3.1 Exterior Products of Operators Let us take [σ] ∈ K0 (C0 (T ∗ M )Γ ). In this subsection, we assume that there exists a matrix1 nonlocal elliptic symbol σ of order zero on T ∗ M that represents the class [σ]. Let D be a matrix nonlocal elliptic operator of order zero on M whose symbol is equal to σ

(see (5.1)). Without loss of generality, we can assume (adding to D a Λ-compact operator if necessary) that D is the direct sum of an invertible operator and a zero homomorphism between ﬁnitely generated projective Λ-modules. Further, let A be a family of zero-order Γ-invariant operators in the ﬁbers of E, multiplicatively trivial at inﬁnity, of index 1 and trivial cokernel, which is obtained if in each ﬁber F Rn we take the operator in the family to be equal to the Euler that was constructed in Sec. 6.3. Recall that its symbol is given by operator Q construction (4.2) applied to the function (6.7). Since this function is invertible everywhere except for the origin in T ∗ F R2n , the number R in this construction can be chosen arbitrarily small. Let us deﬁne the analytic exterior product D#a A as follows. Let ϕ(ξ) be a compactly supported real Γ-invariant function such that ϕ(ξ) = 1 in the domain |ξ| ≤ 3R/2, outside which A is multiplicatively trivial, and ϕ(ξ) = 0 in the domain |ξ| ≥ 2R. We choose a cover {Uj } of M by charts such thatthe bundle E is trivial over the union of any two charts. We write operator D = (T (g) ⊗ L(g))D(g) as D=

ψj Dψk ≡

Djk

where {ψj } is a partition of unity subordinate to the cover {Uj }. The operator Djk is then lifted to E. To this end, choose an orthonormal frame in Fx , x ∈ U , over U = Uj ∪ Uk and let ξ denote the coordinates in the ﬁbers Fx with respect to this frame. (They are naturally linear functions of the coordinates canonically dual to x.) These coordinates give a trivialization of the bundle E over U . We use this trivialization to lift the operator Djk to π −1 (U ), where π : E → M . (The operators T (g) are lifted to the operators T (g) = T (g) ⊗ (ρ(g)∗ )−1 where ρ(g)∗ stands for the shift operator corresponding to the action ρ of Γ on

jk . We set E.) Denote the lifted operator by D

= D

$

jk , D

D#a A =

ϕD A

−A∗

∗ ϕD

% .

jk , A∗ ] = 0.

jk , A] = 0, [ϕD Lemma 7.10. [ϕD 1 This

means that the symbol acts between sections of trivial bundles.

7.3. Proof of Theorem 7.7

81

Proof. It suﬃces to do the computation for separate terms in trivializing charts. In this case, the lifts T (g) of the shift operators T (g) commute with A, since A is O(n)-equivariant. In turn, the lift of a pseudodiﬀerential operator on M in a trivializing neighborhood commutes with A, because they act along disjoint sets of variables. Finally, the function ϕ commutes with all the operators occurring in the formula. This completes the proof of the lemma. Lemma 7.11. The operator D#a A is Λ-Fredholm. Moreover, indΛ D#a A = indΛ D. Proof. 1. Let us check the Λ-Fredholm property of D#a A. It suﬃces to show that (D#a A)∗ (D#a A) and (D#a A)(D#a A)∗ possess this property. For example, let us compute

∗D

+ A∗ A, ϕ2 DD∗ + AA∗ ). (D#a A)∗ (D#a A) = diag(ϕ2 D The operator AA∗ is strongly positive deﬁnite, which implies that the second diagonal entry is invertible. The operator A∗ A is strongly positive deﬁnite on the orthogonal complement of ker A. The restriction to the subspace ker A is equal to

∗ D)

∗D

=D

∗D

= (P + D

−P . ϕ2 D ker D ker D Here the ﬁrst term is strongly positive deﬁnite, and the second term is Λ-compact, which implies that (D#a A)∗ (D#a A) is Λ-Fredholm. The Λ-Fredholm property for (D#a A)(D#a A)∗ can be proved along the same lines. 2. Let us prove that the indices are equal. The commutator lemma gives

ker D#a A = ker A ∩ ker ϕD. Using the properties of A, one shows that ker A = L2 (M, ΛN ) ⊗ ker Q. If u ∈ ker A, then ϕu = u, and therefore, ker D#a A ker D ⊗ ker A. Thus, we obtain ker D#a A ker D. A similar proof gives the relation ker(D#a A)∗ ker D∗ .

82

Chapter 7. Direct Image and Index Formulas in K-Theory

7.3.2 Homotopy The operator D#a A is not a nonlocal pseudodiﬀerential operator on E.2 Let us make a homotopy of this operator in the class of Λ-Fredholm operators to a multiplicatively trivial pseudodiﬀerential operator on E whose symbol is invertible at inﬁnity. Let us choose smooth nonnegative Γ-invariant scalar symbols χ1 (x, ξ, p, η) and χ2 (x, ξ, p, η) on the cosphere bundle S ∗ E such that 1. χ21 + χ22 = 1 everywhere. 2. If |ξ| ≥ 2R, then χ1 = 0, χ2 = 1. 3. If |ξ| ≤ 3R/2, then χ1 = 0 for |p| ≤ |η|/2 and χ2 = 0 for |η| ≤ |p|/2. (The existence of symbols with these properties is easy to prove by averaging over the compact Lie group Γ. The coordinates (x, ξ, p, η), as well as (x, ξ, px , pξ ), on T ∗ E were deﬁned in Sec. 1.4.) By Φ1 and Φ2 we denote some self-adjoint Γinvariant pseudodiﬀerential operators on E with symbols χ1 and χ2 , respectively. Let us state and prove one technical proposition. Proposition 7.12. For R suﬃciently small,

1 and AΦ2 are nonlocal Λ-pseudodiﬀerential operators (i) The operators DΦ 2 N in Lloc (E, Λ ) modulo Λ-compact operators; the symbols of these operators are

1 ) = χ1 σ(D),

σ(DΦ

σ(AΦ2 ) = χ2 σ(A).

(ii) The commutators

[Φ1 , D],

[Φ1 , A],

[Φ2 , D],

[Φ2 , A]

are Λ-compact in the space L2loc (E, ΛN ). Proof. (a) It suﬃces to prove the statement for D with symbol in the dense subset MatN (C ∞ (S ∗ M )Γ ), since one can then pass to arbitrary symbols by continuity. On this dense set, the series over g ∈ Γ are rapidly converging, and the operators Φj commute with operators T (g). Thus, it remains to prove the statement only for the case in which D is a Λ-pseudodiﬀerential operator without shifts.

jk , which we Moreover, it suﬃces to prove the statement for a separate term D

in what follows, omitting the subscripts for short. denote by D (b) We carry out subsequent computations in local coordinates. To this end, we localize also the operators Φ1,2 using a partition of unity. 2 For example, the component A of this operator is a family of pseudodiﬀerential Euler operators of order zero in the ﬁbers. This family is not a pseudodiﬀerential operator on the total space. (Its symbol has a discontinuity.)

7.3. Proof of Theorem 7.7

83

is a Λ-pseudodiﬀerential operator (with compactly sup(c) The operator D ported Schwartz kernel) in Rn and hence can be exactly represented in the form 1

∂ 2

D = f x, −i , ∂x

where the function f (x, px ) is everywhere smooth, compactly supported in x, and asymptotically homogeneous of order zero in px . To be deﬁnite, let us prove that

1 is a Λ-pseudodiﬀerential operator on E. Take some summand in C in the DΦ representation of Φ1 as a sum over the partition of unity. Then this summand can be written in local coordinates in the form 1 1

∂ ∂ 2 2 C = g x, ξ, −i , −i , ∂x ∂ξ

where g(x, ξ, px , pξ ) is an everywhere smooth function of (px , pξ ) asymptotically homogeneous of order zero in (px , pξ ) and asymptotically homogeneous of order −1 with respect to the same variables in the cone |p| < |η|/2.

(7.11)

Using formulas (1.29), we obtain p = px and η = pξ − A(x, ξ)p, where A(x, ξ) ≤ L|ξ|. Hence in the domain |ξ| < 2R, which is alone of interest to us, the cone (7.11) contains the cone 1 |px | ≤ |pξ |. (7.12) 2(1 − LR) We shall suppose that R is chosen so small that LR < 1. Then the cone (7.12) is not empty and the function g(x, ξ, px , pξ ) is asymptotically homogeneous of order −1 in this cone.

(d) Let us compute the composition DC. By the composition formula of noncommutative analysis [60, p. 106], we obtain 1 1

∂ ∂ 2 2

DC = h ≡ h x, ξ, −i , −i , ∂x ∂ξ

where 1

∂ 2 [g(x, ξ, px , pξ )] h(x, ξ, px , pξ ) = f x, px − i ∂x

= f (x, px )g(x, ξ, px , pξ ) + R(x, ξ, px , pξ )

84

Chapter 7. Direct Image and Index Formulas in K-Theory

and the remainder term has the form 1 1 dτ fpx (x, px + τ p) g (p, ξ, px , pξ )eipx dp. R(x, ξ, px , pξ ) = (2π)n/2 0 Rn In the last formula, g (p, ξ, px , pξ ) stands for the Fourier transform of g(x, ξ, px , pξ ) with respect to x. It follows from the properties of the function g that the derivatives of the remainder term R can be estimated as (αβγδ) Rx,ξ,px ,pξ ≤ Cαβγδ (1 + |px | + |pξ |)−1 , which, in turn, means that the operator with symbol R is Λ-compact in L2 (E, ΛN ).

is a Λ-pseudodiﬀerential Hence, modulo Λ-compact operators, the operator DC operator with the desired symbol. A similar computation shows that the composi is a Λ-pseudodiﬀerential operator modulo a Λ-compact remainder term. tion C D

C] is Λ-compact. This implies the Λ-compactness of Hence the commutator [D,

Φ2 ],

Φ1 ]. To prove the Λ-compactness of the commutator [D, the commutator [D, we decompose the operator 1 − Φ2 instead of Φ2 , using a partition of unity. Then in a narrow cone around px = 0 its complete symbol is homogeneous of order −1, and the above reasoning applies. (e) For the operator AΦ2 and the commutators containing A, the reasoning is similar. The proof of Proposition 7.12 is complete. Let us deﬁne the operator $

A = Φ1 D D# a Φ2 A

−Φ2 A∗

∗ Φ1 D

% ,

called the smoothed exterior product. A is a Λ-pseudodiﬀerential operator on E with Lemma 7.13. The operator D # a smooth symbol equal to % $

1 −σ(A∗ )χ2 σ(D)χ σ(D #a A) = (7.13)

∗ )χ1 σ(A)χ2 σ(D on the cosphere bundle S ∗ (E). Proof. This follows from Proposition 7.12.

A are homotopic in the class of Lemma 7.14. The operators D#a A and D # a Λ-Fredholm operators. In particular, one has A) = indΛ (D#a A). indΛ (D # a

7.3. Proof of Theorem 7.7

85

Proof. Let us show that the operators A) + (D # A)∗ (D#a A), (D#a A)∗ (D # a a A)∗ + (D # A)(D#a A)∗ (D#a A)(D # a

a

deﬁne positive elements in the Λ-Calkin algebra (the quotient of the algebra of adjointable operators modulo the ideal of Λ-compact operators). Consider the ﬁrst element. Its summands are $ %$ % ∗ ∗ ∗

ϕ D DΦ A −A Φ 1 2 A) = (D#a A)∗ (D # a

∗ Φ1

−A ϕD AΦ2 D $ %

1 + A∗ AΦ2 −ϕD

∗ Φ1

∗ DΦ

∗ A∗ Φ2 + A∗ D ϕD = ,

1 + ϕDAΦ

D

∗ Φ1 −ADΦ AA∗ Φ2 + ϕD 2 %$ $ % ∗ ∗ ∗

ϕ D −A D Φ Φ A 1 2 A)∗ (D#a A) = (D # a

∗ −Φ2 A Φ1 D A ϕD % $

∗ Dϕ

+ Φ2 A∗ A −Φ1 D

∗ A∗ + Φ2 A∗ D

∗ϕ Φ1 D . =

D

∗ϕ

+ Φ1 DA Φ2 AA∗ + Φ1 D −Φ2 ADϕ By summing the last two expressions in the Calkin algebra, we obtain A) + (D # A)∗ (D#a A) (D#a A)∗ (D # a a $ %

∗ DΦ

1 + A∗ AΦ2 ) 2(ϕD 0 =

D

∗ Φ1 + AA∗ Φ2 ) . 0 2(ϕD The oﬀ-diagonal terms cancel, since the operators Φ1,2 commute in the Calkin

and D

∗ commute with A and

D

∗ , A, and A∗ , while the operators D algebra with D, A∗ . The latter matrix is obviously nonnegative, being a sum of two nonnegative matrices. In a similar way, one proves that the operator A)∗ + (D # A)(D#a A)∗ (D#a A)(D # a a deﬁnes a nonnegative element in the Λ-Calkin algebra. Now the following lemma completes the proof of Lemma 7.14.

Lemma 7.15 (cf. Proposition 17.2.7 in [22]). Let U and V be Λ-Fredholm operators such that the operators U ∗V + V ∗U

UV ∗ + V U∗

deﬁne nonnegative elements in the Λ-Calkin algebra. Then the operators U and V are homotopic in the class of Λ-Fredholm operators. In particular, indΛ U = indΛ V.

86

Chapter 7. Direct Image and Index Formulas in K-Theory

Proof of Lemma 7.15. 1. We know that an adjointable operator is Λ-Fredholm if and only if its image in the Λ-Calkin algebra is invertible. (See item 2 in Theorem 3.21.) This implies the following criterion for the Λ-Fredholm property. An adjointable operator W is Λ-Fredholm if and only if for some ε > 0 the operators W W ∗ − ε1 and W ∗ W − ε1 deﬁne nonnegative elements in the Λ-Calkin algebra. 2. Now consider the linear homotopy Wt = U t + V (1 − t),

t ∈ [0, 1].

We claim that Wt is Λ-Fredholm for all parameter values. Indeed, Wt Wt∗ = U U ∗ t2 + V V ∗ (1 − t)2 + (U V ∗ + V U ∗ )t(1 − t). The last summand is nonnegative in the Λ-Calkin algebra. We apply the Λ-Fredholm property criterion to the operators U and V and see that for small ε > 0 the operator Wt Wt∗ deﬁnes an element in the Λ-Calkin algebra equal to the sum of ε1 and some nonnegative element. 3. The composition Wt∗ Wt can be considered in a similar way. Hence by the criterion we ﬁnd that Wt is Λ-Fredholm.

7.3.3 Completion of the Proof A) repUsing Proposition 7.3 and Corollary 7.4, it is easy to show that σ(D # a resents the class j! [σ(D)]. On the other hand, by Lemmas 7.11 and 7.14, we have the following equalities for analytic indices: A = indΛ j! [σ(D)]. indΛ D = indΛ D#a A = indΛ D # a Thus, we have proved Theorem 7.7 for classes [σ] that have a matrix representative. Let us prove the theorem for general classes. Let B be an elliptic pseudodiﬀerential operator on the circle acting on func be the Γ-bundle over M × S1 equal tions and such that ind B = 1. Further, let E 1 to the pullback of E → M . (On S , we consider the trivial action of Γ.) It is clear

= E ×S1 (as total spaces), while for the above-mentioned orthogonal direct that E sum decomposition of T ∗ E and for the corresponding decomposition of the bundle

one has (we omit all kinds of π ∗ ) T ∗E

T ∗ (M × S1 ) ⊕ T ∗ F T ∗E = T ∗ (M ) ⊕ T ∗ S1 ⊕ T ∗ F = T ∗ (M ) ⊕ T ∗ F ⊕ T ∗ S1 T ∗ E ⊕ T ∗ S1 , (7.14) α

where the left- and rightmost establish these orthogonal decompositions and the mapping α permutes the second and third components in the sum. Now let [σ] ∈ K0 (C0 (T ∗ M )Γ . Then indΛ [σ] = indΛ [σ#σ(B)].

7.3. Proof of Theorem 7.7

87

Indeed, by analogy with the above, consider the exterior product A#B, where A is an elliptic operator with symbol σ

. Since M × S1 is a direct product, the operator A can be lifted to this product globally, and no choice of partition of unity is necessary. (It is precisely the appearance of a partition of unity in trivializing charts of E that did not allow us to lift the operator from M to E as an operator intertwining nontrivial projections.) As earlier, it turns out that indΛ A#B = indΛ A. Using homotopies of Lemma 7.15, one can show that indΛ A#B = indΛ [σ#σ(B)]. Further, the symbol σ#σ(B) is an endomorphism (since the projections intertwined by it coincide). Adding to it the complementary projection P as a direct summand, we obtain a matrix symbol (σ#σ(B)) ⊕ P that lies in the same stable homotopy class and satisﬁes indΛ [(σ#σ(B))] = indΛ [(σ#σ(B)) ⊕ P ]. Applying the result for matrix operators, we obtain indΛ [(σ#σ(B)) ⊕ P ] = indΛ [((σ#σ(B)) ⊕ P )#Q]. But (∼ denotes equality in K-theory) ((σ#σ(B)) ⊕ P )#Q ∼ ((σ#σ(B))#Q) ⊕ (P #Q) α

∼ (σ#σ(B))#Q ∼ (σ#Q)#σ(B). We ﬁnally obtain indΛ [σ] = indΛ [(σ#Q)#σ(B)] = indΛ [(σ#Q)] ≡ indΛ j! [σ]. (Here we have again used the exterior product by B.) This ﬁnishes the proof of Theorem 7.7.

Chapter 8

Chern Character The main ingredients of the cohomological formula for the Fredholm index of nonlocal elliptic operators are the Γ-Todd class of the complexiﬁed cotangent bundle of the manifold M and the Chern character of the symbol. (Both are collections of cohomology classes deﬁned on cotangent bundles of ﬁxed-point manifolds Mg of the action of Γ.) In this chapter, we describe the Chern character. First, let us give a general construction of the Chern character for projections over the algebra of nonlocal functions. In Secs. 8.1 and 8.2, we assume that Γ is a countable dense subgroup of polynomial growth in a compact Lie group Γ of isometries of a compact Riemannian manifold, which we denote in these sections by X.

8.1 Diﬀerential Forms and Graded Traces 8.1.1 Noncommutative Diﬀerential Forms #∗ Let (X) be the algebra of diﬀerential forms on a manifold X. Then Γ acts on forms by # the induced mappings g ∗ , which#permits us to deﬁne the smooth crossed ∗ ∗ product (X)Γ . Its elements are just (X)-valued functions # f (g) on Γ with rapid decay as |g| → ∞ in the Fr´echet topology on the space ∗ (X), and the product is deﬁned by the formula1 (f1 f2 )(g) =

[l∗ (f1 (k))] ∧ f2 (l).

(8.1)

k,l∈Γ kl=g 1 Here

we denote the exterior product on forms by ∧. However, we omit this sign in what follows and write ω1 ω2 instead of ω1 ∧ ω2 ; this will not lead to confusion.

90

Chapter 8. Chern Character

#∗ (X)Γ is naturally a diﬀerential graded algebra over the algebra The algebra C ∞ (X)Γ . Indeed, there is a natural grading #

(X)Γ =

n #j

(X)Γ

j=0

by degree of diﬀerential forms, C ∞ (X)Γ coincides with the component of degree zero, #0 (X)Γ = C ∞ (X)Γ , #∗ (X)Γ ; it is given by and ﬁnally, there is a diﬀerential on d:

#j

(X)Γ −→

#j+1

(X)Γ ,

(df )(g) = d(f (g)).

(The veriﬁcation of Leibniz’s rule is elementary.) #∗ Note also that for each positive integer N the matrix algebra MatN ( (X)Γ ) is a diﬀerential graded algebra over the algebra MatN (C ∞ (X)Γ ).

8.1.2 Graded Trace on the Algebra of Noncommutative Diﬀerential Forms In each conjugacy class of Γ, we arbitrarily choose some representative g0 and denote the corresponding class by g0 .2 Let us deﬁne a mapping #∗ # τ : MatN ( ∗ (X)Γ ) −→ (Xg0 ) (8.2)

g0

(where Xg0 is the ﬁxed point manifold of the diﬀeomorphism g0 ) that has the following properties: (i) τ is linear and preserves the grading.

#∗ (ii) τ is a graded trace on the algebra MatN ( (X)Γ ), which means that τ [ω1 , ω2 ] = 0, #∗ for all ω1 , ω2 ∈ MatN ( (X)Γ ), where [ · , · ] stands for the supercommutator [ω1 , ω2 ] = ω1 ω2 − (−1)deg ω1 deg ω2 ω2 ω1 . (iii) The trace τ is closed, i.e., commutes with the exterior diﬀerential: τ (dω) = d[τ (ω)]. 2 In

the following, g0 always denotes such representatives, while notation like etc. means sums, products, etc. over the set of all conjugacy classes of Γ.

g0 ,

&

g0 ,

8.1. Diﬀerential Forms and Graded Traces

91

A mapping satisfying these three properties is called a closed graded trace on the # algebra MatN ( ∗ (X))Γ . To deﬁne the desired τ , we introduce some additional notation. Let Cg0 be the centralizer of g0 in the Lie group Γ: Cg0 = {h ∈ Γ : hg0 h−1 = g0 }. It is a closed Lie subgroup in Γ equipped with a normalized bi-invariant Haar measure. For each g ∈ g0 , consider the set Γg0 ,g of elements h ∈ Γ conjugating g0 and g, i.e., satisfying the relation hg0 h−1 = g. This set is a compact submanifold (a left coset of the subgroup Cg0 in Γ); namely, Γg0 ,g = kCg0 , where k ∈ Γg0 ,g is arbitrary. This space is equipped with the pullback of the Haar measure on Cg0 via the diﬀeomorphism of left multiplication by k. (By the invariance of the Haar measure the result does not depend on the choice of k.) We denote this measure by dh. By construction, each h ∈ Γg0 ,g gives a diﬀeomorphism h : Xg0 → Xg . Now let us deﬁne the mapping (8.2) as #∗ [τ f ](g0 ) = tr [h∗ f (g)]Xg dh, (8.3) f ∈ MatN ( (X))Γ . g∈ g0

0

Γg0 ,g

Here ω|Xg0 is the restriction of ω to Xg0 (i.e., its preimage under the embedding Xg0 ⊂ X), and tr stands for the matrix trace in CN . Proposition 8.1. The mapping (8.2), (8.3) is a well-deﬁned closed graded trace on # MatN ( ∗ (X))Γ . Proof. The convergence of the series follows from the rapid decay property for f (g) as |g| → ∞ and the polynomial growth of Γ. Conditions (i) and (iii) are satisﬁed, which is straightforward from the formula deﬁning τ . Let us check the graded trace property (ii). According to (8.1), we obtain ' ( τ (f1 f2 )(g0 ) = tr [h∗ l∗ f1 (k)]Xg [h∗ f2 (l)]Xg dh. (8.4) kl∈ g0

0

Γg0 ,kl

Note that for h ∈ Γg0 ,kl one has

[h∗ l∗ f1 (k)]X

g0

0

= [h∗ (k ∗ )−1 f1 (k)]X . g0

Indeed, h−1 klh = g0 , and therefore, h∗ l∗ = (h−1 klh)∗ h∗ (k ∗ )−1 = g0∗ h∗ (k ∗ )−1 ,

(8.5)

92

Chapter 8. Chern Character

while the mapping g0 is the identity mapping on Xg0 . Let us substitute (8.5) into (8.4) and perform the change of variable h = kh in the integral. Then h ∈ Γg0 ,lk . Since the conditions kl ∈ g0 and lk ∈ g0 are equivalent, we obtain ' ( dh . (8.6) τ (f1 f2 )(g0 ) = tr [(h )∗ f1 (k)]X [(h )∗ k ∗ f2 (l)]X lk∈ g0

g0

Γg0 ,lk

g0

Transposing the factors under the trace sign and taking into account their degrees, we see that τ (f1 f2 )(g0 ) = (−1)deg f1 deg f2 τ (f2 f1 )(g0 ).

This completes the proof of the proposition.

Example 8.2. If the action of Γ on X is free, then the trace is identically zero unless g0 = e. For g0 = e, the formula for the trace is #∗ τ (f )(e) = tr[h∗ f (e)] dh, f ∈ MatN ( (X))Γ ; (8.7) Γ

i.e., τ (f )(e) can be obtained as follows: one averages the trace of the value of a function at the unit element e ∈ Γ over Γ.

8.2 Chern Character of Projections Once we have a diﬀerential graded algebra with a closed graded trace on it, we can deﬁne the Chern character of projections over C ∞ (X)Γ in a standard way. Let p be a projection over the algebra C ∞ (X)Γ (i.e., a projection in the algebra MatN (C ∞ (X)Γ ) for some N ). Deﬁnition 8.3. The Chern character of the projection p is the collection of diﬀerential forms )

* #ev dp dp (Xg0 ). chΓ p = τ p exp − ∈ 2πi

g0

Proposition 8.4. The form chΓ p is closed, and its cohomology class is determined by the class of the projection p in K0 (C ∞ (X)Γ ). Proof. The components of chΓ p are equal, modulo some numerical factors, to ωn = τ [p(dp)2n ],

n = 0, 1, 2, . . . .

To prove the desired properties, we mention the following identities dp = p dp + dp p,

(8.8)

p dp p = 0,

(8.9)

2

2

p(dp) = (dp) p,

(8.10)

8.2. Chern Character of Projections

93

of which the ﬁrst is obtained by diﬀerentiation of the identity p2 = p and the second and third follow from the ﬁrst. 1. First, let us show that ωn is closed. Since τ commutes with the diﬀerential (property (iii)), we have dωn = τ [(dp)2n+1 ]. We substitute the expression for dp given in (8.8) into the latter equation. For n = 0, by the cyclic invariance of the trace (property (ii)) and Eq. (8.9), we obtain dω0 = τ [p dp + dp p] = τ [p2 dp + dp p2 ] = τ [2p dp p] = 0. For n > 0, we multiply all the terms and obtain the trace of a sum of products of 2n + 1 factors each of which is either p dp or dp p. Since the number of factors is odd, there exists (possibly after a cyclic permutation, which does not change the value of the trace) a pair of neighboring factors that coincide, and then the product is zero by (8.9). 2. Let us establish the homotopy invariance of the cohomology class. We should show that the cohomology classes of the forms ωn do not change (a) under direct sum of the projection p with a trivial projection and (b) under homotopies of projections. The invariance in (a) is obvious, since the trace τ is additive for direct sums. To prove the invariance in (b), we note that it suﬃces to consider smooth homotopies. Any smooth homotopy can be represented in the form pt = Ut p0 Ut−1 , where Ut is a smooth family of invertible elements and U0 = 1. Let ωn = ωn (t) be the form corresponding to pt . It suﬃces to show that ∂ωn /∂t is exact for all t. We set ) * ∂Ut −1 Ut A= . p = pt0 , ∂t t=t0 Then ∂pt /∂t = [A, p] at t = t0 , and + , 2n ∂ωn 2n j−1 2n−j = τ [A, p](dp) + p(dp) d([A, p])(dp) . ∂t t=t0 j=1

(8.11)

Note that d([A, p]) = [A, dp] + [dA, p]. Substituting this into the right-hand side of (8.11), we see that the sum of terms containing the commutator [A, dp] is equal to τ ([A, ωn ]) = 0; hence it contains only terms where [dA, p] occurs. By the cyclic invariance of the trace and (8.10), we obtain ∂ωn = nτ (p[dA, p] dp + p dp[dA, p])(dp)2n−2 ∂t t=t0 (8.12) 2n−2 = nτ (p dp dA − p dA dp)(dp) .

94

Chapter 8. Chern Character

(The remaining terms are zero by cyclic invariance, (8.9), and (8.10).) Finally, by cyclic permutations we place dp in the second summand into the leftmost place (producing a sign change) and obtain ∂ωn = nτ (p dp dA + dpp dA)(dp)2n−2 ∂t t=t0 = nτ dp dA(dp)2n−2 = −d[nτ (A(dp)2n−1 )].

This completes the proof of the proposition. Now recall that C ∞ (X)Γ is a dense local subalgebra in C(X)Γ and hence K0 (C(X)Γ ) = K0 (C ∞ (X)Γ ). Thus, we have deﬁned the Chern character chΓ : K0 (C(X)Γ ) −→

H ev (Xg0 , C).

(8.13)

g0

Example 8.5. Let the group Γ = Z act on the circle X = S1 by rotations by angles that are integer multiples of 2π/θ, where θ is an irrational number. In this case, the action extends to the action of the circle Γ = S1 . Let us compute the range of the Chern character (8.13) chZ : K0 (C(S1 )Z ) −→ H ev (S1 , C) = H 0 (S1 , C) = C. It is well known (see [64]) that the K-group of the noncommutative torus C(S1 )Z is generated by the class of the unit element 1 and the Rieﬀel projection pθ (see Eq. (1.10)). The Chern characters of these elements are dϕ = 1, chZ (1) = S1

chZ (pθ ) =

θ{1/θ}

dϕ = 1 − θ[1/θ],

f (ϕ)dϕ = S1

0

where [x] ∈ Z and {x} ∈ [0, 1) stand for the integer and fractional part of a real number x. This computation shows that in this example the range of the Chern character is equal to the subgroup Z + θZ ⊂ C.

8.3 Chern Character of Symbols Let D = (D, P1 , P2 ) be a nonlocal pseudodiﬀerential operator in spaces deﬁned by projections P1 and P2 . The Chern character of its symbol σ(D) = (σ(D), P1 , P2 )

8.4. How to Compute the Chern Character

95

can be deﬁned as follows. Let us assume that the symbol is deﬁned over T ∗ M . Then, by Proposition 4.1, the stable homotopy class of the symbol has a representative of the form (1, Q1 , Q2 ), where Q1 and Q2 are projections that coincide at inﬁnity and are equal there to a trivial projection and 1 stands for the identity mapping. We set Hcev (T ∗ Mg0 , C). (8.14) chΓ σ(D) = chΓ Q1 − chΓ Q2 ∈

g0

(The right-hand side consists of well-deﬁned compactly supported cohomology classes.) Remark 8.6. Clearly, this construction is just the standard extension of the Chern character (8.13) to the case of the noncompact space X = T ∗ M . Thus, we obtain the Chern character mapping chΓ : K0 (C0 (T ∗ M )Γ ) −→ Hcev (T ∗ Mg0 , C),

g0 (8.15) [σ(D)] −→ chΓ σ(D).

8.4 How to Compute the Chern Character 8.4.1 Computation in Terms of the Symbol The deﬁnition of Chern character in Sec. 8.3 can be made more explicit if one writes out some speciﬁc expression for the homotopies that occur in the deﬁnition. Let us give such homotopies. Let us assume the symbol σ(D) = (σ(D), P1 , P2 ) to be homogeneous of order zero on the cotangent bundle T ∗ M \ 0 minus the zero section. Let ψ = ψ(|ξ|) ∈ C ∞ (T ∗ M ) be a nondecreasing Γ-invariant function equal to −π/2 for small |ξ| and +π/2 for large |ξ|. Proposition 8.7. The Chern character of [σ(D)] is equal to )

* 0 0 , chΓ [σ(D)] = chΓ [p] − chΓ 0 P2 where p is the matrix projection over C ∞ (T ∗ M )Γ given by

1 (1 − sin ψ)P1 σ −1 (D) cos ψ p= . σ(D) cos ψ (1 + sin ψ)P2 2

(8.16)

(8.17)

Proof. To prove the equivalence of (8.14) and (8.16), it suﬃces to note that both expressions are invariant under homotopies of the symbol. Using stabilization, we can assume that P2 is a trivial projection. In this case, one can insert Q1 = p and Q2 = diag(0, P2 ) into Eq. (8.14).

96

Chapter 8. Chern Character

8.4.2 Computation in Terms of Connections In applications, one naturally deals with operators acting on sections of classical vector bundles. In this case, it is natural to represent the Chern character in terms of connections in these bundles. Let us describe, without proof, how this is done. Graded Trace Let E ∈ Vect(X) be a classical vector bundle3 over a compact Γ-manifold X. #∗ Let us deﬁne the algebra (X, End E)Γ of nonlocal End E-valued forms on the manifold X as #∗ #∗ (X, End E)Γ = PE MatN ( (X)Γ )PE , where PE is the projection onto the subbundle E ⊂ CN . Let us also #∗note that, by analogy with the case of matrix operators, a nonlocal form ω ∈ (X, End E)Γ can be viewed as a rapidly decaying function on Γ whose value ω(g) at a point ∞ ∗ g ∈ Γ lies in the space #∗ C (X, Hom(E, g E)). The algebra (X, End E)Γ is equipped with a graded trace #∗ # τE : ∗ (X, End E)Γ −→ (Xg0 ) (8.18)

g0 ⊂Γ

deﬁned as

τE (ω, g0 ) =

g∈ g0

h∗ tr(ω(g)|Xg )dh.

Γg0 ,g

This is well deﬁned. Indeed, since g|Xg = id, we see that the restriction of ω(g) to Xg is an End E|Xg -valued form on Xg . The trace tr in the latter formula is the ﬁberwise (matrix) trace of an endomorphism of the vector bundle E. Chern Character of Projections Let us choose some connection ∇E :

#∗

(X, E) −→

#∗

(X, E)

in E. Let p ∈ C ∞ (X, End E)Γ be a projection. Consider the nonlocal ﬁrst-order diﬀerential operator #∗ #∗ (X, E) −→ (X, E). (8.19) ∇ = p∇E p : The operator Ω = ∇2 :

#∗

(X, E) −→

#∗

(X, E)

is the operator of multiplication by a nonlocal 2-form # Ω ∈ 2 (X, End E)Γ , 3 We

do not assume that Γ acts on E.

8.4. How to Compute the Chern Character

97

R 2B ∗ M ξ

T ∗M

ψ S∗M

Figure 8.1: Atiyah–Bott–Patodi space 2B ∗ M with coordinates ψ ∈ (−π/2, π/2), ξ ∈ S∗M which is called the curvature form of the projection p and the connection ∇E . The Chern character of the projection p ∈ C ∞ (X, End E)Γ is the following collection of diﬀerential forms:

#ev Ω chΓ p = τE p exp − (Xg0 ). (8.20) ∈ 2πi

g0 ⊂Γ

By the graded trace property and the identity dτE (A) = τE ([∇, A]) , # which holds for an arbitrary A ∈ ∗ (X, End E)Γ such that pA = A = Ap, one can show, using the standard techniques of the theory of characteristic classes, that the form chΓ p is closed and its cohomology class does not depend on the choice of a connection in E and is determined by the class of p in K0 (C ∞ (X, End E)Γ ) alone. Chern Character of Symbols Now let σ(D) ∈ C ∞ (S ∗ M, Hom(π ∗ E, π ∗ F ))Γ be the symbol of a nonlocal elliptic operator. (Here π : S ∗ M → M .) Then the Chern character of σ(D) is deﬁned in terms of the trace τE and connections in E

98

Chapter 8. Chern Character

and F by the formula ) 0 chΓ [σ(D)] = chΓ [p] − chΓ 0 where

1 p= 2

(1 − sin ψ)1π∗ E σ(D) cos ψ

0

*

1π ∗ F

σ −1 (D) cos ψ (1 + sin ψ)1π∗ F

,

(8.21)

(cf. the projection in (8.17)) is an element of the algebra C ∞ (T ∗ M , End(π ∗ E ⊕ π ∗ F ))Γ . Here T ∗ M = 2B ∗ M (8.22) is the Atiyah–Bott–Patodi compactiﬁcation of the cotangent bundle T ∗ M . One obtains the space 2B ∗ M by ﬁberwise one-point compactiﬁcation of T ∗ M , i.e., by adding a point at inﬁnity to each ﬁber; see Fig. 8.1. The notation hints at the second representation of this compactiﬁcation: it can be obtained by gluing two copies of the unit ball bundle in T ∗ M along the boundary. The action of Γ naturally extends to 2B ∗ M .

Chapter 9

Cohomological Index Formula 9.1 Todd Class Apart from the Chern character of the symbol of an elliptic operator, the second main ingredient of our cohomological index formula is the Todd class Td(T M ⊗ C; Γ) of the complexiﬁed tangent bundle of the manifold M with a Γ-action. Let us construct this class. It belongs to the product -

H ev (Mg0 , C)

g0

of even degree cohomology groups of the ﬁxed point submanifolds Mg0 , where g0 runs over representatives of all conjugacy classes in Γ. (Recall that the ﬁxed point set of an isometry is a smooth submanifold; see Sec. 1.2.) To deﬁne this element, consider the normal bundle N Mg0 . Since g0 preserves the metric, its diﬀerential dg0 preserves the ﬁbers of N Mg0 and therefore induces a well-deﬁned automorphism dg0 : N Mg0 −→ N Mg0

(9.1)

# #even #odd (N Mg0 ) = (N Mg0 ) ⊕ (N Mg0 )

(9.2)

of this bundle. Let

be the Z2 -graded exterior algebra bundle of N Mg0 . The automorphism (9.1) extends to this bundle in a standard way. Let Ω be the curvature form of the connection induced in the bundle (9.2) by the restriction to N Mg0 of the Riemannian connection on the tangent bundle T M . Following Atiyah and Singer [11], we deﬁne the Chern character localized at g0 ∈ Γ, # ch (N Mg0 ⊗ C)(g0 ) ∈ H ev (Mg0 , C),

(9.3)

100

Chapter 9. Cohomological Index Formula

of the complexiﬁcation of the Z2 -graded bundle (9.2) as the cohomology class of the form

# 1 Ω , (9.4) ch (N Mg0 ⊗ C)(g0 ) = str dg0 exp − 2πi where str stands for the ﬁberwise supertrace of endomorphisms of a Z2 -graded vector bundle. This cohomology class rewritten without mentioning connections. # can readily be# Namely, the bundles ev (N Mg0 ) and odd (N Mg0 ) can be decomposed as direct sums #ev #odd (N Mg0 ) Eλj , (N Mg0 ) Fμk μk

λj

of the eigensubbundles of the diﬀerential dg0 :

# # (N Mg0 ) → (N Mg0 )

corresponding to eigenvalues λj and μk , respectively; i.e., dg0 |Ej = λj · 1, Then

dg0 |Fk = μk · 1.

# ch (N Mg0 )(g0 ) = λj ch Eλj − μk ch Fμk . λj

(9.5)

μk

This expression will be useful in the proof of the invertibility of the Chern character (see below). # Note that the zero component of the Chern character ch (N Mg0 ⊗ C)(g0 ) is invertible [9], and thus the entire Chern character is invertible (both as an element of the algebra of diﬀerential forms and as an element of the cohomology ring). It is clear that a cohomology class is invertible (as an element of the cohomology ring) if and only if its component of degree zero is invertible. Hence it suﬃces to compute this component at some point x ∈ Mg0 . At this point, the action of the diﬀerential dg0 in the normal plane is a direct sum of reﬂections and rotations by nonzero angles # in 2-dimensional planes. Thus, the component of degree zero of the class ch (N Mg0 )(g0 ) is the product of the corresponding expressions for the components of the direct sums. Therefore, to prove the invertibility, one has to consider two examples, namely, reﬂection and rotation in a 2-dimensional plane. For the reﬂection α : R → R, α(x) = −x, using (9.5), we ﬁnd that ch0 (R)(α) = tr∧0 (R) α − tr∧1 (R) α = 1 − (−1) = 2 = 0.

9.2. Index Theorem

101

(In these computations, the zero-degree component of the Chern character is denoted by ch0 .) For the rotation by an angle ϕ, β : R2 −→ R2 , (x, y) −→ (x cos ϕ − y sin ϕ, y cos ϕ + x sin ϕ), we obtain ch0 (R2 )(β) = tr∧0 (R2 ) β−tr∧1 (R2 ) β+tr∧2 (R2 ) β = 1−2 cos ϕ+1 = 2(1−cos ϕ) = 0, as desired. Now the component of the Todd class localized at the conjugacy class g0 is deﬁned as Td(T Mg0 ⊗ C) # ∈ H ev (Mg0 , C), (9.6) Td(T M ⊗ C; Γ)(g0 ) = ch (N Mg0 ⊗ C)(g0 ) where the denominator is just the usual Todd class of the complexiﬁed tangent bundle of Mg0 . (This expression was introduced by Atiyah and Singer [11]; we call it the Todd class following Baum and Connes [15].) Remark 9.1. If g ∈ g0 , then the ﬁxed point manifolds Mg0 and Mg are diﬀeomorphically mapped onto each other by diﬀeomorphisms conjugating g and g0 . One can readily show that in the cohomology these diﬀeomorphisms induce mappings that take Td(T M ⊗ C; Γ)(g0 ) to Td(T M ⊗ C; Γ)(g). This shows that the speciﬁc choice of representatives g0 of the conjugacy classes is irrelevant.

9.2 Index Theorem 9.2.1 Topological Index Let D be a nonlocal elliptic operator on M . We deﬁne its topological index indt D ∈ C| Γ|

(9.7)

(this index ranges in the product of |Γ| copies of C labeled by conjugacy classes of Γ) by the formula (indt D)(g0 ) = chΓ [σ(D)](g0 ) Td(T M ⊗ C; Γ)(g0 ), [T ∗ Mg0 ; Γ] ,

(9.8)

where g0 runs over the set of representatives of conjugacy classes of Γ, [T ∗ Mg0 ; Γ] ∈ Hev,c (T ∗ Mg0 ) is the fundamental class (here and in the following, we equip the cotangent bundles with the orientation given by the following ordering of canonical coordinates: ξ1 , x1 , . . . , ξn , xn ), the Todd class is lifted from Mg0 to T ∗ Mg0 via the natural projection, and the brackets denote the natural pairing of homology and cohomology. In the following subsections, we establish the relation between the topological index and the analytic invariants introduced earlier.

102

Chapter 9. Cohomological Index Formula

9.2.2 Chern Character on the Group K0 (Λ) Let Γ be the set of conjugacy classes in Γ. Here we deﬁne the Chern character chΛ : K0 (Λ) −→ C| Γ| .

(9.9)

Informally speaking, we extend deﬁnition (8.15) of the Chern character on the group K0 (C0 (T ∗ M )Γ ) to the case in which M is a point. The extension considered here is far from being the most general. Actually, we deﬁne only the “degree zero” component of the Chern character, having in mind that this suﬃces for computing the Fredholm index of an elliptic operator. A more general construction is considered in Chap. 10. The mapping (9.9) is constructed as follows. The group K0 (Λ) is generated by equivalence classes of projections over the algebra Λ. Since the algebra Λ∞ of rapidly decaying functions on Γ is a local subalgebra in Λ (this is a special case of Theorem 1.23), each such class has a representative given by a formal diﬀerence of projections over Λ∞ . Let P = P (g) be such a projection. For γ ∈ Γ, deﬁne tr P (g) (9.10) chγ (P ) = g∈γ

and set chΛ [P ] = {chγ (P )}γ∈ Γ . This is well deﬁned, since the mapping τγ : Λ∞ −→ C,

a −→

(9.11)

a(g)

g∈γ

is a trace on Λ∞ , and consequently, Eqs. (9.10) and (9.11) deﬁne homotopy invariants of projections.

9.2.3 Statement of the Theorem The main theorem about the homotopy invariants introduced earlier can be stated as follows. Theorem 9.2. Let D be a nonlocal elliptic operator on the manifold M . Then chΛ (indΛ D) = indt D.

(9.12)

This theorem describes the relation between the analytic and topological invariants introduced earlier. The proof is given in Secs. 9.4 and 9.5. Corollary 9.3. One has the index formula indt (D)(g0 ) ≡ chΓ [σ(D)] Td(T M ⊗ C; Γ), [T ∗ M ; Γ] , inda (D) =

g0

where the series on the right-hand side is absolutely convergent.

(9.13)

9.3. Vanishing Theorem

103

Proof. The proof follows from Proposition 5.2 and the easy fact that the homomorphism α : Λ → C has the representation τ g0 . α=

g0

To prove the absolute convergence of the series in (9.13), we note that its terms are just the components of the Chern character chΓ of some element in K0 (Λ). Each such element is represented as a formal diﬀerence of projections over Λ∞ (since Λ∞ is a local subalgebra), and for elements in Λ∞ the absolute convergence of the series follows from the rapid decay of these elements viewed as functions on the group.

9.3 Vanishing Theorem Equation (9.13) represents the index as a sum over conjugacy classes g0 of the group Γ. Obviously, the terms for which the ﬁxed-point manifold Mg0 is empty are zero by deﬁnition; however, even with this fact taken into consideration, the series (9.13) can, depending on the complexity of the action of Γ, contain quite a few (or even inﬁnitely many) terms. It turns out, however, that in some situations many of the terms are zero automatically (even if the ﬁxed point manifold Mg0 is nonempty) and the formula is simpliﬁed dramatically. (For example, this is the case if Γ is a ﬁnitely generated Abelian group.) Let us state and prove the corresponding general result. Proposition 9.4 (Vanishing theorem). Let χ : Γ −→ Z be a group homomorphism, and let g0 ∈ Γ be an element such that χ(g0 ) = 0. Then indt (D)(g0 ) = 0 for an arbitrary nonlocal elliptic operator D. Proof. By Theorem 9.2 and deﬁnition (9.10), (9.11) of the Chern character on the group K0 (Λ), it suﬃces to show that chγ (P ) = 0 for any projection P over Λ∞ provided that γ = g0 , where g0 ∈ Γ is an element satisfying the assumptions of the proposition. Let P = P (g) be such a projection. Since χ is a group homomorphism, we can deﬁne a deformation of our projection by the formula Pt = Pt (g) ≡ P (g)eitχ(g)

104

Chapter 9. Cohomological Index Formula

for any real t. (One can readily verify that Pt = Pt∗ = Pt2 .) Thus, we have a continuous family of projections over Λ∞ with parameter t. Let us compute the Chern character chγ (Pt ). Since Z is Abelian, we obtain for any g ∈ γ.

χ(g) = χ(g0 ) Using Eq. (9.10), we obtain chγ (Pt ) =

tr Pt (g) =

g∈γ

=

eitχ(g) tr P (g)

g∈γ

e

itχ(g0 )

tr P (g) = eitχ(g0 ) chγ (P ).

g∈γ

On the other hand, by the homotopy invariance of the Chern character chγ , we see that chγ (Pt ) = chγ (P ) for all t ∈ R. Since χ(g0 ) = 0 by assumption, the last two equations cannot be valid simultaneously for all t unless chγ (P ) = 0. An important example where this proposition applies is given by the case in which Γ is a ﬁnitely generated Abelian group (and hence a ﬁnite direct sum of cyclic groups by the well-known structure theorem). Then for any element g ∈ Γ of inﬁnite order there exists a homomorphism χ : Γ → Z such that χ(g) = 0. To construct such a homomorphism, one uses the above-mentioned structure theorem and notes that the projection of g onto at least one of the inﬁnite cyclic summands is nonzero. This projection can be chosen as the desired homomorphism χ. Proposition 9.4 implies that the index formula contains only terms corresponding to the torsion subgroup: inda (D) = indt (D)(g). g∈Tor(Γ)

(We replaced the sum over conjugacy classes by the sum over elements, since in the Abelian case each conjugacy class consists of one element.) Thus, the index formula in this case contains only ﬁnitely many terms. The simplest index formula holds for the group Γ = Zn . Indeed, in this case all components of the Chern character of a nonlocal elliptic operator are zero, possibly except for the component corresponding to the zero element in Zn . Thus, Theorem 9.2 gives the following simple index formula. Theorem 9.5. Let D be a Zn -nonlocal elliptic operator on a manifold M . Then the following index formula is valid: inda (D) = indt (D)(0) ≡ ch0 [σ(D)] Td(T M ⊗ C), [T ∗ M ] .

(9.14)

9.4. Proof of the Index Theorem

105

Here we have integration over the fundamental cycle of T ∗ M , Td is the usual Todd class, and the Chern character of projections ch0 is deﬁned by the formula )

* dP dP ch0 (P ) = tr P exp − 2πi 0 where the subscript 0 on the right-hand side means evaluation at zero (g = 0), and tr, as before, is the usual matrix trace. Remark 9.6. Formula (9.14) looks very much like the Atiyah–Singer formula. The only distinction is that the product of projections and their diﬀerentiation uses the new rules of noncommutative algebra, and after the computation of the matrix trace one leaves only the zeroth component. (The last step, however, is not really signiﬁcant, since the remaining components, as we have seen in Proposition 9.4, vanish anyway.)

9.4 Proof of the Index Theorem The proof of Theorem 9.2 follows the standard scheme due to Atiyah and Singer. In Chap. 7, we proved a K-theoretical index formula, which expresses the index as the direct image of the operator under the projection of M into a point. To obtain the cohomological formula stated in Theorem 9.2, it remains to make the ﬁnal step, i.e., show that not only the analytical index but also the topological index does not change under the direct image mapping. The aim of this section is to prove this result. We ﬁx an element g ∈ Γ and denote the localized Chern character chΓ x(g) of an element x ∈ K0 (C0 (T ∗ M )Γ ) by chg (x) ∈ Hcev (T ∗ Mg ) ⊗ C for brevity. Let E ∈ VectΓ (M ) be a real vector Γ-bundle over M . Lemma 9.7. For arbitrary g ∈ Γ, the ﬁxed point set Eg ⊂ E is a vector bundle over the set Mg of ﬁxed points on the base M . Proof. We know that Eg and Mg are smooth submanifolds (see Sec. 1.2). It remains to prove that Eg is a smooth subbundle in E|Mg . To this end, it sufﬁces to show that the dimension of the ﬁber is locally constant as a function on Mg . Indeed, the tangent bundle Tx M at x ∈ Mg decomposes as the direct sum Tx Mg ⊕ Nx Mg ⊕ (Eg )x ⊕ (Eg )⊥ x , with regard to which the diﬀerential is dg = 1 ⊕ a ⊕ 1 ⊕ b, where the endomorphisms a and b have no eigenvalue equal to unity. It follows that the dimension of the spaces (Eg )⊥ x and (Eg )x is locally constant. The proof of the lemma is complete.

106

Chapter 9. Cohomological Index Formula

Proposition 9.8. Let dim M > 0. Then indt [σ](g) = indt (i! [σ])(g),

(9.15)

where i! : K0 (C0 (T ∗ M )Γ ) → K0 (C0 (T E)Γ ) is the direct image mapping for the embedding i : M → E of the zero section. Remark 9.9. If M is a point, then E = Rn is a ﬁnitely generated Γ-module and the direct image mapping is a homomorphism i! : K0 (Λ) −→ K0 (C0 (R2n )Γ ). (We suppose that the action of Γ in Rn is topologically free.) In this case, the topological index is also preserved under the embedding. Here the topological index is by deﬁnition the value of the localized Chern character of a diﬀerence of projections deﬁning an element in K0 (Λ). The proof given below can be carried out in this case as well with the following changes: 1. Since in this case M = Mg = {pt}, and Eg and Eg⊥ = N Eg are ﬁnitedimensional subspaces, the Todd classes are equal to 1, and the Chern character of the normal bundle is nonzero in zero degree alone. 2. The multiplicative property of the Chern character is stated in the form chg [(σ : Im P → 0)#QE ] = (chg P ) ch i∗ (QE )(g). The remaining computations are similar. In conjunction with Proposition 9.8, this proves that the topological index is preserved under the direct image associated with the projection into the one-point space. Proof of Proposition 9.8. By deﬁnition, the right-hand side of the equation is equal to indt i! [σ](g) = indt [σ#QE ](g), where QE denotes the symbol of the Euler operator in the ﬁbers of E → M . In more detail, the latter expression is . indt [σ#QE ](g) = chg [σ#QE ]

/ Td(T (Eg ) ⊗ C) # , [T ∗ (Eg )] . ch (N (Eg ) ⊗ C)(g)

(9.16)

Note that the cotangent bundle T ∗ (Eg ) π0∗ T ∗ Mg ⊕ Eg , where π0 : T ∗ (Eg ) → T ∗ Mg , is ﬁbered over T ∗ Mg with ﬁber Tx∗ Eg , x ∈ Mg . Denote the projection of this bundle by π. Let us integrate in (9.16) over the ﬁbers. (On

9.4. Proof of the Index Theorem

107

Tx∗ Eg we ﬁx the orientation given by the ordering ξ1 , x1 , ξ2 , x2 , . . . of the canonical coordinates.) We have . chg [σ#QE ]

/ Td(T (Eg ) ⊗ C) # , [T ∗ (Eg )] ch (N (Eg ) ⊗ C)(g) . / Td(T (Eg ) ⊗ C) # = π∗ chg [σ#QE ] , [T ∗ Mg ] . (9.17) ch (N (Eg ) ⊗ C)(g)

(The fraction need not be integrated over the ﬁber, since, as we shall see below in Eqs.(9.18) and (9.19), it is the pullback of a cohomology class deﬁned on the base.) Let us substitute T (Eg ) π0∗ T Mg ⊕ Eg into the fraction in Eq. (9.17). We obtain Td(T (Eg ) ⊗ C) = Td(T Mg ⊗ C) Td(Eg ⊗ C)

(9.18)

by virtue of the identity Td(E1 ⊕ E2 ) = Td(E1 ) Td(E2 ) for the Todd class of complex vector bundles. In a similar way, for the localized Chern character we obtain # # # ch (N (Eg ) ⊗ C)(g) = ch (N Mg ⊗ C)(g) · ch (Eg⊥ ⊗ C)(g).

(9.19)

(Here we use the decomposition E|Mg Eg ⊕ Eg⊥ , the Cartan formula

# # # (E1 ⊕ E2 ) = (E1 ) ⊗ (E2 )

—the tensor product of Z2 -graded spaces—and also the fact that the action of g on the tensor product is diagonal.) Finally, the Chern character is multiplicative with respect to exterior product, as stated in the following lemma. Lemma 9.10. One has chg [σ#QE ] = chg [σ] ch i∗ (QE )(g),

(9.20)

where i∗ (QE ) is the restriction of the symbol QE to the subbundle T Eg ⊂ T E.

108

Chapter 9. Cohomological Index Formula Substituting (9.18), (9.19), and (9.20) into (9.17), we obtain

indt i! [σ](g) = . / Td(Eg ⊗ C) Td(T Mg ⊗ C) # ⊥ chg [σ] # π∗ ch i∗ QE (g), [T ∗ Mg ] . ch (N Mg ⊗ C)(g) ch (Eg ⊗ C)(g) To complete the proof of Proposition 9.8, we apply the following lemma to the last expression and make the corresponding cancellations. Lemma 9.11.

# ∗ ch (Eg⊥ ⊗ C)(g) . π∗ ch i QE (g) = Td(Eg ⊗ C)

This completes the proof of Proposition 9.8.

(9.21)

9.5 Proofs of Auxiliary Statements The proofs of Lemmas 9.10 and 9.11 occupy the following two subsections.

9.5.1 Multiplicativity of the Chern Character Here we prove Lemma 9.10. Case 1. Compact Manifold Let X be a compact Γ-manifold, and let the action of Γ on X be topologically free. Remark 9.12. The results of this subsection are true without the assumption that the action is topologically free. The proofs in this case are essentially the same. Let p ∈ MatN (C ∞ (X)Γ ) be a matrix projection, and let F ∈ VectΓ (X) be a Γ-bundle. We shall deﬁne the product [p] × [F ] ∈ K0 (C ∞ (X)Γ ) of elements

[p] ∈ K0 (C ∞ (X)Γ ), [F ] ∈ KΓ0 (X). To this end, we write p as a sum p = g T (g)p(g). Let p ⊗ 1F : C ∞ (X, CN ⊗ F ) −→ C ∞ (X, CN ⊗ F ) be the following projection: p ⊗ 1F =

g

(1N ⊗ T (g))(p(g) ⊗ 1F ),

(9.22)

9.5. Proofs of Auxiliary Statements where

109

T (g) : C ∞ (X, F ) → C ∞ (X, F )

is the translation operator induced by the mapping x → g(x), Fx → Fg(x) . Then we deﬁne the product (9.22) as [p] × [F ] = [p ⊗ 1F ] ∈ K0 (C ∞ (X)Γ ). Remark 9.13. The projection p ⊗ 1F is deﬁned on sections of a possibly nontrivial bundle CN ⊗ F . Of course, by embedding this bundle in some trivial bundle, we obtain a matrix projection. Since the Chern character is preserved under this procedure, we omit the embedding. Lemma 9.14. For a matrix operator p ∈ MatN (C ∞ (X)Γ ) and a Γ-bundle F ∈ VectΓ (X), one has chg ([p] × [F ]) = chg (p) · ch F (g) ∈ H ev (Xg ) ⊗ C.

(9.23)

Proof. Denote the projection p ⊗ 1F by p0 for brevity. Let us choose some Γ-invariant connection ∇F in F . Then the Chern character of p0 is equal to

Ω chg (p0 ) = τCN ⊗F,g p0 exp − 2πi (see Sec. 8.4.2), where the curvature form is equal to Ω = (p0 (1N ⊗ ∇F )p0 )2 . By the Γ-invariance of ∇F , we obtain Ω = p0 (1N ⊗ ∇F )p0 (1N ⊗ ∇F )p0 = p0 (1N ⊗ ∇F )2 p0 + p0 [1N ⊗ ∇F , p0 ](1N ⊗ ∇F )p0 = p0 (1N ⊗ ΩF )p0 + p0 dp0 (1N ⊗ ∇F )p0 = (1N ⊗ ΩF )p0 + p0 dp0 [1N ⊗ ∇F , p0 ]p0 = (1N ⊗ ΩF )p0 + p0 dp0 dp0 p0 . (Here we have used the identities [1N ⊗ ∇F , p0 ] = dp0 and p0 dp0 p0 = 0.) Since the last two summands commute (by virtue of the Γ-invariance of the curvature form ΩF = (∇F )2 ), it follows that the Chern character is equal to

Ω τCN ⊗F,g p0 exp − 2πi

1 N ⊗ ΩF p0 dp0 dp0 p0 = τCN ⊗F,g p0 exp − exp − 2πi 2πi

1 N ⊗ ΩF dp0 dp0 = τCN ⊗F,g p0 exp − exp − . 2πi 2πi

110

Chapter 9. Cohomological Index Formula

It is now clear that to complete the proof of the lemma, it remains to prove the relation1 τCN ⊗F,g p0 (1N ⊗ ΩF )k (dp0 dp0 )l = trF T (g)(ΩF )k |Xg τCN ,g p0 (dp0 dp0 )l (9.24) for all k, l ≥ 0. Let us prove (9.24). According to the deﬁnitions, we have τCN ⊗F,g (p0 (1N ⊗ ΩF )k (dp0 dp0 )l ) = h∗ trCN ⊗F (1N ⊗ T (g ))(1N ⊗ ΩF )k [p(dpdp)l ⊗ 1F ]g g ∈ g

Γg,g

= trF T (g)(ΩF )k |Xg g ∈ g

Γg,g

dh Xg

h∗ trCN [p(dpdp)l ]g |Xg dh

= trF T g (ΩF )k |Xg τF,g p(dpdp)l . Here [x]g stands for the coeﬃcient of the element x corresponding to the shift T (g), and to obtain the second equality we have used the relation tr(A ⊗ B) = tr A tr B and observed that the class h∗ trF T (g )(ΩF )k |Xg ∈ H ev (Xg ) ⊗ C is independent of the choice of h ∈ Γg,g , g ∈ g (since ΩF is Γ-invariant) and hence can be taken out of the integral and the sum. This completes the proof of Lemma 9.14. Case 2. Noncompact Manifold Now let E1 , E2 ∈ VectΓ (M ) be two Γ-bundles. Then the exterior product induces a product (9.25) # : K0 (C0 (E1 )Γ ) × KΓ0 (E2 ) −→ K0 (C0 (E1 ⊕ E2 )Γ ) in K-theory. (The product of the class of a triple (σ, P0 , P1 ) in K0 (C0 (E1 )Γ ) by the class of an equivariant triple (σ , Q0 , Q1 ) in KΓ0 (E2 ) is the class of the exterior product σ#σ .) The proof of Lemma 9.10, i.e., the veriﬁcation of the multiplicative property of the Chern character with respect to product (9.25), can be carried out by reduction to the compact case. To this end, we shall use the Atiyah–Bott–Patodi 1 In

the ﬁrst factor on the right-hand side, we consider the restriction to the ﬁxed point set; thus, the shift operator acts only in the ﬁbers of F , and the trace is well deﬁned.

9.5. Proofs of Auxiliary Statements

111

compactiﬁcation (see Sec.8.4.2). Namely, for a real vector bundle E, let 2BE be the space obtained from E by ﬁberwise one-point compactiﬁcation. In other words, 2BE = S(E ⊕ R) is the unit sphere bundle in the real vector bundle E ⊕ R over M . It is clear that the exterior product also deﬁnes products in K-theory: K0 (C0 (E1 )Γ ) × KΓ0 (2BE2 ) −→ K0 (C0 (E1 ×M 2BE2 )Γ ), K0 (C(2BE1 )Γ ) × KΓ0 (2BE2 ) −→ K0 (C(2BE1 ×M 2BE2 )Γ ), where one or two factors correspond to the compact spaces 2BE1 either 2BE2 .2 More precisely, the symbol σ in the ﬁrst product can be assumed to be zero, and the symbol σ is simply twisted by a Γ-bundle deﬁning a class in KΓ0 (2BE2 ). In the second product, both symbols σ and σ are zero, and the mapping is induced by the tensor product of projections. (Of course, the shift operators in the ﬁrst factor act on sections of the Γ-bundle.) These two products will be denoted by #. Lemma 9.15. One has the commutative diagram #

/ K0 C0 (E1 ⊕ E2 )Γ

K0 (C0 (E1 )Γ ) × KΓ0 (2BE2 )

#

/ K0 C0 (E1 ×M 2BE2 )Γ

K0 (C(2BE1 )Γ ) × KΓ0 (2BE2 )

#

/ K0 C(2BE1 ×M 2BE2 )Γ ,

K0 (C0 (E1 )Γ ) × KΓ0 (E2 )

(9.26)

where the vertical mappings are induced by the embeddings E1,2 ⊂ 2BE1,2 . Proof. 1. First, let us prove the commutativity of the lower square. Consider a Γ-bundle F deﬁning a class in KΓ0 (2BE2 ) and a triple (σ, P0 , P1 ) deﬁning a class in K0 (C0 (E1 )Γ ). We can assume that the projections P0,1 coincide at inﬁnity with the trivial projection 1N ⊕ 0N and the symbol σ also coincides with 1N ⊕ 0N at inﬁnity. By deﬁnition, the exterior product of these classes is equal to the direct sum of the triples (σ ⊗ 1F , P0 ⊗ 1F , P1 ⊗ 1F ), 2 Here

(σ ∗ ⊗ 1F , P1 ⊗ 1F , P0 ⊗ 1F ).

E1 ×M E2 = {(e1 , e2 ) : π1 (e1 ) = π2 (e2 )} is the ﬁbered product of the bundles π1,2 : E1,2 → M .

112

Chapter 9. Cohomological Index Formula

The mapping K0 C0 (E1 ×M 2BE2 )Γ → K0 C(2BE1 ×M 2BE2 )Γ takes this sum to the element3 ([P1 ] − [P0 ])#[F ]. On the other hand, we arrive at the same element if we pass through the lower left corner of diagram (9.26). This shows that the lower square is commutative. 2. Let us prove the commutativity of the upper square. Let (σ, P0 , P1 ) be the same triple as in the ﬁrst part of the proof; i.e., the projections and the symbol are trivial at inﬁnity. For a representative of an element in the group KΓ0 (E2 ) we take some triple (σ , Q0 , Q1 ) for which the projections and the symbol σ are trivial at inﬁnity. If we pass through the lower left corner of the square, then we obtain the element [(σ ⊗ 1Q0 , P0 ⊗ 1Q0 , P1 ⊗ 1Q0 )] + [(σ ∗ ⊗ 1Q1 , P1 ⊗ 1Q1 , P0 ⊗ 1Q1 )]. We deﬁne a homotopy of the corresponding triples by the formula

σ ⊗ 1Q0 cos(ϕt) P0 ⊗ 1Q1 sin(ϕt)

−P1 ⊗ 1Q0 sin(ϕt) , σ ∗ ⊗ 1Q1 cos(ϕt)

(we change only the symbols), where t ∈ [0, 1] is the parameter of the homotopy and ϕ is a function on E1 ⊕ E2 that depends only on |e2 |, is identically zero for |e2 | < 1, and is monotone increasing from 0 to π/2. At t = 1 we obtain a symbol that, for increasing |e2 |, is equal to a homotopy connecting the symbols σ ⊗ 1Q0 0

0 , σ ∗ ⊗ 1Q1

0 P0 ⊗ 1Q1

−P1 ⊗ 1Q0 . 0

On the other hand, it is clear that this homotopy deﬁnes the same class in K0 C0 (E1 ×M 2BE2 )Γ as the exterior product of σ#σ symbols trivial at inﬁnity. Thus, we have proved that the upper square commutes. The proof of the lemma is now complete. Now we can state the following lemma. Lemma 9.16. One has the following commutative diagram, in which the vertical 3 Since the projections are trivial at inﬁnity, they can naturally be extended to the space 2BE1 ×M 2BE2 . To avoid excessively clumsy notation, we denote the extensions by the same symbol as the original projections.

9.5. Proofs of Auxiliary Statements

113

mappings in the right column are monomorphisms: chg

/ Hcev (E1 )g ×Mg (E2 )g ⊗ C

K0 (C0 (E1 ×M 2BE2 )Γ )

chg

/ H ev (E1 )g ×Mg 2B(E2 )g ⊗ C c

K0 (C(2BE1 ×M 2BE2 )Γ )

chg

K0 (C0 (E1 ⊕ E2 )Γ )

(9.27)

/ H ev 2B(E1 )g ×Mg 2B(E2 )g ⊗ C

Proof. 1. First, let us prove that the mapping i∗ Hcev (E1 )g ×Mg 2B(E2 )g Hcev (E1 )g ×Mg (E2 )g −→ induced by embedding i

(E1 )g ×Mg (E2 )g ⊂ (E1 )g ×Mg 2B(E2 )g of an open subset has trivial kernel. The exact sequence of the pair in the last formula is · · · → Hcev (E1 )g ×Mg (E2 )g −→ Hcev (E1 )g ×Mg 2B(E2 )g −→ Hcev (E1 )g → · · · (The ﬁrst mapping is induced by the embedding, and the second mapping is induced by the restriction to the set of points at inﬁnity in 2B(E2 )g .) Moreover, the projection π : (E1 )g ×Mg 2B(E2 )g → (E1 )g splits the sequence. The splitting gives the desired monomorphism in K-theory. A similar reasoning proves that the second mapping is a monomorphism. 2. Let us prove that diagram (9.27) commutes. This is actually obvious, since in the deﬁnition of Chern character on K-groups of noncompact spaces we use projections that coincide at inﬁnity and are equal to a trivial projection like 1N ⊕ 0N outside some compact set. Thus, the form representing the diﬀerence of Chern characters of projections has compact support. This completes the proof of Lemma 9.16. Proof of Lemma 9.10. Combining the commutative diagrams (9.26) and (9.27), we obtain the commutative square K0 (C0 (E1 )Γ ) × KΓ0 (E2 ) K0 (C(2BE1 )Γ ) × KΓ0 (2BE2 )

chg ◦#

/ H ev (E1 )g ×Mg (E2 )g ⊗ C c

/ H ev 2B(E1 )g ×Mg 2B(E2 )g ⊗ C,

chg ◦#

(9.28)

114

Chapter 9. Cohomological Index Formula

in which the right vertical mapping has trivial kernel. On the other hand, by the multiplicative property of the Chern character for compact spaces (Lemma 9.14), we obtain the commutative square #

K0 (C(2BE1 )Γ ) × KΓ0 (2BE2 ) chg × ch(g)

H

ev

2B(E1 )g ⊗ C × H

ev

2B(E2 )g ⊗ C

/ K0 (C(2BE1 ×M 2BE2 )Γ )

/H

ev

chg

2B(E1 )g ×Mg 2B(E2 )g ⊗ C.

Finally, note that the commutativity of these two squares and a monomorphism in the right column of (9.28) imply the desired multiplicative property of the Chern character. The proof of Lemma 9.10 is now complete.

9.5.2 Chern Character of the Symbol of the Euler Operator In this subsection, we prove Lemma 9.11. 1. First, for the restriction of the symbol of the Euler operator to the ﬁxed point set we have # i∗ QE = (Eg⊥ ⊗ C) ⊗ QEg , # where (Eg⊥ ⊗C) is treated as a Z2 -graded space. In the proof, one uses the Cartan formula # # # (Eg ⊕ Eg⊥ ) = (Eg ) ⊗ (Eg⊥ ) as well as the identity QEg ⊕Eg⊥ = QEg #QEg⊥ ; see [10]. Note that the action of g is nontrivial only on the exterior algebra of the normal bundle Eg⊥ . Hence the localized Chern character is equal to # ch(i∗ QE )(g) = ch (Eg⊥ ⊗ C)(g) · ch(QEg ). 2. The integral of the latter expression over the ﬁbers of the bundle π : T Eg → T Mg is well known (the Riemann–Roch formula for the embedding Mg ⊂ Eg ): # # π∗ (ch (Eg⊥ ⊗ C)(g) · ch QEg ) = ch( (Eg⊥ ⊗ C)(g)) · π∗ (ch QEg ) # 1 = ch( (Eg⊥ ⊗ C)(g)) · Td(Eg ⊗ C) (e.g., see [52]). The proof of Lemma 9.11 is now complete.

Chapter 10

Cohomological Formula for the Λ-Index In Chap. 7, we obtained a K-theoretical index formula indΛ (D) = p! [σ(D)], for nonlocal elliptic operators, where 1. Λ = C ∗ (Γ) is the group C ∗ -algebra of group Γ. 2. indΛ (D) ∈ K0 (Λ) is the Λ-Fredholm index of an operator in Hilbert Λmodules associated with symbol σ(D) (see Sec. 5.2). 3.

p! : K0 (C0 (T ∗ M ) Γ) −→ K0 (Λ) is the direct image mapping corresponding to the projection p : M → {pt} of the manifold M into the one-point space.

However, the cohomological formula was obtained only for the usual Fredholm index, which can be extracted from the Λ-Fredholm index by the formula ind D = α∗ indΛ (D), where α : Λ → C is the “forgetful” homomorphism associated with the trivial representation of Γ in C (see Proposition 5.2). The reason for this is that we have only constructed the Chern character corresponding to traces of “zero order” on the algebra Λ. Although these traces permitted us to compute the Fredholm index, they are not suﬃcient to compute the Λ-Fredholm index (even modulo torsion). To obtain additional information about the Λ-Fredholm index, one needs “higher traces.” In this chapter, we give formulas for such traces and the corresponding Chern character. The proof of the corresponding cohomological ΛFredholm index formula basically reproduces the proof of Theorem 9.2; one also

116

Chapter 10. Cohomological Formula for the Λ-Index

has to show that the invariants deﬁned in terms of the new Chern character are invariant under direct image mappings.

10.1 Noncommutative Diﬀerential Forms Let Λ∞ ⊂ Λ be the local subalgebra of elements g∈Γ ag g, where the coeﬃcients ag ∈ C rapidly decay as |g| → ∞. Consider the enveloping diﬀerential algebra associated with Λ∞ (cf. Appendix C). As a linear space, the enveloping algebra is just the direct sum ∞ # ∞ #j ∞ (Λ ) = (Λ ), j=0

where

#0

(Λ∞ ) = Λ∞ , and

#j

(Λ∞ ) consists of formal sums ag0 g1 ...gj g0 dg1 · · · dgj

g0 ,g1 ,...,gj ∈Γ

with rapidly decaying coeﬃcients ag0 g1 ...gj ∈ C. The product of “formal diﬀerentials” dgs and elements gs is uniquely determined by the “Leibniz rule” dg h = dk − g dh, for all g, h ∈ Γ and k = gh. (From now on, elements of the form (ae)dg1 · · · dgj , a ∈#C, where e is the unit element in Γ, are written as adg1 · · · dgj .) The diﬀerential in (Λ∞ ) is deﬁned as follows: d[ag0 g1 ...gj g0 dg1 · · · dgj ] = ag0 g1 ...gj dg0 dg1 · · · dgj ,

de = 0.

Obviously, d2 = 0 (since de = 0). One can show that the diﬀerential satisﬁes the Leibniz rule d(uv) = (du)v + (−1)deg u u(dv). # Deﬁnition 10.1. The diﬀerential graded algebra (Λ∞ ) is called the algebra of noncommutative diﬀerential forms on the group Γ. # The algebra (Λ∞ ) is actually bigraded. The second grading degΓ (unlike the ﬁrst Z-valued grading) takes values in Γ. Namely, we deﬁne degΓ (g) = degΓ (dg) = g,

degΓ (λ) = e

(λ ∈ C).

Now we extend this construction to the algebra C ∞ (X) Γ, where X is a Γ-manifold, and deﬁne the diﬀerential algebra ∞ # ∞ #j ∞ (C (X) Γ) = (C (X) Γ) j=0

10.2. Graded Traces over Λ∞

117

as follows. Forms of degree j are just linear combinations (with rapidly decaying coeﬃcients) of monomials of the form ω k g0 dg1 · · · dgs , where ω k is a usual diﬀerential form on X of degree k and k + s = j. For this monomial, we deﬁne degΓ [ω k g0 dg1 · · · dgs ] = degΓ [g0 dg1 · · · dgs ] ≡ g0 g1 · · · gs . The commutation relations for elements of Γ and diﬀerentials dg are the same as above, while their commutators with diﬀerential forms on X satisfy ωg0 dg1 · · · dgs = (−1)s deg ω g0 dg1 · · · dgs [degΓ (g0 dg1 · · · dgs )∗ ω] ≡ (−1)s deg ω g0 dg1 · · · dgs [(g0 g1 · · · gs )∗ ω]. The diﬀerential is d[ωg0 dg1 · · · dgs ] = (dω)g0 dg1 · · · dgs + (−1)deg ω ωdg0 dg1 · · · dgs . As earlier, we have d2 = 0.

# Deﬁnition 10.2. The diﬀerential graded algebra (C ∞ (X)Γ) is called the algebra of noncommutative diﬀerential forms on the Γ-space X. Remark 10.3. Unlike forms on manifolds, noncommutative forms can have arbitrarily large degree.

10.2 Graded Traces over Λ∞ Deﬁnition 10.4. A linear functional # τ : (Λ∞ ) −→ C is called a closed graded trace of degree m(τ ) ∈ Z+ on following conditions

# ∞ (Λ ) if it satisﬁes the

1. τ (f ) = 0 if deg f = m(τ ). 2. τ (df ) = 0 for all forms f . 3. τ ([f1 , f2 ]) = 0, where [f1 , f2 ] is the supercommutator. # ∞We shall also assume that the traces are continuous on the Fr´echet space (Λ ). The following lemma is straightforward. #m ∞ Lemma 10.5. For a continuous linear functional τ : (Λ ) → C, there exist numbers k, C such that |τ (g0 dg1 · · · dgm )| ≤ C(|g0 | + |g1 | + · · · + |gm |)k for all g0 , . . . , gm . The converse is also true: if the estimate is valid, then the corresponding linear functional is continuous.

118

Chapter 10. Cohomological Formula for the Λ-Index

Let us give some examples of graded traces. To this end, recall the construction (e.g., see [27]) of closed graded traces over the smooth group algebra Λ∞ starting from group cocycles. Let Γ be a discrete group.

10.2.1 Cohomology of Groups A function c : Γn+1 → C is an n-dimensional cochain if c(gg0 , gg1 , . . . , ggn ) = c(g0 , g1 , . . . , gn ) for all g, gi ∈ Γ. Remark 10.6. One can also consider cocycles with values in some Γ-module. We shall not need this generalization. Denote by C n (Γ) the vector space of all n-dimensional cochains over Γ. We deﬁne the diﬀerential d : C n (Γ) → C n+1 (Γ) by the formula (dc)(g0 , . . . , gn+1 ) =

n+1

(−1)i c(g0 , . . . , gi−1 , gi+1 , . . . , gn+1 ).

i=0

A straightforward computation shows that d ◦ d = 0. The cohomology of the complex d

d

d

d

· · · −−−−→ C n (Γ) −−−−→ C n+1 (Γ) −−−−→ C n+2 (Γ) −−−−→ · · · is called the cohomology of the group Γ and is denoted by H ∗ (Γ, C). An element c for which dc = 0 is called a group cocycle. This algebraic construction plays an important role in topology. Theorem 10.7. There exists an isomorphism H ∗ (Γ, C) H ∗ (BΓ, C), where BΓ denotes the classifying space of Γ. Remark 10.8. By [39], isomorphic cohomology groups are obtained if instead of C ∗ (Γ) we consider the subcomplex Cλn (Γ) = {c ∈ C n (Γ) : σ(c) = (−1)|σ| c,

σ ∈ Sn+1 },

of antisymmetric cochains. (Here Sn+1 is the permutation group on n+1 elements.)

10.2. Graded Traces over Λ∞

119

10.2.2 Construction of Traces Let c ∈ Cλn (Γ) be a group cocycle. Consider the functional τc :

#n

(Λ∞ ) −→ C,

deﬁned on the generators by 0 0, τc (g0 dg1 dg2 . . . dgn ) = c(e, g1 , g1 g2 , . . . , g1 g2 . . . gn ),

g0 g 1 . . . gn = e, g0 g1 . . . gn = e.

(10.1)

This functional is well deﬁned, because it takes monomials in which one of the gi is equal to the unit element e ∈ Γ to zero. (This follows from the skew commutativity of the cocycle # c.) One can verify that this functional is a closed graded trace (of degree n) on (Λ∞ ). (The proof of this fact is left to the reader as a simple exercise: the graded trace property follows from the cocycle condition.)

10.2.3 Examples Using the described construction we can write out examples of closed graded traces. Traces of Degree 0 Such traces were considered in the previous chapter. These are the traces τγ associated with various conjugacy classes γ in the group Γ. It is clear that an arbitrary trace of degree zero is a linear combination of such traces. We also note that if Γ is ﬁnite, then these traces classify the K-theory. (In other words, two elements in K0 (Λ) are equal if and only if the values of each trace τγ on these elements coincide.) One can readily prove this using the following facts: the K-group of the group algebra coincides with the ring of virtual representations, K(Λ) R(Γ); moreover, the mapping K(Λ) −→ C

g

that takes a projection to the collection of all its traces is equivalent to the homomorphism C (10.2) R(Γ) −→

g

that takes a representation to its character (which is a function on the group). One ﬁnally notes that (10.2) is an isomorphism.

120

Chapter 10. Cohomological Formula for the Λ-Index

Trace of Degree 1 for the Group Z By Theorem 10.7, the one-dimensional cohomology group of Z has one generator (since BZ = S1 ). Let us write out the corresponding generator and a closed graded trace associated with it. The cochain c(g0 , g1 ) = g0 − g1 is a cocycle. One can readily show using the deﬁnition that it is not a coboundary. By Eq. (10.1), the associated closed graded trace is

τ ag0 g1 g0 dg1 = ag0 g1 g1 . (10.3) g0 ,g1

g0 g1 =e

In a similar way, one deﬁnes traces on all free Abelian groups Zk . More precisely, this construction gives k linearly independent traces of degree 1 over C ∞ (Zk ). Trace of Degree 2 for the Group Z2 The cochain c(g0 , g1 , g2 ) = [g0 − g1 , g0 − g2 ] (here [a, b] ∈ R stands for the exterior product of vectors a, b ∈ R2 ) is a cocycle on the group Z2 . The corresponding trace is

τ ag0 g1 g2 g0 dg1 dg2 = ag0 g1 g2 [g1 , g2 ] g0 ,g1 ,g2

g0 g1 g2 =e

=

ag0 g1 g2 (g1x g2y − g2x g1y ),

g0 g1 g2 =e

where we use notation g = (gx , gy ) ∈ Z2 . Similar traces can be deﬁned on all free Abelian groups Zk . More precisely, this construction gives k(k − 1)/2 linearly independent traces of degree 2 over the algebra C ∞ (Zk ).

10.3 Graded Traces over C ∞(X) Γ Let Γ be the set of conjugacy classes g0 in Γ, and ﬁx an arbitrary element g0 in each conjugacy class g0 . # In this subsection, we associate a closed graded trace over (C ∞ (X) Γ) to a closed graded trace τ over Λ∞ . We deﬁne the mapping # # τ : MatN ( (C ∞ (X) Γ)) −→ (10.4) (Xg0 )

g0

10.3. Graded Traces over C ∞ (X) Γ

121

on monomials as follows. Let # f = ϕ0 dϕ1 · · · dϕs ω ∈ MatN ( (C ∞ (X) Γ)), where ω is a matrix-valued diﬀerential form on X and ϕ0 , ϕ1 , . . . , ϕs ∈ Γ. Let g = degΓ (ϕ0 dϕ1 · · · dϕs ) ≡ ϕ0 ϕ1 · · · ϕs ∈ g0 . Then all components of τ (f ) are zero except for the component corresponding to the conjugacy class g0 , and we deﬁne this component as s deg ω [ τ f ](g0 ) = (−1) τ (ϕ0 dϕ1 · · · dϕs ) tr{[h∗ ω] Xg } dh. 0

Γg0 ,g

We extend the mapping τ to arbitrary elements f by linearity. (The corresponding series converge by Lemma 10.5.) # Theorem 10.9. The mapping (10.4) is a closed graded trace on MatN ( (C ∞ (X) Γ)); i.e., 1. It is linear and preserves (modulo shift) the Z-grading: deg τ (f ) = deg f − m(τ ). 2. τ (df ) = (−1)m(τ ) d[ τ (f )]. 3. τ [f1 , f2 ] = 0, where [f1 , f2 ] is the supercommutator of f1 and f2 . Proof. Property 1 is obvious, because a similar property is valid for τ . To prove property 2, we note that if f = ϕ0 dϕ1 · · · dϕs ω, then τ (df ) = τ (dϕ0 dϕ1 · · · dϕs ω) + (−1)s τ (ϕ0 dϕ1 · · · dϕs dω). The ﬁrst summand has a factor τ (dϕ0 dϕ1 · · · dϕs ) and is zero by property 2 of the trace τ . The second summand is (−1)s τ (ϕ0 dϕ1 · · · dϕs dω)(g0 ) s

s(deg ω+1)

= (−1) (−1)

τ (ϕ0 dϕ1 · · · dϕs ) Γg0 ,g

tr{[h∗ dω] Xg } dh 0

= d[ τ (f )]. Let us check property 3. We omit the factor of −1 to some power, since it is the same for τ (f1 f2 ) and for τ (f2 f1 ). Let f1 = ϕ0 dϕ1 · · · dϕs ω1 ,

degΓ f1 = k,

f2 = ψ0 dψ1 · · · dψn ω2 ,

degΓ f2 = l,

122

Chapter 10. Cohomological Formula for the Λ-Index

where kl ∈ g0 . Then the only (potentially) nontrivial element both for τ (f1 f2 ) and τ (f2 f1 ) corresponds to the conjugacy class g0 . Let us compute these two elements. We have τ (f1 f2 ) = (−1)n deg ω1 τ (ϕ0 dϕ1 · · · dϕs ψ0 dψ1 · · · dψn ) × tr{[(h∗ l∗ ω1 )(h∗ ω2 )] Xg } dh, 0

Γg0 ,kl

τ (f2 f1 ) = (−1)s deg ω2 τ (ψ0 dψ1 · · · dψn ϕ0 dϕ1 · · · dϕs ) × tr{[( h∗ k ∗ ω2 )( h∗ ω1 )] Xg } d h. 0

Γg0 ,lk

Further, Γg0 ,kl

tr{[(h∗ l∗ ω1 )(h∗ ω2 )] Xg } dh 0

deg ω1 deg ω2

= (−1)

Γg0 ,lk

tr{[( h∗ k ∗ ω2 )( h∗ ω1 )] Xg } d h 0

(this computation is completely similar to that in Chap. 8), and it remains to apply property 3 of the trace τ and add the exponents of (−1), to conclude that τ satisﬁes property 3. This completes the proof of the theorem.

10.4 Chern Character and the Index Formula 10.4.1 Chern Character

# Now for a closed graded trace τ on (Λ∞ ) we can deﬁne the Chern character of projections over C ∞ (X) Γ by the formula )

* #ev dP dP (Xg0 ) (10.5) chτ (P ) = τ P exp − ⊂ 2πi

g0

# (where d is the diﬀerential in (C ∞ (X) Γ)). One shows, as in Chap. 8, that this is a closed diﬀerential form whose cohomology class depends only on the class of P in the corresponding K-group. Thus, our Chern character is well deﬁned on K0 (C(X) Γ). (Here we use Theorem 1.23, which implies the locality of the subalgebra C ∞ (X) Γ ⊂ C(X) Γ.) In particular, for X = {pt} Eq. (10.5) deﬁnes the Chern character chτ ([P ]) for elements of the group K0 (Λ) (represented by projections P ): )

* dP dP chτ ([P ])(g0 ) = τ Πg0 P exp − ∈ C, (10.6) 2πi

10.4. Chern Character and the Index Formula

123

# where Πg0 is the projection in (Λ∞ ) onto the subspace of elements f such that degΓ f ∈ g0 : ) * Πg0 ag0 g1 ...gj g0 dg1 · · · dgj = ag0 g1 ...gj g0 dg1 · · · dgj . g0 ,g1 ,...,gj ∈Γ

g0 g1 ···gj ∈ g0

Note the following property. Lemma 10.10. If τ is a closed graded trace on τ ◦ Πg0 .

# ∞ (Λ ), then so is the composition

Proof. Condition 1 in Deﬁnition 10.4 is trivially satisﬁed; condition 2 is also easy, because Πg0 commutes with d. To prove condition 3, we can choose two monomials f1 and f2 . Since degΓ (f1 f2 ) and degΓ (f2 f1 ) automatically lie in the same conjugacy class g0 , we have either τ ◦ Πg0 ([f1 , f2 ]) = τ ([f1 , f2 ]) or τ ◦ Πg0 ([f1 , f2 ]) = τ (0) = 0.

10.4.2 Index Formula For a nonlocal # elliptic operator D on a manifold M and an arbitrary closed graded trace τ on (Λ∞ ), we deﬁne the Chern character chτ (σ(D)) as the Chern character of the diﬀerence construction of the symbol, i.e., of the virtual bundle [σ(D)] ∈ K0 (C0 (T ∗ M ) Γ) with compact support over T ∗ M . Theorem 10.11 (cohomological formula for Λ-Fredholm index). One has chτ (indΛ (D)) = chτ (σ(D)) Td(T M ⊗ C, Γ), [T ∗ M, Γ] ∈ C| Γ| .

(10.7)

Proof. To prove the theorem, it suﬃces, by analogy with the previous chapter, to verify that the invariant on the right-hand side in Eq. (10.7) is preserved under the direct image mapping and apply the index formula in K-theory, which we have proved earlier (Theorem 7.8). This veriﬁcation literally coincides with a similar one in the proof of Theorem 9.2, except Lemma 9.14, which (together with its proof) can be generalized to the present situation.

10.4.3 Proofs of Auxiliary Statements Let us write out an analog of Lemma 9.14 and its proof. Lemma 10.12. For a matrix projection P ∈ MatN (C ∞ (X)Γ ) and a Γ-bundle F ∈ VectΓ (X), one has chτ ([P ] × [F ])(g) = chτ (P )(g) · ch F (g) ∈ H ev (Xg ) ⊗ C, where [P ] × [F ] ∈ K0 (C ∞ (X)Γ ) stands for the product of elements [P ] ∈ K0 (C ∞ (X)Γ ),

[F ] ∈ KΓ0 (X).

(10.8)

124

Chapter 10. Cohomological Formula for the Λ-Index

Proof. 1. Recall that if P = g Tg Pg , then the product [P ] × [F ] is equal to the class of the projection (1N ⊗ T g )(Pg ⊗ 1F ) : C ∞ (X, CN ⊗ F ) −→ C ∞ (X, CN ⊗ F ), P ⊗ 1F = g

where T g : C ∞ (X, F ) → C ∞ (X, F ) is the shift operator induced by the mapping x → g(x), Fx → Fg(x) . We denote the projection P ⊗ 1F by P0 for brevity. 2. As in the proof of Lemma 9.14, we choose a Γ-invariant connection ∇F in

F we denote the operator F . By ∇

F : #(C ∞ (X, F ⊗ CN ) Γ) −→ #(C ∞ (X, F ⊗ CN ) Γ) ∇ equal to

F (uk g0 dg1 · · · dgl ) = (∇F uk )g0 dg1 · · · dgl + (−1)k udg0 dg1 · · · dgl ∇ ≡ (∇F + dΓ )(uk g0 dg1 · · · dgl ) (a noncommutative-geometric connection). Then the Chern character of P0 is equal to

Ω chτ (P0 )(g) = τ CN ⊗F,g P0 exp − , 2πi

F )P0 )2 . where the curvature form is equal to Ω = (P0 (∇ 3. By the Γ-invariance of the connection ∇F , we obtain

F )P0 (∇

F )P0 Ω = P0 (∇

F )2 P0 + P0 [∇

F , P0 ](∇

F )P0 = P0 (ΩF ⊗ 1N )P0 + P0 dP0 (∇

F )P0 = P0 (∇

F , P0 ]P0 = (ΩF ⊗ 1N )P0 + P0 dP0 dP0 P0 . = (ΩF ⊗ 1N )P0 + P0 dP0 [∇ #

F , P0 ] = (Here d is the diﬀerential in (C ∞ (X) Γ), and we use the identities [∇ dP0 and P0 dP0 P0 = 0.) The last two summands commute (by the Γ-invariance of the curvature form ΩF = (∇F )2 ), and for the Chern character we obtain

Ω τ F ⊗CN ,g P0 exp − 2πi

ΩF ⊗ 1 N P0 dP0 dP0 P0 = τ F ⊗CN ,g P0 exp − exp − 2πi 2πi

ΩF ⊗ 1 N dP0 dP0 = τ F ⊗CN ,g P0 exp − exp − . 2πi 2πi

10.4. Chern Character and the Index Formula

125

It is now clear that to complete the proof of the lemma we should prove the relation1 τ F ⊗CN ,g P0 (ΩF ⊗ 1N )k (dP0 dP0 )l = trF T g (ΩF )k |Xg τCN ,g P0 (dP0 dP0 )l (10.9) for all k, l ≥ 0. Let us prove (10.9). By the deﬁnition of the trace τ F ⊗CN ,g , we have τ F ⊗CN ,g (P0 (ΩF ⊗ 1N )k (dP0 dP0 )l ) h∗ trF ⊗CN (T g ⊗ 1N )(ΩF ⊗ 1N )k [1F ⊗ P (dP dP )l ]g = g ∈ g

Γg,g

= trF T g (ΩF )k |Xg g ∈ g

Γg,g

dh Xg

h∗ trCN [P (dP dP )l ]g |Xg dh

= trF T g (ΩF )k |Xg τF,g P (dP dP )l . Here [x]g denotes the component of x of grading degΓ equal to g, and to obtain the second equality we have used the relation tr(A ⊗ B) = tr A tr B and also observed that the class h∗ trF T g (ΩF )k |Xg ∈ H ev (Xg ) ⊗ C is independent of h ∈ Γg,g , g ∈ g (since ΩF is invariant) and hence can be taken outside the integral and the sum. This completes the proof of Lemma 10.12 and also of Theorem 10.11.

1 In

the ﬁrst factor on the right-hand side in this relation, we take the restriction to the ﬁxed point set; thus, the shift operator acts in the ﬁbers of F , and the trace is well deﬁned.

Chapter 11

Index of Nonlocal Operators over C ∗-Algebras In this chapter, we obtain index formulas for nonlocal elliptic operators D = (D, P1 , P2 ) over some C ∗ -algebra Λ.1 As in the Mishchenko–Fomenko index theory [57], where one considers the index of local elliptic operators over C ∗ -algebras and whose ideology we follow to a large extent here, the main role in the structure and the proof of such index formulas is played by the K¨ unneth formula describing unneth forthe K-group of tensor products of C ∗ -algebras. To ensure that the K¨ mula applies, we assume throughout the chapter that the algebra Λ belongs to the so-called class N (bootstrap class) in the sense of [66]. Let us recall the deﬁnition of this class. Deﬁnition 11.1. The class N is the minimal class of separable nuclear C ∗ -algebras such that 1. N contains the complex numbers C. 2. N is closed under countable inductive limits and KK-equivalence. 3. N satisﬁes the “2 out of 3 property”; i.e., if in an exact sequence 0 −→ A −−−−→ B −−−−→ C −→ 0 two terms are in N , then the third term is in N as well. It is known that class N contains all separable commutative C ∗ -algebras as well as all C ∗ -algebras of type I and their inductive limits. Many of the constructions in this chapter can be almost literally obtained from the constructions in the previous chapters. Whenever this is the case, we simply give the corresponding explanation and a reference rather than reproduce the material. 1 From

now on, Λ is an arbitrary C ∗ -algebra.

128

Chapter 11. Index of Nonlocal Operators over C ∗ -Algebras

11.1 Classiﬁcation of Nonlocal Elliptic Operators 11.1.1 Ell-Group Let M be a smooth Γ-manifold that is either a compact space or the noncompact total space of some real Γ-bundle over a compact base Y . Denote by EllΓ (M, Λ) the group of stable homotopy classes of elliptic pseudodiﬀerential operators of order zero over the algebra Λ acting in sections of nonlocal Λ-bundles over M . (If M is noncompact, we consider only pseudodiﬀerential operators multiplicatively trivial at inﬁnity.) The deﬁnition of this group is similar to the corresponding deﬁnition in Chap. 4: two nonlocal Λ-elliptic operators are equivalent if their direct sums with suitable trivial Λ-elliptic operators (i.e., isomorphisms of nonlocal Λ-bundles) are homotopic.

11.1.2 Diﬀerence Construction Let E be the noncompact total space of a real Γ-vector bundle. Let P1 , P2 ∈ MatN (C(E, Λ)Γ ) be two projections over C(E, Λ)Γ , and let a ∈ MatN (C(E, Λ)Γ ) be a homomorphism of nonlocal bundles deﬁned by these projections, i.e., an element such that P2 aP1 = a. We denote such a triple by a = (a, P1 , P2 ).

(11.1)

The following statement is valid (cf. Chap. 4). Proposition 11.2. The group of stable homotopy classes of elements (11.1) invertible at inﬁnity coincides with K0 (C0 (E, Λ)Γ ). The class [a] ∈ K0 (C0 (E, Λ)Γ ) deﬁned by an element a invertible at inﬁnity is called the diﬀerence construction of a.

11.1.3 Isomorphism of the Groups Ell and K Theorem 11.3 (Homotopy classiﬁcation). The mapping χ : EllΓ (M, Λ) −→ K0 (C0 (T ∗ M, Λ)Γ ) that takes a Λ-elliptic operator D to the diﬀerence construction [σ(D)] of its symbol is an isomorphism. Proof. The proof is completely similar to the corresponding proof in Chap. 4 and follows from an assertion similar to Proposition 4.6.

11.1. Classiﬁcation of Nonlocal Elliptic Operators

129

11.1.4 K¨ unneth Formula and Classiﬁcation Modulo Torsion Our group Γ acts trivially on the C ∗ -algebra Λ. This enables us to express (modulo torsion) the K-group in Theorem 11.3 in terms of the K-group of the usual nonlocal symbols and K-group of the algebra Λ. The computation is based on the K¨ unneth formula. Lemma 11.4. For a compact space X with a topologically free action of the group Γ, one has (the K¨ unneth formula) K0 (C(X, Λ)Γ ) ⊗ Q K0 (C(X)Γ ) ⊗ (K0 (Λ) ⊗ Q) ⊕ K1 (C(X)Γ ) ⊗ (K1 (Λ) ⊗ Q). (11.2) Proof. 1. By the isomorphism theorem (we can apply this theorem because the action of Γ on Prim(C(X, Λ)) = X × Prim(Λ) is topologically free), we obtain Λ. C(X, Λ)Γ C(X)Γ ⊗ (Since C(X)Γ is nuclear, the tensor product is unique.) 2. Since Λ is in the class N , we can apply the K¨ unneth formula in [66]. This gives us the exact sequence Λ) → Tor1 (K∗ (C(X)Γ ), K∗ (Λ)) → 0. 0 → K∗ (C(X)Γ ) ⊗ K∗ (Λ) → K∗ (C(X)Γ ⊗ (11.3) Recall the deﬁnition of the functor Tor1 (G, H) of two Abelian groups G and H: Tor1 (G, H) = ker((G ⊗ F2 ) → (G ⊗ F1 )), where H = F1 /F2 is a representation of H as the quotient of a free Abelian group by a free subgroup. Lemma 11.5. For any Abelian groups G and H, one has Tor1 (G, H) ⊗ Q 0. In other words, Tor1 (G, H) is a torsion group. Proof. Let eα be a basis in F1 , and let fβ be a basis in F2 . Then we decompose fβ in the basis eα with integer coeﬃcients cβα eα . fβ = Let a=

gβ ⊗ fβ ∈ ker (G ⊗ F2 ) −→ (G ⊗ F1 ) .

(The sum is ﬁnite.) This condition means that gβ ⊗ cβα eα = 0.

Chapter 11. Index of Nonlocal Operators over C ∗ -Algebras

130 It follows that

gβ cβα = 0

for all α. Consider the last equation as a system of equations for the unknowns gβ . (We consider ﬁnitely many values of the indices β and α.) The rank of this system is equal to the number of unknowns. Thus, its solution is a torsion element. In other words, the element a is a torsion element. This completes the proof of the lemma. 3. The tensor product of the sequence (11.3) by Q and the last lemma give us the desired statement of Lemma 11.4. Combining Lemma (11.4) with the homotopy classiﬁcation theorem, we obtain a rational isomorphism χ

: EllΓ (M, Λ) ⊗ Q −→ K0 (C0 (T ∗ M, Λ)Γ ) ⊗ (K0 (Λ) ⊗ Q) ⊕ K1 (C0 (T ∗ M, Λ)Γ ) ⊗ (K1 (Λ) ⊗ Q), (11.4) which can be viewed as the homotopy classiﬁcation of nonlocal elliptic operators over Λ modulo torsion.

11.2 Chern Character and the Index Theorem 11.2.1 Chern Character Let D be a nonlocal Λ-elliptic operator on M . For an element [σ(D)] ⊗ 1 ∈ K0 (C0 (T ∗ M, Λ)Γ ) ⊗ Q, denote its decomposition according to the K¨ unneth formula (11.2) by [σ(D)] ⊗ 1 = a0 ⊗ a0 + a1 ⊗ a1 . Deﬁnition 11.6 (cf. [57]). The Chern character of the symbol σ(D) is the class chΓ σ(D) = (chΓ a0 ) ⊗ a0 ∈

H ev (T ∗ Mg0 ) ⊗ K0 (Λ) ⊗ C,

g0

where chΓ : K0 (C0 (T ∗ M )Γ ) −→

H ev (T ∗ Mg0 ) ⊗ C

g0

is the Chern character (8.13), (8.14) in Chap. 8.

11.2. Chern Character and the Index Theorem

131

11.2.2 Index Theorem Theorem 11.7. Let D be a nonlocal Λ-elliptic operator on manifold M . Then the following index formula holds: ind D = chΓ σ(D) Td(T ∗ M ⊗ C; Γ), [T ∗ M ; Γ] ∈ K0 (Λ) ⊗ C.

(11.5)

Proof. The left and right-hand side of the index formula deﬁne the analytical and topological index, respectively, as homomorphisms inda , indt : K0 (C0 (T ∗ M, Λ)Γ ) ⊗ Q −→ K0 (Λ) ⊗ C. To prove the theorem, it suﬃces to show that the analytical index is equal to the topological index on the image in the Ell-group of each of the summands on the right-hand side in Eq. (11.2). Case 1. Product of Elements of Even K-Groups Consider the element σ ] ⊗ [q] ∈ K0 (C0 (T ∗ M )Γ ) ⊗ (K0 (Λ) ⊗ Q), [σ(D)] ⊗ 1 = [σ where

σ ] ∈ K0 (C0 (T ∗ M )Γ ), [σ

[q] ∈ K0 (Λ)

are the elements associated with a nonlocal elliptic symbol σ = (σ, P1 , P2 ) on T ∗ M , which deﬁnes an operator in L2 -sections of nonlocal bundles on M , and a projection q over Λ. The embedding of the tensor product of K-groups in the K¨ unneth formula (11.3) is deﬁned in terms of the product in K-theory, which in this case (i.e., for elements of even K-groups) is given by σ ] × [q] = [σ σ ⊗ 1Im q ] ∈ K0 (C0 (T ∗ M, Λ)Γ ), [σ where σ ⊗ 1, P1 ⊗ q, P2 ⊗ q), σ ⊗ 1Im q = (σ

acting in Hilbert Λ-modules is and the corresponding Λ-Fredholm operator D given by

= σ ⊗ 1Im q : Im P1 ⊗ Im q −→ Im P2 ⊗ Im q. D This operator is Λ-Fredholm. It is also obvious that σ ⊗ 1Im q ]) = [q] inda σ ∈ K0 (Λ) ⊗ C inda ([σ (cf. the proof of [57, Lemma 3.5]).

Chapter 11. Index of Nonlocal Operators over C ∗ -Algebras

132

On the other hand, the topological index of the symbol σ ⊗ 1Im q is equal to σ ⊗ 1Im q ]) = chΓ σ Td(T ∗ M ⊗ C; Γ), [T ∗ M ; Γ][q] = [q] inda σ ∈ K0 (Λ) ⊗ C. indt ([σ (Here we have used the index formula for nonlocal elliptic operators; see Corollary 9.3.) Thus, we have proved the desired equality of the topological and the analytical index in this case. Case 2. Product of Elements of Odd K-Groups Now consider [σ(D)] ⊗ 1 = a ⊗ b ∈ K1 (C0 (T ∗ M )Γ ) ⊗ (K1 (Λ) ⊗ Q), which is the product of elements a ∈ K1 (C0 (T ∗ M )Γ ),

b = [u] ∈ K1 (Λ)

of odd K-groups. In this case, the topological index is zero by deﬁnition. Let us show that the analytical index is zero as well. 1. One can readily prove that the index is zero if the element a satisﬁes the assumptions of the following lemma. To state it, consider the exact sequence l C(S ∗ M )Γ → 0 → C0 (0, 1) ⊗ C0 (T ∗ M )Γ ) → C(M )Γ → 0.

(11.6)

(The embedding l is induced by homeomorphisms T ∗ M \ 0 R+ × S ∗ M (0, 1) × S ∗ M .) Lemma 11.8. Let a ∈ K1 (C0 (T ∗ M )Γ ) be the image of some [p] ∈ K0 (C(S ∗ M )Γ ) under the mapping l∗ : K0 (C(S ∗ M )Γ ) → K1 (C0 (T ∗ M )Γ ) corresponding to the exact sequence (11.6) in K-theory. Then ind(a × b) = 0. Proof. It is well known that, for unital algebras A and B, the product of an element [p] ∈ K0 (A) of the even K-group by an element [u] ∈ K1 (B) of the odd K-group can be written as2 [p] × [u] = [p ⊗ u + (1 − p) ⊗ 1] ∈ K1 (A ⊗min B) (the minimal tensor product). 2 See,

for example, [41, Exercise 4.10.20].

11.2. Chern Character and the Index Theorem

133

Consider the diagram ×

K0 (C(S ∗ M )Γ ) ⊗ K1 (Λ)

/ K1 (C(S ∗ M )Γ ⊗ Λ) s

β

×

×

K1 (C(S ∗ M )Γ ⊗ C0 (0, 1)) ⊗ K1 (Λ)

K1 (C0 (T ∗ M )Γ ) ⊗ K1 (Λ)

/ K0 (C(S ∗ M )Γ ⊗ Λ) C0 (0, 1) ⊗ / K0 (C0 (T ∗ M )Γ ⊗ Λ).

Here the horizontal mappings are products in K-theory. The Bott periodicity isomorphism is denoted by β, and s stands for the suspension isomorphism. Finally, the mappings without labels are induced by embeddings of the corresponding C ∗ -algebras (e.g., the suspension over C(S ∗ M )Γ is a subalgebra in C0 (T ∗ M )Γ ). The diagram commutes, because products in K-theory commute with suspension and periodicity isomorphisms. This diagram shows that, under the assumptions of our lemma, the product a × b, where a = l∗ [p] and b = [u], is determined by the element p ⊗ u + (1 − p) ⊗ 1. Let P : L2 (M, Cn ) −→ L2 (M, Cn ) be a projection with symbol equal to p. Then ind(a × b) = ind[p ⊗ u + (1 − p) ⊗ 1] = ind(P ⊗ u + (1 − P ) ⊗ 1). However, the latter operator is invertible. (The inverse is P ⊗ u−1 + (1 − P ) ⊗ 1.) Hence the index is equal to zero. This completes the proof of the lemma. Lemma 11.9. The mapping l∗ : K0 (C(S ∗ M )Γ ) → K1 (C0 (T ∗ M )Γ ) in Lemma 11.8 is surjective if there exists a Γ-invariant nondegenerate vector ﬁeld on M . Proof. Indeed, the sequence of K-groups induced by (11.6) has the form · · · → K0 (C(S ∗ M )Γ ) → K1 (C0 (T ∗ M )Γ ) ∗ K1 (C(S ∗ M )Γ ) → · · · . → K1 (C(M )Γ ) →

l

The mapping l∗ is induced by the embedding C(M )Γ → C(S ∗ M )Γ . (In turn, this embedding is induced by the natural projection S ∗ M → M .) On the other hand,

Chapter 11. Index of Nonlocal Operators over C ∗ -Algebras

134

a nondegenerate Γ-invariant vector ﬁeld deﬁnes a section C(S ∗ M )Γ → C(M )Γ . Hence the sequence of K-groups splits, l∗ has trivial kernel, and we obtain the desired statement: the mapping K0 (C(S ∗ M )Γ ) −→ K1 (C0 (T ∗ M )Γ )

is surjective.

We have proved that the analytical index is zero for elements of EllΓ (M, Λ) equal to products of elements of the odd K-groups K1 (C0 (T ∗ M )Γ ) and K1 (Λ), provided that the manifold M has a nondegenerate Γ-invariant vector ﬁeld. 2. Let us reduce the general case to the special situation just described by taking the product of the manifold M by the circle S1 . We shall consider the trivial action of Γ on the second factor in M × S1 . Obviously, the vector ﬁeld ∂/∂ϕ on the product manifold is Γ-invariant. Let [A] ∈ EllΓ (M, Λ) be an element deﬁned by an Λ-elliptic operator A. Let [B] ∈ Ell(S1 ) be an element deﬁned by an elliptic pseudodiﬀerential operator on the circle which acts on functions and satisﬁes ind B = 1. To obtain an operator on the product of manifolds, we shall use the exterior products of elliptic operators, which we already discussed in Chap. 6. Lemma 11.10. The operator A#B =

A −B

B∗ A∗

is a Λ-Fredholm operator, and ind(A#B) = ind A · ind B = ind A ∈ K0 (Λ). Proof is standard. Indeed, we can proceed as follows. 1. The exterior product is additive; i.e., one has (A ⊕ A )#B = (A#B) ⊕ (A #B), and a similar formula holds for the second factor. 2. If A or B is invertible, then A#B is invertible. Indeed, ∗

A −B ∗ A B∗ ∗ (A#B) (A#B) = B A −B A∗ ∗

A A + B∗B 0 = , 0 AA∗ + BB ∗ and similarly for the product in reverse order. Thus, the problem is reduced to showing that a sum C + D of positive self-adjoint elements in a C ∗ -algebra

11.2. Chern Character and the Index Theorem

135

is invertible provided that one of the terms (say, C) is invertible. But this is trivial, since C + D = C 1/2 (1 + C −1/2 DC −1/2 )C 1/2 , and the invertibility of elements of the form “1 + a positive self-adjoint element” is known. 3. Now, by deﬁnition, A = M ⊕ K,

B = N ⊕ O,

where K : N1 → N2 is an operator in ﬁnitely generated projective modules and m − n = ind B. O = 0 : Cm → Cn , Using the properties mentioned earlier, we obtain A#B = (M #N ) ⊕ (M #O) ⊕ (K#N ) ⊕ (K#O). The ﬁrst three terms are invertible, while the last term acts in ﬁnitely generated projective modules, K#O : mN1 ⊕ nN2 −→ mN2 ⊕ nN1 , and therefore, ind(A#B) = [mN1 ⊕ nN2 ] − [mN2 ⊕ nN1 ] = (m − n)([N1 ] − [N2 ]) = ind B ind A.

Let (x, ξ) be canonical coordinates on T ∗ M , and let (y, η) be the canonical coordinates on T ∗ S1 . Let χ1 (x, y, ξ, η) and χ2 (x, y, ξ, η) be smooth nonnegative Γ-invariant scalar symbols on S ∗ (M × S1 ) such that 1. χ21 + χ22 = 1 everywhere. 2. χ1 = 0 in a neighborhood of the set ξ = 0, and χ2 = 0 in a neighborhood of the set η = 0. The existence of Γ-invariant symbols of this form can be proved by averaging over the compact Lie group Γ; the symbols after the averaging still vanish where they are supposed to vanish, because the group acts along (x, ξ) only. By Φ1 and Φ2 we denote self-adjoint Γ-invariant pseudodiﬀerential operators on M × S1 with symbols χ1 and χ2 , respectively. Consider the operator

∗ B = Φ1 A Φ 2 B ∗ . A# a −Φ2 B Φ1 A

Chapter 11. Index of Nonlocal Operators over C ∗ -Algebras

136

B is a pseudodiﬀerential operator on M × S1 Lemma 11.11. The operator A # a with smooth symbol equal to

σ(A)ϕ1 σ(B ∗ )ϕ2 σ(A #a B) = (11.7) −σ(B)ϕ2 σ(A∗ )ϕ1 on the cosphere bundle S ∗ (M × S1 ). The proof is similar to that of Lemma 7.13.

B are homotopic in the class of Lemma 11.12. The operators A#B and A # a Λ-Fredholm operators. In particular, one has B) = ind(A#B). ind(A # a Proof is similar to that of Lemma 7.14.

By applying Lemmas 11.10–11.12 to the pair (A, B), we obtain B). ind A = ind(A # a B is an elliptic pseudodiﬀerential operator on M × S1 . Moreover, the product A # a B deﬁnes the element in K0 equal to the Note that the symbol of A # a product of diﬀerence constructions of the symbols of A and B, B)] = [σ(A)] × [σ(B)], [σ(A # a where × : K0 (C0 (T ∗ M, Λ)Γ ) × K0 (C0 (T ∗ S1 )) → K0 (C0 (T ∗ (M × S1 ), Λ)Γ ) is the product in K-theory. Indeed, the product of symbols σ(A) and σ(B) in the sense of K-theory is obtained if in Eq. (7.13) we replace ϕ1 and ϕ2 by |ξ| and |η|, respectively. Clearly, the functions ϕ1,2 in the above construction can be reduced to this form by a homotopy. This gives the desired result: these two symbols deﬁne the same element in K-theory. It follows that if [σ(A)] lies in the subgroup generated by the product of odd K-groups (see the decomposition in Eq. (11.2)), then so does the element B)]. However, we know that the analytical index is trivial on the product [σ(A # a M × S1 , B) = 0. ind(A # a Hence ind A = 0. The proof of the index theorem is complete.

Part III

Examples

Chapter 12

Index Formula on the Noncommutative Torus In this chapter, we show that index formulas for nonlocal elliptic operators obtained in the preceding chapters can be applied to Connes’ problem [27] concerning the index of operators on the noncommutative torus.

12.1 Operators on the Noncommutative Torus Consider the symmetric Sobolev spaces H s (R) of functions on the real line R with the norm

s/2 d2 2 u us = 1 + x − 2 2, dx L

and let V1 and V2 be the operators in these spaces deﬁned as [V1 u](x) = u(x + 1),

[V2 u](x) = e−2πix/θ u(x).

(12.1)

Clearly, V1 and V2 satisfy the commutation relation V2 V1 = e2πi/θ V1 V2 . Denote by A∞ θ , where θ = 1/θ, the algebra of operators of the form

c=

∞

ajk V1j V2k ,

j,k=−∞

where the coeﬃcients ajk ∈ C rapidly decay, |ajk | ≤ C(1 + |j| + |k|)−l ,

j, k = 0, ±1, ±2, . . .

(12.2)

140

Chapter 12. Index Formula on the Noncommutative Torus

Operators of the form D=

α+β≤m

β d cαβ x −i : H s (R) −→ H s−m (R), dx α

(12.3)

where cαβ ∈ A∞ θ , are called diﬀerential operators of order ≤ m on the noncommutative torus. (For simplicity, we consider only scalar operators.) They are continuous in the Sobolev spaces introduced above. The symbol of the operator (12.3) is the operator cαβ ξ α η β : L2 (R) −→ L2 (R), ξ 2 + η 2 = 1, (12.4) σ(D)(ξ, η) = α+β=m

which is naturally considered as an element of the algebra C ∞ (S1 , A∞ θ ). The operator (12.3) is said to be elliptic if its symbol (12.4) is invertible. In this case, the operator has the Fredholm property. (In other words, the standard ﬁniteness theorem is true.)

12.2 Index Computation 12.2.1 Reduction to the Two-Dimensional Torus Lemma 12.1. The mapping I : S(R) −→ C ∞ (T2 , γ), f (ϕ + θn)e2πinψ (If )(ϕ, ψ) = n

is an isomorphism of the Schwartz space on the real line and the space C ∞ (T2 , γ) = {g ∈ C ∞ (R × S1 ) | g(ϕ + θ, ψ) = g(ϕ, ψ)e−2πiψ }

(12.5)

of smooth sections of the Bott bundle on the 2-torus. Proof. The inverse mapping I −1 is deﬁned as 1 (I −1 g)(x) = g(θ{x/θ}, ψ)e−2πi[x/θ]ψ dψ, 2π S1 where [a] and {a} are respectively the integer and fractional part of a real number a. Let

= a V1 , V2 , −i d , 2πx : S(R) −→ S(R) A dx θ

12.2. Index Computation

141

−1 , which is be an elliptic operator. Conjugation by I gives the operator A = I AI a nonlocal elliptic operator on the 2-torus. It is equal to

2π ∂ ∂ −i (12.6) A = a V1 , V2 , −i , ϕ : C ∞ (T2 , γ) −→ C ∞ (T2 , γ), ∂ϕ θ ∂ψ where the operators V1 and V2 now act as the shift by 1 in the variable ϕ and the product by e−2πiϕ/θ , respectively.

12.2.2 Index of Operators on the 2-Torus Let us compute the index of the last operator on the 2-torus by using the index formula for nonlocal elliptic operators (Theorem 9.5 for Γ = Z). The index formula (9.14) on the torus expresses the index of A in terms of the Chern character of its symbol σ as

chΓ [σ](0). (12.7) ind A = ind A = T ∗ T2

(Note that the Todd class of the torus is equal to 1.) To compute the Chern character of σ, we shall use connections (see Subsect. 8.4.2), because the operator A acts in sections of a nontrivial bundle. In the Bott bundle γ, we take the connection ∇γ = d + ϕdψ

#1 2 2πi (T )). : C ∞ (T2 , γ) −→ C ∞ (T2 , γ ⊗ θ

The curvature form is equal to (∇γ )2 =

2πi dϕdψ. θ

Let us parametrize T ∗ T2 T2 × R2 as ϕ ∈ [0, θ],

ψ ∈ [0, 1],

u ∈ (−π/2, π/2),

ξ = (cos t, sin t).

Substituting the expression for the Chern character given in Subsect. 8.4.2 into Eq. (12.7), we arrive at the following index formula in terms of the symbol σ: 1 1

= ind A = ind A τ (Ω2 ), (12.8) (−2πi)2 2! T ∗ T2 where Ω = (p∇γ⊕γ p)2 and the projection p ∈ C(T ∗ T2 , End(γ ⊕ γ))Z is equal to

1 1 − sin u (cos u)σ −1 p= , 1 + sin u 2 (cos u)σ while the trace τ:

#∗

(T ∗ T2 , End(γ ⊕ γ))Z −→

#∗

(T ∗ T2 )

142

Chapter 12. Index Formula on the Noncommutative Torus

is deﬁned as τ

V1k ωk

=

k∈Z

1 θ

S1

tr(ω0 ).

Finally, T ∗ T2 is equipped with the orientation such that the form dtdψdudϕ is positive. A clumsy computation of the right-hand side in (12.8), which we omit, gives 1 1 −1 d ind A = − σ)dt + τ0 (σ τ0 ((σ −1 dσ)2 σ −1 [ϕ, σ]), (12.9) 2πi S1 dt 4πiθ S1t ×S1ϕ where τ0 : A∞ θ −→ C is the trace equal to j 1 θ V1 fj ) = f0 (ϕ)dϕ. τ0 ( θ 0 One can also show that this formula coincides up to notation with the formula obtained in [27].

12.3 Special Cases Consider the ﬁrst-order operator

= x − (2P − 1) d , A dx

(12.10)

where P ∈ A∞ θ is a projection. (Such operators are automatically elliptic.) The symbol of the corresponding operator on the 2-torus (deﬁned using (12.6)) is homotopic to the symbol σ = sin t − (2P − 1)i cos t. For this symbol, the index formula (12.9) yields (after some computations) the following expression for the index (in terms of invariants of P ):

= −τ0 (2P − 1) + 1 τ0 (P [δ (P ), δ (P )]), ind A 1 2 πiθ

(12.11)

where δ1 , δ2 are the diﬀerentiations of the algebra A∞ θ deﬁned as δ1 (u) = [−2πiϕ, u],

δ2 (u) = [−θ

∂ , u]. ∂ϕ

(12.12)

For example, for P one can take the so-called Rieﬀel projection P = Pθ ∈ A∞ θ (see Eq. (1.10) in Example 1.9).

12.3. Special Cases

143

Let us compute the index of the operator (12.10) with the Rieﬀel projection P using formula (12.11). Clearly, + , 1 τ0 (P ) = τ0 (1) = 1, , θ where {1/θ} stands for the fractional part of 1/θ. The second term is equal to τ0 (P [δ1 (P ), δ2 (P )]) = 2πi.

(ﬁrst obtained in [27]): Thus, we obtain the following expression for the index of A + ,

= 2 1 − 2 1 + 1 = 1 + 2[1/θ], ind A θ θ where [1/θ] stands for the integer part of 1/θ.

Chapter 13

An Application of Higher Traces 13.1 Index with Values in Odd K-Groups Let P =

T (g)P (g) : C ∞ (M, CN ) −→ C ∞ (M, CN )

(13.1)

g

be a nonlocal elliptic operator with shifts generating the group Γ. Suppose that P is a projection; i.e., P = P ∗ = P 2. Let us deﬁne its index indΛ P ∈ K1 (Λ)

(13.2)

with values in the odd K-group of the group C ∗ -algebra Λ = C ∗ (Γ). This can be done, for example, as follows. First, extend the action of Γ on M to the action of the direct product Γ × Z on M trivial on the second factor Z. Denote the C ∗ -algebra of Γ × Z by Λ . Then Λ C(S1 , Λ).

(13.3)

(The isomorphism is deﬁned via Fourier series.) Let P : L2 (M, Λ) −→ L2 (M, Λ) be the standard lifting of the operator P to an operator in Hilbert Λ-modules (see Eq. (5.1)). We now deﬁne the operator B = uP + (1 − P) : L2 (M, Λ ) −→ L2 (M, Λ ), N

N

(13.4)

acting in Λ -modules, where u ∈ Λ is the operator of multiplication by the generator u ∈ Z.

146

Chapter 13. An Application of Higher Traces

Using the isomorphism (13.3), one can equivalently represent the operator B as the family eiϕ P + (1 − P) of operators in Λ-modules parameterized by ϕ ∈ S1 . The operator B is Λ -Fredholm. (Its almost inverse is u−1 P + (1 − P).) Now let us deﬁne indΛ P = indΛ B ∈ K0 (Λ ). The homomorphism Λ → Λ takes the latter index to the zero element of K0 (Λ): 1 2 indΛ B ∈ ker K0 (Λ ) −→ K0 (Λ) . Finally, note that 2 1 ker K0 (Λ ) −→ K0 (Λ) K0 (Λ ⊗ C0 (R)) K1 (Λ). (The last “” stands for the suspension isomorphism.) Thus, the index of the operator B can be interpreted as the desired index with values in K1 (Λ). Proposition 13.1. If the projection P can be lifted to a projection P in Λ-modules, then the index (13.2) is zero. Proof. Suppose that there exists such a projection, P. Then the corresponding operator B is invertible, and B−1 = u−1 P + (1 − P), since u and P commute. Hence indΛ B = 0.

Below we shall give an example of an operator with nontrivial index. But before that, we obtain cohomological formulas for the pairing of this index with traces of odd degree over Λ∞ .

13.2 Odd Index Formula One can extract numerical invariants from the Λ-index (13.2) by pairing it with cycles of odd degree over Λ. In this section, we deﬁne these numerical invariants and express them in topological terms. Let (Ω, τ ) be a cycle over the algebra Λ, where Ω is a diﬀerential graded algebra such that its component of grading zero Ω0 is just Λ and τ : Ωn −→ C is a closed graded trace of degree n.

13.2. Odd Index Formula

147

13.2.1 Suspension Let

Ω(Z) = C ∞ (Z) ⊕ C ∞ (Z)du

be the algebra of exterior forms on the group Z. Let us construct a cycle (Ω , τ ) over Λ . For the diﬀerential graded algebra we take the graded tensor product Ω = Ω(Z) ⊗ Ω. Recall that the product of elements in the graded tensor product is deﬁned by the formula (ω1 ⊗ ω2 )(ω3 ⊗ ω4 ) = (−1)deg ω2 deg ω3 ω1 ω3 ⊗ ω2 ω4 , where the diﬀerential is d(ω1 ⊗ ω2 ) = dω1 ⊗ ω2 + (−1)deg ω1 ω1 ⊗ dω2 . (Cf. Sec. 10.1.) We deﬁne a trace

τ : Ω −→ C

by setting

τ (u

0 −l

⊗ ω) = 0 for all l,

τ (u

−k

du ⊗ ω) =

0, 2πiτ (ω),

k = 1, k = 1.

Obviously, this trace has degree n + 1. The veriﬁcation of the fact that τ is a closed graded trace is left to the reader as an easy exercise.

13.2.2 Index Theorem Let P be a projection as above. Its principal symbol is a projection over the algebra C(S ∗ M ) Γ. Denote its class in K-theory by [σ(P )] ∈ K0 (C(S ∗ M ) Γ). Theorem 13.2 (Cohomological formula for Λ-Fredholm index). Let P be a nonlocal projection with symbol σ(P ), and let τ be a closed graded trace of odd degree over Λ∞ . Then 3 4 chτ (indΛ (P )) = chτ (σ(P ))Td(T M ⊗ C, Γ), [S ∗ M, Γ] ∈ C| Γ| , (13.5) where is the fundamental class.

[S ∗ M, Γ](g0 ) ∈ Hodd (S ∗ Mg0 )

148

Chapter 13. An Application of Higher Traces

Remark 13.3. The Chern character on the left-hand side of the equation ranges in the space C| Γ×Z| corresponding to the set of conjugacy classes of Γ × Z. However, it follows from the deﬁnition of the trace τ that the components of the Chern character corresponding to elements (g, k) ∈ Γ × Z are zero unless k = 0. Proof. The Chern character on the left-hand side in Eq. (13.5) can be expressed via the symbol of B by the index formula (10.7). Hence it remains to show that the corresponding expression can be converted to the form occurring on the right-hand side in (13.5). The computation is divided into two steps. Step 1. Reduction to the Cosphere Bundle and the Chern–Simons Character Lemma 13.4. Let σ ∈ MatN (C ∞ (S ∗ M ) Γ) be an elliptic symbol, and let τ be a closed graded trace of even degree over C ∞ (Γ). Then for any g0 ∈ Γ and x ∈ H ev (Mg0 ) one has chτ (σ)(g0 )x, [T ∗ Mg0 ] = csτ (σ)(g0 )x, [S ∗ Mg0 ], where

(13.6)

csτ (g0 ) : K1 (C(S ∗ M ) Γ) −→ H odd (S ∗ Mg0 )

is the Chern–Simons character ) csτ [a](g0 ) = τ Πg0 n≥0

of an element

* n! −1 2n+1 (a da) ∈C (2πi)n+1 (2n + 1)!

[a] ∈ K1 (C(S ∗ M ) Γ)

represented by an invertible matrix a over C ∞ (S ∗ M ) Γ. Proof. First, we represent the left-hand side of Eq. (13.6) as an integral over the Atiyah–Bott–Patodi space (see (8.22)), where chτ (σ) is replaced by the Chern character of the projection deﬁned in Eq. (8.17):

1 (1 − sin ψ)I σ −1 cos ψ . (13.7) p= σ cos ψ (1 + sin ψ)I 2 Then the integration over ψ gives precisely the expression on the right-hand side in Eq. (13.6). We omit this rather standard computation. The proof is complete. Step 2. Desuspension and Integration over S1 Lemma 13.5. For the symbol σ = up + (1 − p) ∈ MatN (C ∞ (S ∗ M ) (Γ × Z)),

13.3. Example of Λ-Index Computation

149

one has csτ (σ)(g0 , e) = chτ (p)(g0 ),

(13.8)

where e = 0 is the unit element of the group Z. Proof. 1. Before computing the Chern–Simons character of this symbol, we mention a couple of useful formulas: b−1 db = u−1 du p + (1 − p + u−1 p)(u − 1)dp ≡ A + B; 1 2 B 2 = (u − 1)2 u−1 dp p dp + p dp dp , where

b = u−1 p + (1 − p).

2. We obtain (for brevity, we omit the subscripts (g0 , e) ∈ Γ × Z on τ ) 1 2 1 2 τ (b−1 db)2n+1 = (2n + 1) τ AB 2n 1 2 (u − 1)2n p(dp dp)n = (2n + 1) τ u−1 du n u 2n (n) 1 2 −1 n ((u − 1) ) = (2n + 1) τ u du p(dp dp) n! u=0 2 1 (2n + 1)!(−1)n = 2πi τ p(dp dp)n . (n!)2 3. Substituting the last expression into the Chern–Simons character, we obtain csτ (b)(g0 , e) =

n! (2πi)n+1 (2n

1 2 τ (b−1 db)2n+1

+ 1)! )

* n 1 2 1 dp dp 1 n τ p(dp dp) = τ p exp − = − = chτ (p). n! 2πi 2πi n≥0

n≥0

The proof of the lemma is complete.

Now, to conclude the proof of Theorem 13.2, it remains to apply Lemmas 13.4 and 13.5.

13.3 Example of Λ-Index Computation Consider the operator Q = pθ H : C ∞ (S1 ) −→ C ∞ (S1 ), where H is the Hardy projection on the space of boundary values of holomorphic functions in the unit disc and p = pθ = V −1 g + f + gV

150

Chapter 13. An Application of Higher Traces

is the Rieﬀel projection (see Example 1.9). The symbol of Q is a projection: 0 p, ξ > 0, σ(Q) = 0, ξ < 0. (Recall that the symbol of the Hardy projection is equal to 1 on the circle {ξ = 1} in S ∗ S1 and is zero on the circle {ξ = −1}.) Let us show that the index indΛ Q ∈ K1 (Λ) is not zero. To this end, we take the closed graded trace ag0 g1 τ ag0 g1 g0 dg1 = g0 g1 =e

of degree 1 over the group algebra of Z (this trace was introduced in Sec. 10.2) and compute the number chτ [indΛ Q](e). Theorem 13.2 gives the following formula: 1 τ p dp dp (e). chτ [indΛ Q](e) = − 2πi S1 Let us compute the right-hand side of this formula. Multiplying out p dp dp and leaving only the terms of zero degZ grading, we obtain 1 2 τ (p dp dp) = τ (V −1 g + f + gV )((dV −1 g + V −1 dg) + df + (dg V + g dV ))2 1 = τ V −1 g(df g dV + g dV df ) + f (dg V dV −1 g + g dV V −1 dg 2 + dV −1 g dg V + V −1 dg g dV ) + gV (dV −1 g df + df dV −1 g) . Making use of the identity τ (V −1 dV ) = τ (dV V −1 ) = 1, we can rewrite the latter expression as τ (p dp dp) = V −1 (g 2 df ) − df V −1 (g 2 ) − 2f g dg + g dg V (f ) + f V −1 (g dg) + g 2 df − g 2 V (df ). (Here V (f ) = V f V −1 is the induced action of the shift operator on functions.) When integrating the last expression over Γ = S1 , we can apply the shift operator to each of the terms. Thus, the integral is equal to τ (p dp dp) = 2g 2 df + 2g dg V (f ) − 2f g dg − 2g 2 V (df ).

13.3. Example of Λ-Index Computation

151

The integrand vanishes outside the interval (0, ε), since the function g is identically zero outside this interval. On this interval, we use the identity f + V (f ) = 1 to transform the integral to the form ε ε 2 (4g df + 2g dg − 4f g dg) = (4g 2 df − 4f g dg) τ (p dp dp) = 0 0 ε ε 2 [4(f − f )df − 2f (df − 2f df )] = 2f df = 0

0

= f 2 (ε) − f 2 (0) = 1. Thus, we have shown that the index of our projection P is nonzero.

Chapter 14

Index Formula for a Finite Group Γ 14.1 Trajectory Symbol Let Γ be a ﬁnite group acting topologically freely on a smooth closed manifold M , and suppose that we are given a nonlocal elliptic operator1 A= T (g)A(g) : C ∞ (M, Cn ) −→ C ∞ (M, Cn ). g∈Γ

The nonlocal equation Au = f can be reduced to a system of local equations. To this end, we replace the functions u and f by the vector functions U = {Ug }g∈Γ ,

F = {Fg }g∈Γ

whose components Ug = T (g)u,

Fg = T (g)f

are obtained as shifts of the original functions by the elements of Γ. For U , we obtain the system AU = F, where A = (Agh )g,h∈Γ 1 For

simplicity, we consider operators in trivial bundles.

(14.1)

154

Chapter 14. Index Formula for a Finite Group Γ

is the matrix operator whose entries Agh = T (h)A(g −1 h)T (h)−1 are (pseudo)diﬀerential operators. Deﬁnition 14.1. The trajectory symbol σT (A) of the nonlocal operator A is the symbol of the operator A: (14.2) σT (A)gh = T (h)σ(A(g −1 h))T (h)−1 ∈ C ∞ S ∗ M, Matn|Γ| (C) . The trajectory symbol is invariant under the action (T (k)U )(h, x) = U (k −1 h, k −1 x),

k, h ∈ Γ,

x ∈ S ∗M

of Γ on C ∞ (S ∗ M, Cn|Γ| ). This follows from the invariance of A under a similar action of Γ on C ∞ (M, Cn|Γ| ). Note that we already met the trajectory symbol in the “abstract” situation in Chap. 1. The following statements are valid. 1. The subspace of Γ-invariant sections in C ∞ (M, Cn|Γ| ) is naturally isomorphic to C ∞ (M, Cn ). (This isomorphism takes a function u to the vector U with components Ug = T (g)u.) Moreover, the restriction of A to this invariant subspace is equal to the original nonlocal operator A. 2. An application of the isomorphism theorem (Theorem 1.12) allows us to identify the algebra of symbols of nonlocal operators and the algebra of Γ-invariant matrix symbols by j : C ∞ (S ∗ M )Γ −→ C ∞ (S ∗ M, Mat|Γ| (C))Γ , T (g)a(g) −→ agh = T (h)a(g −1 h)T (h)−1 .

(14.3)

g∈Γ

14.2 Index Formula Let us obtain an index formula for operator A in terms of its trajectory symbol using the index formula obtained in Chap. 9. Note that isomorphisms similar to (14.3) hold also for algebras of diﬀerential forms on the cotangent bundle T ∗ M , with the standard diﬀerential. (It acts on matrices componentwise and commutes with shift operators.) In particular, the # # trace τ on (T ∗ M )Γ induces a trace on the algebra (T ∗ M, Mat|Γ| (C))Γ . Lemma 14.2. One has τ ω, g0 = where ω ∈

# 1 tr(T (g0 )j(ω)|T ∗ Mg0 ) ∈ (T ∗ Mg0 ), |Cg0 |

(14.4)

# ∗ (T M )Γ and g0 ∈ Γ are arbitrary and Cg0 is the centralizer of g0 .

14.2. Index Formula

155

Proof. Since Γ is ﬁnite and, in particular, compact, we have τ

g

1 , gωg , g0 = h∗ (ωg |T ∗ Mg ) |Cg0 | g∈ g0 Γg0 ,g

where Γg0 ,g = {h ∈ Γ | hg0 h−1 = g}. Note that the sets Γg0 ,g for diﬀerent g ∈ g0 are disjoint and cover the entire group Γ. Moreover, the element h runs over the entire group Γ. Thus, the latter formula can be rewritten as 1 ∗ gωg , g0 = h (ωhg0 h−1 )|T ∗ Mg0 . (14.5) τ |Cg0 | g h∈Γ

On the other hand, tr(T (g0 )j(ω)|T ∗ Xg0 ) =

(T (g0 ))gh ωh,g |T ∗ Mg0 =

g,h∈Γ

=

(g −1 )∗ (ωg−1 g0 g )|T ∗ Mg0 =

g∈Γ

ωg−1 g,g |T ∗ Mg0 0

g∈Γ

h∗ (ωhg0 h−1 )|T ∗ Mg0 .

h∈Γ

Comparing the latter expression with (14.5), we obtain the desired equation (14.4). The proof of the lemma is complete. Now we can give an index formula for an elliptic operator A in terms of its trajectory symbol a = σT (A). Consider the localized Chern character

6 5 # dp dp 0 0 ∗

∈ ev ch a(g0 ) = tr T (g0 ) p exp − − c (T Mg0 ), 0 1 2πi on T ∗ Mg0 (the trace tr is well deﬁned, because Γ is ﬁnite), where the projection p is equal to

1 (1 − sin ψ) a−1 cos ψ p= . a cos ψ (1 + sin ψ) 2 Proposition 14.3. Let A be a nonlocal elliptic operator with trajectory symbol a. Then 1 ch a(g0 ) Td(T M ⊗ C, g0 ), [T ∗ Mg0 ] . (14.6) ind A = |C | g 0 g 0

Proof. This formula is obtained from the index formula (9.13) if we replace the symbol σ(A) by the trajectory symbol a and the trace τ occurring in the deﬁnition of the Chern character chΓ by the trace tr using Eq. (14.4).

156

Chapter 14. Index Formula for a Finite Group Γ

Remark 14.4. Formula (14.6) contains the expressions ch a(g0 ) Td(T M ⊗ C, g0 ), [T ∗ Mg0 ] , which can be rewritten (using results due to Atiyah–Segal–Singer [11]) as the Lefschetz number ch a(g0 ) Td(T M ⊗ C, g0 ), [T ∗ Mg0 ] = L(A, g0 ) of the operator A and the transformation T (g0 ), which is deﬁned as follows: L(A, g) = tr T (g)|ker A − tr T (g)|ker A∗ . Thus, we obtain the index formula ind A =

1 L(A, g) |Γ|

(14.7)

g∈Γ

due to Antonevich [5]. (Here we have made use of the identity |Γ| = | g0 ||Cg0 | from the theory of ﬁnite groups and the simple fact that Lefschetz numbers of conjugate elements coincide.) Remark 14.5. In this chapter, we assigned a Γ-equivariant operator to a nonlocal operator A. (The group is assumed to be ﬁnite.) Of course, a similar mapping is deﬁned for symbols as well. Obviously, this induces a well-deﬁned homomorphism K1 (C(S ∗ M ) Γ) −→ KΓ1 (S ∗ M )

(14.8)

of the K-group of the crossed product algebra into the equivariant K-group. It can be shown that this mapping coincides with the well-known Green–Julg isomorphism (e.g., see [22, Th. 11.7.1.]).

Part IV

Appendices

The results presented in the main text of the book are essentially based on notions and methods of the theory of C ∗ -algebras and K-theory of operator algebras. For the reader’s convenience, some basic deﬁnitions and facts concerning these notions and methods are given below in Appendices A and B. As a rule, the proofs of propositions and theorems are omitted. (They can be found in the relevant literature, including [6,22,35,41,62] etc.) The appendix does not claim to be a complete exposition of anything, and it should be treated not as an introduction to the topic but just as reference material intended possibly to decrease the reader’s need to consult other sources when reading the book. In Appendix C, we brieﬂy describe the basic concepts of the theory of cyclic homology and cohomology. Although this material is practically not used in the book (except for the notions of enveloping diﬀerential algebra and closed cyclic traces), we still include it in the book, so as to give the reader the impetus to further profound study of the literature.

Appendix A

C ∗-Algebras A.1 Basic Notions A.1.1 Deﬁnitions and Examples Let H be a Hilbert space, and let A = B(H) be the set of bounded linear operators in H. This set has the following properties. 1. A is an algebra over the ﬁeld C of complex numbers. 2. A is a Banach space (i.e., a complete normed vector space). The norm in B(H) is given by the formula a = sup u∈H u=0

au , u

a ∈ B(H).

It is called the operator norm. 3. The multiplication in A is norm continuous, and moreover, the following inequality holds: ab ≤ a b , a, b ∈ A. 4. In the algebra A, an involution (i.e., an antilinear antihomomorphism) a → a∗ is deﬁned, and moreover, a = a∗ for any a ∈ A. The involution in B(H) takes each operator to the adjoint operator. 5. Each element a ∈ A satisﬁes the relation a = a∗ a . 2

The norm satisfying such a relation is called a C ∗ -norm. It is precisely properties 1–5 that characterize C ∗ -algebras. In addition, we give the following deﬁnition.

Appendix A. C ∗ -Algebras

162 Deﬁnition A.1. An algebra A is called 1. A Banach algebra if properties 1–3 are satisﬁed.

2. An involutive Banach algebra if properties 1–4 are satisﬁed. 3. A C ∗ -algebra if properties 1–5 are satisﬁed. A subalgebra B of a C ∗ -algebra A is itself a C ∗ -algebra if it is closed and invariant under involution; in this case, it is called a C ∗ -subalgebra. In what follows, unless otherwise speciﬁed, we only deal with C ∗ -algebras, and speaking of subalgebras, we always mean C ∗ -subalgebras. The algebra B(H) and its subalgebras are examples of C ∗ -algebras. (They are called operator C ∗ -algebras.) It turns out that, in fact, there are no other examples. Theorem A.2 (Gelfand–Naimark). Any C ∗ -algebra is isomorphic to an operator C ∗ -algebra. Remark A.3. The Hilbert space H where the elements of a given C ∗ -algebra are realized as operators need not be separable. Example A.4. We present several examples of C ∗ -algebras. 1. The algebra K(H) of all compact operators in a Hilbert space H. This is a subalgebra in B(H). 2. The algebra Mn (C) of complex n × n matrices and its subalgebras. (Of course, these algebras are realized by operators in Cn .) 3. More generally, the algebra Mn (A) of n × n matrices whose entries are elements of some C ∗ -algebra A. 4. The algebra C(X) of complex-valued bounded continuous functions on a locally compact space X and its subalgebra C0 (X) consisting of functions tending to zero at inﬁnity (i.e., of functions f (x) such that for each ε > 0 the inequality |f (x)| < ε holds outside some compact subset in X depending on ε). Under additional conditions, such an algebra can be realized by operators in the space L2 (X, μ) of functions on X square integrable with respect to a given measure μ on X.

A.1.2 Unital Algebras and Units A C ∗ -algebra A is said to be unital if it contains a unit element, i.e., an (obviously, unique) element 1 such that a1 = 1a

for all a ∈ A.

One can readily verify that, in addition, this element is also self-adjoint, i.e., satisﬁes the relation 1∗ = 1. In the above examples, the algebras Mn (C) and C(X) are unital, and the algebras K(H) and C0 (X) (in the case of an inﬁnite-dimensional space H and

A.1. Basic Notions

163

noncompact space X) are not unital. The algebra Mn (A) is unital if and only if the algebra A itself has this property. Although a C ∗ -algebra may not contain a unit, it always contains so-called approximate units. An approximate unit is deﬁned to be a net en of self-adjoint elements of a C ∗ -algebra A such that lim en a = lim aen = a

n→∞

n→∞

for each element a ∈ A. Approximate units replace the “true” unit in the study of many problems. If A is a possibly nonunital C ∗ -algebra, then it can always be transformed into a unital one by adjoining the unit to it, i.e., by considering the algebra A+ whose elements are pairs (λ, a), where λ ∈ C and a ∈ A, and the algebraic operations are deﬁned by the formulas (λ, a) + (λ , a ) = (λ + λ , a + a ),

(λ, a)(λ , a ) = (λλ , λ a + λa + aa ),

(λ, a)∗ = (λ, a∗ ). The unit of the algebra A+ is the element (1, 0). The algebra A is naturally embedded in A+ by the mapping a → (0, a). Proposition A.5. On the algebra A+ , there exists a norm making it a C ∗ -algebra. If the algebra A itself is already unital, then A+ is the direct sum of the algebras A and C in the sense of the following deﬁnition. Deﬁnition A.6. The direct sum of C ∗ -algebras A and B is the algebra A⊕B whose elements are pairs (a, b), a ∈ A, b ∈ B, and the algebraic operations are deﬁned componentwise. Proposition A.7. On the direct sum A⊕B, there is a norm making it a C ∗ -algebra. If both algebras A and B are unital, then A ⊕ B is also unital; the unit in it is the element (eA , eB ), which is the direct sum of the units in A and B.

A.1.3 Homomorphisms, Ideals, Quotient Algebras, and Extensions A mapping f : A → B of a C ∗ -algebra A into a C ∗ -algebra B is called a homomorphism (of C ∗ -algebras) if it preserves the algebraic operations and the involution, i.e., if ∗ f (a + λb) = f (a) + λf (b), f (ab) = f (a)f (b), f (a∗ ) = f (a) , a, b ∈ A,

λ ∈ C.

Note that we do not require the mapping f to be continuous, because, in fact, this property is satisﬁed automatically.

Appendix A. C ∗ -Algebras

164

Proposition A.8. An arbitrary homomorphism f : A → B of C ∗ -algebras does not increase the norm: f (a) ≤ a , a ∈ A. Corollary A.9. If A is a C ∗ -algebra, then the C ∗ -norm on A is uniquely determined. Let f : A → B be a homomorphism of C ∗ -algebras. The kernel J = {a ∈ A : f (a) = 0} of the homomorphism f is a closed linear subspace in A such that aJ ⊂ J,

Ja ⊂ J

∗

for any a ∈ A, ∗

where J = {a∗ : a ∈ J}.

J = J,

Any subspace J ⊂ A satisfying the above conditions is called a two-sided C ∗ -ideal in A. (In what follows, for brevity, such subspaces will simply be called ideals.) Thus, the kernel of a C ∗ -algebra homomorphism is an ideal. Conversely, each ideal J in A is the kernel of a homomorphism into some C ∗ -algebra B. Namely, for B we can take the quotient algebra A/J determined as the set of cosets a + J in the initial algebra A; this set is equipped with the algebraic operations (a + J) + (a + J) = (a + a ) + J,

(a + J)(a + J) = aa + J,

(a + J)∗ = a∗ + J. Proposition A.10. On the quotient algebra A/J, there exists a C ∗ -norm. Thus, A/J is naturally a C ∗ -algebra, and we obtain the exact sequence f

g

0 −→ J −−−−→ A −−−−→ A/J −→ 0 of C ∗ -algebras, where f is the embedding of J in A and g is the natural projection onto the quotient algebra (g(a) = a + J). In general, if an exact sequence1 f

g

0 −→ A −−−−→ B −−−−→ C −→ 0

(A.1)

of C ∗ -algebras is given, then A is naturally identiﬁed with the image of itself, which is automatically an ideal in B, and C is identiﬁed with the quotient algebra of B by this ideal. Deﬁnition A.11. The algebra B in the exact sequence (A.1) is called an extension of the algebra C by the algebra (ideal) A. The sequence (A.1) is said to split if there exists a C ∗ -algebra homomorphism h : C → A such that g ◦ h = idC (the identity homomorphism of the algebra C). Obviously, in this case, B = A ⊕ C, and the extension (A.1) is said to be trivial. Extensions of C ∗ -algebras are considered in more detail in Sec. B.2. 1 Recall

that a sequence of homomorphisms is said to be exact if the kernel of each homomorphism in this sequence coincides with the image of the preceding homomorphism.

A.1. Basic Notions

165

A.1.4 Commutative C ∗ -Algebras Consider the structure of commutative C ∗ -algebras. Let A be a commutative C ∗ -algebra. An ideal J ⊂ A is said to be maximal if it is proper (i.e., is nonzero and does not coincide with A) and is not contained in any other proper ideal of A. If J ⊂ A is a maximal ideal, then the quotient algebra A/J is a Banach ﬁeld (a commutative Banach algebra in which any nonzero element is invertible) and hence A/J = C by the Gelfand–Mazur theorem. Thus, for each maximal ideal J ⊂ A there exists a naturally deﬁned homomorphism ϕJ : A −→ C. (Note that the algebra C does not have nontrivial automorphisms.) Let X be the set of all maximal ideals of the algebra A. Then to each element a ∈ A one can assign a function a on X by setting a(J) = ϕJ (A),

J ∈ X.

The function a is called the Gelfand transform of the element a. Theorem A.12 (Gelfand–Naimark). The set X can be equipped with the topology of a locally compact space such that the Gelfand transform is an isomorphism of the algebra A onto the algebra C0 (X) of functions on X vanishing at inﬁnity. The space X is compact if and only if the algebra A is unital ; in this case, C0 (X) = C(X), so that the Gelfand transform realizes an isomorphism of the algebra A onto the space of all continuous functions on X.

A.1.5 Spectrum and Functional Calculus Deﬁnition A.13. Let A be a unital C ∗ -algebra. The spectrum of an element a ∈ A is deﬁned to be the set of complex numbers λ for which the element a − λ ≡ a − λ1 is not invertible: σ(a) = {λ ∈ C : (a − λ)−1

does not exist}.

In what follows, we sometimes need an explicit indication of the algebra A with respect to which the spectrum of an element is considered. In this case, we write σA (a) instead of σ(a). Deﬁnition A.14. Let A be a nonunital C ∗ -algebra. The spectrum of an element a ∈ A is deﬁned as the spectrum of a in the algebra A+ : σ(a) = σA+ (a). Note that if A is unital, then σA+ (a) = σA (a) ∪ {0}.

Appendix A. C ∗ -Algebras

166

Theorem A.15 (Spectral invariance). Let A be a unital C ∗ -algebra, and let B ⊂ A be a unital subalgebra.2 Then σA (a) = σB (a) for each element a ∈ B. In particular, if an element a ∈ B is invertible in A, then it is also invertible in B (i.e., a−1 ∈ B). Let A be a unital C ∗ -algebra, let a ∈ A, and let f (λ) be a function holomorphic in a neighborhood U of the spectrum of a on the complex plane. Further, let γ be a contour in the complex plane surrounding the spectrum σ(a) in the positive sense and lying entirely in U . Then we can deﬁne the function f (a) of the element a using the Cauchy integral by setting 7 1 f (λ)(λ − a)−1 dλ. f (a) = 2πi γ This integral gives a well-deﬁned (i.e., independent of the choice of the contour) element f (a) of the algebra A. This construction is referred to as holomorphic functional calculus. Theorem A.16 (The spectral mapping theorem). The following relation holds: σ(f (a)) = f (σ(a)). For a given element a ∈ A, the mapping f → f (a) is a homomorphism of the algebra of functions holomorphic in neighborhoods (diﬀerent for diﬀerent functions) of the spectrum of a into the algebra A. Moreover, ∗ f (a) = f (a∗ ), where the holomorphic function f (λ) is deﬁned by the formula f (λ) = f (λ). If only self-adjoint elements a∗ = a of the algebra A are considered, then functional calculus can be deﬁned for a much wider class of functions. In this case, the spectrum σ(a) is a compact subset of the real axis, and the operator f (a) can be deﬁned for any continuous function f on the spectrum σ(a). (To this end, one can use the Gelfand transform for the commutative C ∗ -subalgebra B ⊂ A generated by the element a.) Functional calculi and spectral mapping theorems can also be constructed for elements of nonunital algebras by passing from the algebra A to the algebra A+ . But in this case the condition f (0) = 0 ensuring that f (a) ∈ A should be imposed on admissible functions f . 2 In

this case, we assumed that 1B = 1A .

A.1. Basic Notions

167

A.1.6 Local C ∗ -Algebras In numerous problems of noncommutative geometry, it does not suﬃce to consider only C ∗ -algebras, since, for example, the (co)homology theory may well be trivial for them. Therefore, one should use a diﬀerent, somewhat more complicated class of algebras, which inherit good properties of C ∗ -algebras but do not inherit their bad properties. We mean the class of local C ∗ -algebras, for which we now give the corresponding deﬁnition. This deﬁnition has two versions, one of which can be used for unital algebras, while the other can be used for nonunital algebras. Deﬁnition A.17. Let A be a C ∗ -algebra, and let A0 ⊂ A be a dense subalgebra invariant under the involution and equipped with the structure of a Fr´echet algebra such that the embedding A0 ⊂ A is continuous. We say that A0 is a local C ∗ -algebra if A0 is closed with respect to holomorphic functional calculus in A, i.e., if one of the following two conditions is satisﬁed: 1. A0 is unital, and for any element a ∈ A0 and function f (z) holomorphic in a neighborhood of the spectrum σA (a) of the element a in the algebra A, the element f (a) lies in A0 . 2. A0 is nonunital, and condition 1 holds under the additional assumption that f (0) = 0. Remark A.18. In the unital case, condition 1 is equivalent to any of the following two conditions (e.g., see [67, 68]): (i) If a ∈ A0 , then ea ∈ A0 as well. (ii) If an element a ∈ A0 is invertible in A, then a−1 ∈ A0 ; i.e., it is also invertible in A0 . Example A.19. By way of example, consider the C ∗ -algebra C(X) of continuous functions on a smooth manifold X and the subalgebra C k (X) of k times continuously diﬀerentiable functions (or the subalgebra C ∞ of inﬁnitely diﬀerentiable functions). This subalgebra is a local C ∗ -algebra in the sense of the above deﬁnition.

A.1.7 Positive Elements An element a of a C ∗ -algebra A is said to be positive if it is self-adjoint (i.e., a = a∗ ) and σ(a) ∈ [0, +∞). We denote the set of positive elements of the algebra A by A+ . Each element of the form b∗ b is positive, and each element a ∈ A+ can be represented in the form a = b2 , where b is self-adjoint. Finally, each self-adjoint element a ∈ A can be represented in the form a = a+ − a− , where a+ , a− ∈ A+ and a+ a− = a− a+ = 0 (the Hahn decomposition).

Appendix A. C ∗ -Algebras

168

A.1.8 Projections in C ∗ -Algebras Let us present several technical results about projections in C ∗ -algebras. For details, see [22]. Deﬁnition A.20. An element p ∈ A such that p = p∗ = p2 is called a projection in the C ∗ -algebra A. Obviously, any projection p lies in the set A+ , and σ(p) ∈ {0, 1}. On the set of projections, there is a natural order. It is given by p ≤ q ⇐⇒ q − p ∈ A+ . Proposition A.21. If p and q are projections in a C ∗ -algebra A, then the relation p ≤ q holds if and only if pq = qp = p. In this case, q − p is also a projection in A. On the set of projections in a C ∗ -algebra A, the following three equivalence relations are introduced: 1. (Algebraic equivalence.) There exists an element u ∈ A such that u∗ u = p,

uu∗ = q.

(Such an element u is called a partial isometry.) 2. (Similarity.) There exists a unitary element u ∈ A such that u∗ pu = q. 3. (Homotopy.) In the algebra, there exists a continuous family of projections p(t), t ∈ [0, 1], such that p(0) = p,

p(1) = q.

Proposition A.22. Each of the above equivalence relations is strictly stronger than the preceding one, namely: 1. If projections p and q are similar, then they are algebraically equivalent; the converse statement is generally not true. 2. If projections p and q are homotopic, then they are similar ; the converse statement is generally not true. Moreover, if p − q < 1, then the projections p and q are homotopic.

A.2. Representations of C ∗ -Algebras

169

Although these equivalence relations are not equivalent, it is possible to obtain converses of the above implications if we pass to the matrix algebra over A. Namely, the following statement holds. Proposition A.23. Let p and q be projections in a C ∗ -algebra A, and let P and Q be the projections determined by the formulas

p 0 q 0 P = , Q= 0 0 0 0 in the algebra M2 (A). 1. If p and q are algebraically equivalent, then P and Q are similar. 2. If p and q are similar, then P and Q are homotopic. Let us state two more results that can be useful when studying projections in A. Let A be a unital C ∗ -algebra. By GLn (A) we denote the multiplicative subgroup of invertible elements in the algebra Mn (A), and by Un (A) we denote the subgroup of unitary elements. Proposition A.24. Let u, v ∈ GLn (A). Then the elements uv ⊕ 1,

vu ⊕ 1,

u ⊕ v,

v⊕u

are pairwise homotopic in GL2n (A). Proposition A.25. Let π

0 −→ J −−−−→ A −−−−→ A/J −→ 0 be an exact sequence of C ∗ -algebras. Then for an arbitrary element u ∈ Un (A/J) there exists an element w ∈ U2n (A) that is homotopic to the unit element and satisﬁes the condition π(w) = u ⊕ u∗ .

A.2 Representations of C ∗-Algebras A.2.1 Basic Deﬁnitions Deﬁnition A.26. A homomorphism π : A −→ B(H) of a C ∗ -algebra A into the algebra of bounded linear operators in a Hilbert space H is called a representation of the C ∗ -algebra A.

Appendix A. C ∗ -Algebras

170

In what follows, we denote a representation by π or by (π, H) if we need to indicate the space where the representation acts. A representation is said to be faithful if π(a) = 0 for any nonzero element a ∈ A. Let K ⊂ H be a subspace in H invariant under a representation π. This means that π(a)K ⊆ K for any a ∈ A, or, equivalently, π(a)PK = PK π(a)

for any a ∈ A,

where PK is the orthogonal projection in H onto the subspace K. Then the restriction (π|K , K) of the representation (π, H) to the subspace K is well deﬁned; it is called the subrepresentation of the representation π, and the subspace K is also called a reducing subspace of the representation π. Obviously, if K is a reducing subspace, then its orthogonal complement K ⊥ in the space H is also a reducing subspace. A. Thedirect sum of Let (πj , Hj ), j ∈ J , be representations of the algebra these representations is deﬁned to be the representation ( j πj , j Hj ) of the algebra A in the direct sum of the Hilbert spaces Hj , which is determined by the formula ) * πj (a) ξj = πj (a)ξj , a ∈ A, xj ∈ Hj , j ∈ J. j

j

j

Clearly, each of the representations-summands is a subrepresentation of the direct sum. A representation π is said to be nondegenerate if for each vector ξ ∈ H there exists an element a ∈ A such that π(a)ξ = 0. Since A (and hence π(A)) is a C ∗ algebra, this condition is equivalent to the fact that the closed linear span π(A)H of all vectors of the form π(a)ξ, where ξ ∈ H and a ∈ A, coincides with H. A representation π is said to be cyclic if H = π(A)ξ for some vector ξ ∈ H, which in this case is called a cyclic vector of the representation π. (Of course, the cyclic vector is not unique.) Clearly, any cyclic representation is nondegenerate. The converse statement is not true, but the following statement holds. Proposition A.27. Any nondegenerate representation is a direct sum of cyclic representations. If (π, H) is an arbitrary representation of the algebra A, then we have the orthogonal decomposition H = H0 ⊕ H1 of the space H into the sum of invariant subspaces such that the restriction of the representation π to H1 is the zero representation and its restriction to H2 is a nondegenerate representation. Thus, an arbitrary representation of the C ∗ -algebra A is a direct sum of cyclic representations and possibly the zero representation. Deﬁnition A.28. Two representations (π1 , H1 ) and (π2 , H2 ) of a C ∗ -algebra A are said to be equivalent (more precisely, spatially or unitarily equivalent) if there

A.2. Representations of C ∗ -Algebras

171

exists a unitary operator U : H1 → H2 such that π1 (a) = U ∗ π2 (A)U

for any a ∈ A.

The mapping ξ → U ξ can be treated as a “change of coordinates” in the Hilbert space, and from this standpoint, equivalent representations are simply “forms of writing out a same representation in diﬀerent coordinate systems,” so that they need not be distinguished; accordingly, the main problem of representation theory of C ∗ -algebras is to study representations up to unitary equivalence. The “simplest” representations are those that do not contain nontrivial subrepresentations (i.e., subrepresentations other than the representation itself and the zero representation in the zero-dimensional space), or, which is the same, nontrivial invariant subspaces. Deﬁnition A.29. A representation (π, H) of a C ∗ -algebra A is said to be irreducible if the space H contains no subspaces invariant under π(A) except for {0} and H. In the following statement, we describe several properties of a representation π which are equivalent to its irreducibility. Proposition A.30. Let (π, H) be a nonzero representation of a C ∗ -algebra A. It is irreducible if and only if any of the following conditions holds. 1. The commutant of the algebra π(A) (i.e., the set of all operators in B(H) commuting with all operators in π(A)) consists of scalar operators. 2. The algebra π(A) is strongly dense3 in B(H). 3. Any nonzero vector in H is cyclic for the representation π. 4. If ξ, η ∈ H and ξ = 0, then A contains an element a such that the operator π(a) takes ξ to η; i.e., π(a)ξ = η. Suppose that (π1 , H1 ) and (π2 , H2 ) are two representations of a C ∗ -algebra A. A bounded linear operator T : H1 −→ H2 is called an intertwining operator for the representations π1 and π2 if T π1 (a) = π2 (a)T

for every a ∈ A.

Obviously, the intertwining operators for given representations π1 and π2 form a linear space, which we denote by I(π1 , π2 ). Deﬁnition A.31. Representations (π1 , H1 ) and (π2 , H2 ) are said to be disjoint if I(π1 , π2 ) = {0} (consists only of the zero operator). The following alternative holds for two arbitrary irreducible representations. Proposition A.32. Let (π1 , H1 ) and (π2 , H2 ) be two irreducible representations of a C ∗ -algebra A. Then they are either equivalent or disjoint. 3 Recall

that a sequence of operators An converges to operator A strongly if An x → Ax as n → ∞ for any x ∈ H.

Appendix A. C ∗ -Algebras

172

A.2.2 Existence of Representations We have not yet discussed the problem on the existence of representations (in particular, irreducible) of a C ∗ -algebra A. In this section, we present a method for constructing a suﬃcient supply of representations (and, in particular, for proving Theorem A.2). We consider only the case of a unital algebra A. A linear functional ϕ on the algebra A (i.e., an element of the space A∗ dual to the Banach space A) is said to be positive if ϕ(a) ≥ 0 for any a ∈ A+ . Deﬁnition A.33. A positive functional ϕ of unit norm is called a state on the C ∗ -algebra A. A state ϕ is said to be pure if any positive functional ψ on A such that ϕ − ψ is also positive is proportional to ϕ, i.e., has the form ψ = tϕ for some t ∈ [0, 1]. States can be used to construct representations of the C ∗ -algebra. Theorem A.34 (The Gelfand–Naimark–Segal construction). Let ϕ be a state on a C ∗ -algebra A. There exists a cyclic representation (πϕ , Hϕ ) of the algebra A with cyclic vector ξϕ ∈ Hϕ such that (πϕ (a)ξϕ , ξϕ ) = ϕ(a)

for any a ∈ A.

If ϕ is a pure state, then the representation (πϕ , Hϕ ) is irreducible. It turns out that there are “suﬃciently many” pure states. Proposition A.35. If a ∈ A is a positive element, then there exists a pure state ϕ such that ϕ(a) = a. Using this statement, the Hahn decomposition, and the Gelfand–Naimark– Segal construction, one can show that for each element a ∈ A there exists an irreducible representation πϕ such that πϕ (a) = 0. Corollary A.36. An element a of a unital C ∗ -algebra A is invertible if and only if the operator π(a) is invertible for any irreducible representation π of A. The direct sum (Π, H) =

(πϕ , Hϕ ),

ϕ

where ϕ runs over the set of all pure states on the algebra A, is called the universal representation of the C ∗ -algebra A. It follows from the preceding that the universal representation is faithful (and hence determines a realization of the C ∗ -algebra A as an operator algebra in a Hilbert space, which, however, can be nonseparable).

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173

A.2.3 Representations of Ideals and Quotient Algebras Let A be a C ∗ -algebra, and let J be an ideal in A, so that one has the exact sequence 0 −→ J −−−−→ A −−−−→ A/J −→ 0. (A.2) Now we shall see that studying representations (and especially irreducible representations) of the algebra A can be reduced to studying representations of the algebras J and A/J. Proposition A.37. Let (π, H) be a nondegenerate representation of the ideal J. Then there exists a unique representation ( π , H) of A that continues the representation π, i.e., has the property π

(a) = π(a)

for all a ∈ J.

If π is irreducible, then π

is also irreducible. Conversely, let a representation π

of the algebra A be given. Obviously, its restriction to J determines a representation π of the ideal J. Proposition A.38. If the representation π

is irreducible and its restriction π to the ideal J is nonzero, then the representation π of the ideal J is also irreducible. we denote the set of equivalence classes of irreducible representations of a By A C ∗ -algebra A. (This set is called the spectrum of the algebra A.) It follows from the above that the following statement holds. Proposition A.39. One has = J ∪ A/J. A Indeed, any irreducible representation of the quotient algebra A/J in composition with the natural projection onto the quotient algebra gives an irreducible representation of the algebra A. By Proposition A.37, the irreducible representations of J can be continued to irreducible representations of A. These representations are not equivalent to any representations lifted from the quotient algebra, because the latter are identically zero on J. Further, if π is an irreducible representation of the algebra A, then the following two cases are possible: either π|J ≡ 0, and then π is the pullback of the representation π

of A/J deﬁned by the formula π

(a + J) = π(a), or π|J = 0; in the latter case, by Proposition A.38, the representation π|J is an irreducible representation of the ideal J, and π is obtained from it by lifting according to Proposition A.37. An important special case is obtained if the ideal J is isomorphic to the algebra K(H) of compact operators in a Hilbert space H.

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174

Proposition A.40. Any irreducible representation of the algebra K(H) is equivalent to its identical representation. Corollary A.41. Any irreducible operator C ∗ -algebra that contains at least one nonzero compact operator contains the entire algebra K(H). Corollary A.42. All automorphisms of the algebra B(H) are given by conjugation by unitary operators in B(H), i.e., have the form a −→ U aU −1 ,

where U ∈ B(H) is unitary.

It follows from Proposition A.40 that if the ideal J in the exact sequence (A.2) is K(H), then ∪ {[idK(H) ]}, = A/K(H) A where [idK(H) ] is the equivalence class of the identity representation of the algebra of compact operators. In particular, if A is a subalgebra of the algebra B(H) and J = K(H), then the identity representation of the algebra A is irreducible and all “nontrivial” irreducible representations of the algebra A correspond to irreducible representations of the Calkin algebra A/K(H) of the algebra A.

A.2.4 Primitive Ideals coincides with the (locally) If A is a commutative C ∗ -algebra, then its spectrum A compact Hausdorﬀ space X on which the Gelfand transforms of elements of A are deﬁned, i.e., with the set of maximal ideals of the algebra A: the irreducible representation πJ corresponding to an element J ∈ X is one-dimensional and is determined by the composition of the natural projection A → A/J and the isomorphism A/J C. (In terms of the Gelfand transforms of elements of the algebra A, this representation is the evaluation mapping of functions at the point J ∈ X.) In this case, the ideal J is exactly the kernel of the representation πJ . Maximal ideals do not play a similar role for irreducible representations of a noncommutative C ∗ -algebra A. Instead, primitive ideals should be considered. of irreducible representations of the Namely, to each equivalence class [π] ∈ A ∗ C -algebra A one assigns the ideal I = ker π of the algebra A. (Note that the kernel of π is independent of the speciﬁc choice of π in the equivalence class.) Such ideals are said to be primitive. Proposition A.43. Any primitive ideal in a C ∗ -algebra A is simple. Recall that a (two-sided) ideal I in the algebra A is said to be simple if the relation xAy ⊂ I implies that either x ∈ I or y ∈ I. The equivalent condition is as follows: if I1 , I2 ⊂ A are ideals such that I1 I2 ⊂ I, then either I1 ⊂ I or I2 ⊂ I. If the algebra A is separable (i.e., if it contains a countable everywhere dense set), then the converse statement is also true. Proposition A.44. In a separable C ∗ -algebra A, any closed simple ideal is primitive.

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175

Thus, by deﬁnition, we have the mapping −→ Prim A A of equivalence classes of irreducible representations of the algebra A of the set A onto the set Prim A of primitive ideals. Generally speaking, this is not a one-to-one mapping (i.e., an irreducible representation is not determined by its kernel up to equivalence). But this is the case for a suﬃciently wide class of algebras, which will be considered in the next section. Here we only note that in many cases it is because there is a natural more convenient to use the set Prim A rather than A, topology (the so-called Jacobson topology) on Prim A. Closed sets in this topology have the form hull(I) = {J ∈ Prim A : J ⊃ I}, where I runs over the set of C ∗ -ideals in A. Indeed, the mapping I → hull(I) is a bijection of the set of C ∗ -ideals in A onto the set of closed sets in the Jacobson topology on Prim A. The inverse mapping is given by the formula J, F = F ⊂ Prim A. F −→ ker F = J∈F

This is a consequence of the following assertion. Proposition A.45. Any ideal in a C ∗ -algebra A is the intersection of primitive ideals containing this ideal. We note that the Jacobson topology on Prim A is in general not Hausdorﬀ (unless the algebra A is commutative). But in most cases of interest it is T0 . (Recall that in a T0 -space for each pair of distinct points there exists an open set that contains one of them and does not contain the other.)

A.2.5 Algebras of Type I Deﬁnition A.46. A C ∗ -algebra A is called a CCR-algebra (CCR stands for completely continuous representations) if its image under any irreducible representation consists of compact operators. If A is an arbitrary C ∗ -algebra, then by CCR(A) we denote the ideal in A consisting of elements whose images are compact for any irreducible representation of the algebra A. Obviously, this is the maximal CCR-ideal in A. Deﬁnition A.47. A C ∗ -algebra A is said to be of type I if the ideal CCR(A/J) is nonzero for any proper ideal J ⊂ A. Theorem A.48. If A is a separable algebra of type I, then each equivalence class of irreducible representations of A is uniquely determined by the kernel of the representation.

Appendix A. C ∗ -Algebras

176

→ Prim A is one-to-one for algebras of type I. Thus, the mapping A We present one more criterion for algebras to belong to type I. Proposition A.49. A C ∗ -algebra A is an algebra of type I if and only if its image under any nonzero irreducible representation contains at least one compact operator.

A.3 Tensor Products and Nuclear Algebras A.3.1 Minimal and Maximal Tensor Products Let A and B be C ∗ -algebras. Consider the algebraic tensor product A ⊗ B of the linear spaces A and B. Recall that it is determined as the quotient space of the linear space W of ﬁnite formal linear combinations of elements (a, b) ∈ A × B by the subspace generated by elements of the form (λa, b) − λ(a, b), (a + a , b) − (a, b) − (a , b),

(a, λb) − λ(a, b), (a, b + b ) − (a, b) − (a, b ),

where a, a ∈ A, b, b ∈ B, and λ ∈ C. In other words, the elements of the algebraic tensor product A ⊗ B are ﬁnite linear combinations of the form s

aj ⊗ b j ,

aj ∈ A,

bj ∈ B,

j=1

and (a + a ) ⊗ b = a ⊗ b + a ⊗ b,

a ⊗ (b + b ) = a ⊗ b + a ⊗ b ,

(λa) ⊗ b = a ⊗ (λb) = λ(a ⊗ b). On A ⊗ B, one can introduce the structure of an involutive algebra by setting (a ⊗ b)(a ⊗ b ) = aa ⊗ bb ,

(a ⊗ b)∗ = (a∗ ⊗ b∗ )

on monomials and then by extending multiplication and involution to the entire tensor product by linearity. If the algebras A and B are unital, then A ⊗ B is also unital with unit 1A ⊗ 1B . It turns out that the product A⊗ B can be equipped with a C ∗ -norm, so that the completion of A ⊗ B with respect to this norm is a C ∗ -algebra. For example, this can be done as follows. Let πA and πB be faithful representations of the algebras A and B in Hilbert spaces HA and HB , respectively. The formula τ

s j=1

aj ⊗ b j

=

s j=1

πA (aj ) ⊗ πB (bj )

A.3. Tensor Products and Nuclear Algebras

177

determines a representation τ of the algebra A ⊗ B in the Hilbert space HA ⊗ HB , which is called the tensor product of the representations πA and πB . Now we can set amin = τ (a) , a ∈ A ⊗ B. (A.3) This is a C ∗ -norm on A⊗B independent of the speciﬁc choice of the representations πA and πB , and the completion of the algebra A ⊗ B with respect to this norm is called the minimal (or spatial ) tensor product of A and B and is denoted by A ⊗min B. The C ∗ -norm on the product A ⊗ B is not unique in general. Any such norm satisﬁes the inequality a ≥ amin ,

a ∈ A ⊗ B,

and is automatically a cross-norm, i.e., satisﬁes the relation a ⊗ b = a b , Therefore,

a ∈ A,

b ∈ B.

(A.4)

s s ≤ a ⊗ b aj bj , j j j=1

j=1 ∗

which implies that the set of C -norms on A ⊗ B contains a maximal C ∗ -norm · max (which is equal to the supremum of all these norms). The completion of the algebra A ⊗ B with respect to this norm is called the maximal (or projective) tensor product of the algebras A and B and is denoted by A ⊗max B.

A.3.2 Nuclear Algebras Deﬁnition A.50. A C ∗ -algebra A is said to be nuclear if there exists a unique C ∗ -norm on the product A ⊗ B (or, which is the same, · min = · max ) for each C ∗ -algebra B. In this case, the completion of the product A ⊗ B with respect to this norm is simply called the tensor product of the algebras A and B and is B. denoted by A ⊗ It turns out that the supply of nuclear C ∗ -algebras is rather wide. Proposition A.51. Any ﬁnite-dimensional C ∗ -algebra is nuclear. Proposition A.52. If a C ∗ -algebra A has an inclusion-ordered family {Aj }j∈J of nuclear subalgebras whose union is dense in A, then the algebra A itself is nuclear. Corollary A.53. The algebra K(H) of compact operators in a Hilbert space H is nuclear. Indeed, for the subalgebras Aj we can take the (ﬁnite-dimensional) algebras of operators in ﬁnite-dimensional subspaces of H. On the contrary, the algebra B(H) is not nuclear unless H is ﬁnite-dimensional.

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Appendix A. C ∗ -Algebras

Proposition A.54. Any separable C ∗ -algebra of type I (in particular, any separable commutative C ∗ -algebra) is nuclear.

A.3.3 Primitive Ideals in the Tensor Product Let us ﬁnd the structure of the primitive ideal space of the tensor product of C ∗ -algebras A and B. The tensor product of irreducible representations of A and B can be extended to an irreducible representation of the algebra A ⊗min B; therefore, there is a well-deﬁned mapping ι : Prim A × Prim B −→ Prim(A ⊗min B),

(A.5)

ι(ker πA , ker πB ) = ker(πA ⊗min πB ).

(A.6)

where This mapping turns out to be continuous (in the Jacobson topology), and its image is dense in Prim(A ⊗min B). If one of the algebras A and B is nuclear, then we have the following stronger statement. Proposition A.55 (e.g., see [69]). If A and B are C ∗ -algebras and at least one of them is nuclear, then the mapping (A.5), (A.6) is a homeomorphism of the direct B). product Prim A × Prim B onto Prim(A ⊗

Appendix B

K-Theory of Operator Algebras B.1 Covariant K-Theory Let X be a locally compact Hausdorﬀ space. The correspondence X → C0 (X) determines a contravariant functor from the category of locally compact Hausdorﬀ spaces into the category of C ∗ -algebras, which is an isomorphism onto the subcategory of commutative C ∗ -algebras; the K-groups of a commutative C ∗ -algebra can be deﬁned by the formula K∗ (A) = K ∗ (A),

∗ = 0, 1,

(B.1)

is the space of maximal ideals of the algebra A and K ∗ stands for the where A topological K-theory. K-theory of C ∗ -algebras generalizes topological K-theory to the noncommutative case. In this theory, the K-groups of an arbitrary C ∗ -algebra A are deﬁned in such a way that relation (B.1) is satisﬁed if the algebra A is commutative and the main results of topological K-theory (the six-term exact sequence and the Bott periodicity) also remain valid in the general case.

B.1.1 Topological K-Theory We recall the construction of K-groups of topological spaces. First, let X be a compact Hausdorﬀ space. The group K 0 (X) is deﬁned as the Grothendieck group of the monoid whose elements are classes of isomorphic complex vector bundles over X with addition operation determined by the direct sum of vector bundles. By the Swan theorem, each vector bundle over X is a subbundle (and hence a direct summand) of a trivial bundle; therefore, the group K 0 (X) is generated by classes of stable equivalence of vector bundles over X. More precisely, two bundles E1 , E2 ∈ Vect X deﬁne the same element in the group K 0 (X) if the bundles E1 ⊕ CN and E2 ⊕ CN , where CN is the trivial N -dimensional vector bundle

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Appendix B. K-Theory of Operator Algebras

over X, are isomorphic for some N ≥ 0. If Y and X are compact Hausdorﬀ spaces and f : Y → X is a continuous mapping, then there is a well-deﬁned group homomorphism f ∗ : K 0 (X) −→ K 0 (Y ),

[E] −→ [f ∗ E],

E ∈ Vect X.

(B.2)

Suppose that f : Y → X is an embedding. Then the relative K 0 -group K 0 (X, Y ) of the pair (X, Y ) is deﬁned as follows. Consider the Grothendieck group of the monoid whose elements are classes of isomorphic triples (E1 , E2 , ϕ), where E1 , E2 ∈ Vect X and ϕ is an isomorphism of the restrictions of the bundles E1 and E2 to Y . Here an isomorphism of triples (E1 , E2 , ϕ) and (E1 , E2 , ϕ ) is naturally determined as a pair of isomorphisms αj : Ej → Ej , j = 1, 2, such that the following diagram commutes over Y : ϕ

E1 −−−−→ ⏐ ⏐ α1

E2 ⏐ ⏐α 2

E1 −−−− → E2 ϕ

A triple (E1 , E2 , ϕ) is trivial if ϕ can be extended to an isomorphism of E1 and E2 . The relative K-group is the quotient of the Grothendieck group by the subgroup generated by trivial triples. In this case, we have the exact sequence of K-groups f∗

K 0 (X, Y )

−−−−→

K 0 (X)

−−−−→ K 0 (Y ),

[(E1 , E2 , ϕ)]

−→

[E1 ] − [E2 ].

(B.3)

The construction of the relative group is used to deﬁne the K-group of a locally compact Hausdorﬀ space X: the one-point compactiﬁcation X + = X ∪ {pt} of the space X is considered, and the group K 0 (X) is determined by the formula K 0 (X) = K(X + , {pt}); in this case, the corresponding sequence of the form (B.3) extends to give the exact sequence f∗

0 −→ K 0 (X, {pt}) −−−−→ K 0 (X + ) −−−−→ K 0 ({pt}) ≡ Z −→ 0, so that K 0 (X) is simply the kernel of the homomorphism K 0 (X + ) → K 0 ({pt}). This deﬁnition does not contradict the deﬁnition of the K-group in the case of a compact space X. Indeed, in this case, we have X + = X {pt} (the disjoint union); accordingly, K 0 (X + ) = K 0 (X) ⊕ K 0 ({pt}), the homomorphism f ∗ coincides with the projection onto the second summand, and its kernel is exactly K 0 (X).

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181

If X is a (locally) compact space, then the suspension over X is deﬁned by the formula SX = X × R. The odd K-group of the space X is introduced by the formula K 1 (X) = K 0 (SX). The subsequent suspensions do not lead to new groups; namely, the following statement is true. Theorem B.1 (Bott periodicity). There is an isomorphism β : K ∗ (X) −→ K ∗ (S 2 X),

∗ = 0, 1.

In terms of representatives, the isomorphism β takes the equivalence class [E] ∈ K 0 (X) of a bundle E on X to the equivalence class of the exterior tensor product E τ , where τ is the (virtual) Bott bundle on R2 , which can be described as follows. One has K 0 (R2 ) = K 0 (D2 , S1 ), where D2 is the unit disk in the plane R2 and S1 is its boundary, i.e., the unit circle. Then τ = (C, C, z), where C is the trivial one-dimensional bundle over D2 and z is the automorphism of restriction of this bundle to S1 determined by the multiplication by the function z = x + iy. (Here (x, y) are the standard coordinates in R2 .) Theorem B.2 (Six-term exact sequence). If X is a locally compact space and Y ⊂ X is a closed subspace, then one has the natural exact sequence of K-groups K 0 (X \ Y ) −−−−→ K 0 (X) −−−−→ 8 ⏐ ∂⏐ K 1 (Y )

K 0 (Y ) ⏐ ⏐ ∂

(B.4)

←−−−− K 0 (Y ) ←−−−− K 1 (X \ Y ).

Here we do not consider the deﬁnition of vertical arrows (the connecting homomorphisms ∂) and refer the reader to any standard text on topological K-theory. Bearing in mind the generalization of notions of K-theory to the case of C ∗ -algebras, which is described in the subsequent sections, it is useful to have an alternative description of K-groups, which appeals only to the algebra of continuous functions on the space X rather than directly to the space X. We give such a description for the case of a compact space X. Recall that, by the Swan theorem, each vector bundle E over X is a direct summand in some trivial bundle CN over X and hence can be described as the image of some N × N matrix projection p whose entries are continuous functions on X. In other words, p ∈ MN (C(X)). Thus, the elements of the group K 0 (X) can be treated as equivalence classes of projections in matrix algebras over C(X). (Here we do not describe this equivalence relation, because this will be done later in general form.) To describe the elements of the group K 1 (X), we use the relation K 1 (X) = K 0 (X × R) K 0 X × [0, 1], (X × {0}) ∪ (X × {1}) .

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Appendix B. K-Theory of Operator Algebras

The elements of this group can be described as equivalence classes of quadruples (E1 , E2 , α, β), where E1 , E2 ∈ Vect(X), α and β are isomorphisms of the bundles E1 and E2 , and two quadruples are said to be equivalent if there exist families (with parameter t ∈ [0, 1]) of isomorphisms of the corresponding bundles which conjugate the isomorphisms α for t = 0 and the isomorphisms β for t = 1. Obviously, in each equivalence class there is an element for which β = id and E1 = E2 = CN for suﬃciently large N . In this case, α turns out to be an invertible N × N matrix function on X, so that the elements of the group K 1 (X) can be treated as equivalence classes of invertible matrix functions on X, i.e., of invertible elements in matrix algebras over C(X).

B.1.2 Group K0 (A) Let A be a C ∗ -algebra. We deﬁne an embedding jmn : Mm (A) → Mn (A) for m, n ∈ Z+ , m ≤ n, by the formula a −→

a 0

0 0

(i.e., by adding n − m zero columns and rows to the matrix a on the right and below). The embeddings jmn are isometric (which can readily be veriﬁed if we assume that A is an operator algebra on a Hilbert space) and satisfy the chain rule jnl ◦ jmn = jml , m ≤ n ≤ l. Consider the inductive limit M (A) = lim ind Mn (A). n→∞

Its completion M (A) is a C ∗ -algebra (the inductive limit of the C ∗ -algebras Mn (A)). But we shall use only the algebra M (A) itself, which can be described as the algebra of right and downward inﬁnite matrices over A in which only ﬁnitely many entries are nonzero. For each n, the algebra Mn (A) is naturally embedded into M (A) as a subalgebra, and M (A) =

∞

Mn (A).

n=0

Now assume that A is unital. Let p, q be projections in the algebra M (A). (In this case, one also says that p and q are projections over a.) Recall that this means that p = p∗ = p2 , q = q ∗ = q 2 . The projections p and q are said to be equivalent, p ∼ q, if for some n there exists a unitary element u ∈ Mn (A) such that u−1 pu = q.

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183

Putting this diﬀerently, the projections p and q have ﬁnitely many nonzero entries, and hence p ∈ Mm (A) and q ∈ Ml (A) for some m and l. The projections p and q are equivalent if they can be bordered by zeros so that the resulting matrix projections have the same size n × n and are similar (unitary equivalent); i.e.,

q 0 −1 p 0 u u= . 0 0 0 0 Remark B.3. Instead of similarity, we could use algebraic equivalence or homotopy, but the result would be the same because of Propositions A.22 and A.23. Proposition B.4. The equivalence relation introduced above is consistent with direct sums and homotopies of projections. Moreover, p⊕q ∼q⊕p for any projections p and q over the algebra A. Deﬁnition B.5. The group K0 (A) of a unital C ∗ -algebra A is the Grothendieck group of the monoid of equivalence classes of projections over A with addition in the monoid being induced by the direct sum of projections. If f : A → B is a homomorphism of unital C ∗ -algebras, then the induced homomorphism M (A) → M (B) obtained by applying the homomorphism f term by term takes direct sums to direct sums, projections to projections, and unitary elements in Mn (A) to the unitary elements in the corresponding Mn (B). Therefore, the induced homomorphism of K-groups is well deﬁned: f∗ : K0 (A) −→ K0 (B). Now we deﬁne the group K0 (A) for a nonunital algebra A. We recall that in the commutative case a nonunital algebra corresponds to a noncompact locally compact space X and that the deﬁnition of the K-group of a noncompact space in topological K-theory is based on the use of the one-point compactiﬁcation of the space X. An analog of the one-point compactiﬁcation in C ∗ -algebras is given by unitization, which motivates the following construction. We add the unit to the algebra A and consider the exact sequence 0 −→ A −−−−→ a

−→

A+

π

−−−−→ C −→ 0,

(0, a), (λ, a)

−→

λ.

Deﬁnition B.6. The kernel of the induced mapping π∗ : K0 (A+ ) −→ K0 (C) = Z is called the K0 -group of the algebra A and is denoted by K0 (A).

184

Appendix B. K-Theory of Operator Algebras One can readily see that with this deﬁnition we have the exact sequence 0 −→ K0 (A) −−−−→ K0 (A+ ) −−−−→ Z −→ 0.

(B.5)

Just as in the topological case, the two deﬁnitions do not contradict each other if A is actually unital. Indeed, in this case, A+ = A ⊕ C, K0 (A+ ) = K0 (A) ⊕ Z, and the homomorphism π∗ is simply the projection onto the second factor, so that its kernel exactly coincides with K0 (A). The general element of the group K0 (A) can be represented as the diﬀerence [p] − [In ] of equivalence classes, where p ∈ M (A+ ) is a projection, In ∈ Mn (A+ ) is the unit n × n matrix, and p − In ∈ M (A). Proposition B.7. If A = C(X), where X is a locally compact Hausdorﬀ space, then the following relation holds: K0 (A) = K 0 (X). Indeed, in this case the above deﬁnition is an exact translation of the topological deﬁnition into the language of C ∗ -algebras. If 0 −→ J −−−−→ A −−−−→ A/J −→ 0 is a short exact sequence of C ∗ -algebras, then the corresponding three-term sequence (B.6) K0 (J) −−−−→ K0 (A) −−−−→ K0 (A/J) is also exact. To continue it to either side as an exact sequence, one needs to introduce odd K-groups, just as in the topological case, and this will be done in the next section.

B.1.3 Group K1 (A) In topology, the odd K-group is deﬁned using the suspension, i.e., the passage from the space X to the product X × R. Let us translate this passage into the language of C ∗ -algebras. We have the relation C0 (R) = C0 (R, C0 (X)). C0 (X × R) = C0 (X) ⊗ is unique, since the algebra C0 (R) is Here the tensor product of C ∗ -algebras ⊗ nuclear. This relation motivates the following general deﬁnition. Deﬁnition B.8. The suspension SA over a C ∗ -algebra A is deﬁned to be the algebra A = C0 (R, A). SA = C0 (R) ⊗ The K1 -group of the C ∗ -algebra A is determined by the formula K1 (A) = K0 (SA).

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Taking the above into account, we obtain the following statement. Proposition B.9. If A = C(X), where X is a locally compact Hausdorﬀ space, then K1 (A) = K 1 (X). Let us give an alternative description of elements of the group K1 (A). For simplicity, let A be a unital algebra. By U (A) we denote the direct limit of the groups Un (A) of unitary n × n matrices over A with respect to the embeddings jmn : Um (A) → Un (A),

a 0 jmn (a) = . 0 1 The elements of the group U (A) can be treated as right and downward inﬁnite matrices of the form

u 0 u= ≡ u ⊕ 1, 0 1 where u ∈ Un (A) for some n and 1 is the inﬁnite identity matrix. The mapping u → u gives a natural embedding Un (A) → U (A). Two elements u, v ∈ U (A) are said to be equivalent (u ∼ v) if they are homotopic. Theorem B.10. The group K1 (A) is isomorphic to the Grothendieck group of the monoid of equivalence classes of elements of the group U (A) with respect to the equivalence relation described above. Here the operation of addition in the monoid is determined by the direct sum of elements: if u = u ⊕ 1 and v = v ⊕ 1 are two elements of group U (A), then their direct sum is deﬁned as u ⊕ v ⊕ 1. Now let us explain the structure of this isomorphism. To this end, we need to assign some class in the group K0 (SA) to each element u ∈ Un (A). This can be done as follows. According to Proposition A.25, in the group U2n (A) there exists a homotopy of the element u ⊕ u∗ to the element 1. Let

1 0 u 0 w(t), t ∈ [−∞, ∞], w(−∞) = , w(∞) = 0 1 0 u∗ be such a homotopy. Consider the projections p1 and p2 in the algebra M2n (SA+ ) given by the formulas

1 0 1 0 . w(t), p2 (t) = p1 (t) = w(t)∗ 0 0 0 0 (The second of the projections is a constant function of t.) Note that

1 0 , p1 (−∞) = p1 (∞) = 0 0 so that p1 − p2 ∈ M2n (SA)

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and the pair (p1 , p2 ) gives a well-deﬁned element [p1 ] − [p2 ] ∈ K0 (SA). We can show that this element is independent of the choice of u in the equivalence class and of the ambiguity in the construction itself and that the mapping described above determines the desired isomorphism. Remark B.11. Suppose that elements [u], [v] ∈ K1 (A) are represented by unitary matrices u, v of the same size. Then the class [u] + [v] is equal to the class of composition uv (or vu). This follows from Proposition A.24.

B.1.4 Bott Periodicity Just as in the topological case, further suspensions do not lead to new K-groups. Namely, the following statement holds. Theorem B.12 (Bott periodicity in K-theory of operator algebras). One has the isomorphism (B.7) β : K0 (A) −→ K0 (S 2 A). Let us describe the mapping β : K0 (A) −→ K0 (S 2 A). It is convenient to do this by interpreting K0 (S 2 A) as K1 (SA) and by using Theorem B.10. First, we assume that A is unital. Thus, on the level of representatives of equivalence classes, given a projection p over A, it is required to construct an invertible matrix function over SA+ , or, which is the same, an invertible matrix function Fp (t) on [−∞, ∞] ranging in U (A) and satisfying the condition Fp (−∞) = Fp (+∞) = 1. We deﬁne the mapping z : [−∞, ∞] −→ S1 , t −→ ei(π+2 arctan t) and set Fp (t) = z(t)p + 1 − p. (Here 1 is the identity element of the group U (A).) One can readily verify that this is a unitary matrix function. Then the mapping β is given by the formula β([p]) = [Fp ]. For the case in which A is not unital, the desired function should range in U (A+ ), and the formula for the mapping β becomes β([p] − [q]) = [Fp Fq−1 ], where p, q ∈ M (A+ ) are projections such that p − q ∈ M (A).

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B.1.5 Long Exact Sequence in K-Theory The six-term exact sequence (B.4) in topological K-theory can be generalized to K-theory of operator algebras, where the following result holds. Theorem B.13 (six-term sequence in K-theory of algebras). Let 0 −→ J −−−−→ A −−−−→ A/J −→ 0 be a short exact sequence of C ∗ -algebras. Then the three-term sequence (B.6) is included in the exact sequence −−−−→ K0 (A) −−−−→ K0 (A/J) ⏐ ⏐∂ 2

K0 (J) 8 ⏐ ∂1 ⏐

K1 (A/J) ←−−−− K1 (A) ←−−−−

(B.8)

K1 (J).

As usual, the construction of connecting homomorphisms ∂0 and ∂1 in this sequence is nontrivial (and hence interesting). Let us describe these homomorphisms. Homomorphism ∂1 Let us describe the mapping ∂1 : K1 (A/J) −→ K0 (J). Consider the element [u] ∈ K1 (A/J) corresponding to some unitary element u ∈ Un (A/J). Using Proposition A.25, we lift the element u ⊕ u∗ ∈ U2n (A/J) to a unitary element w ∈ U2n (A) homotopic to the unit. Then the connecting homomorphism ∂1 is given on the element [u] by the formula )

* )

* 1 0 1 0 ∂1 [u] = w w∗ − ∈ K0 (J). 0 0 0 0 To verify that this formula is well deﬁned, note that the element w in the block 2 × 2 representation is diagonal modulo elements of M2n (J) and hence commutes modulo such elements with the projection ( 10 00 ). It follows that

1 0 w w∗ ∈ M2n (J + ) 0 0 and

as desired.

1 w 0

0 1 0 ∗ w − ∈ M2n (J), 0 0 0

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The mapping ∂1 is called the index mapping, because it coincides with the usual (analytic) index of a Fredholm operator in the following situation. Let A be a subalgebra of B(H) containing the ideal J = K(H) of compact operators. Then K0 (J) = Z, and the mapping ∂1 : K1 (A/J) −→ Z has the form ∂1 ([a]) = ind a, where a ∈ GLn (A/J) is an invertible element and a is an arbitrary lift of a to Mn (A) (which is necessarily a Fredholm operator and whose index is independent of the choice of the lift). Remark B.14. In this case, a can be treated as the “symbol” of the operator a. Homomorphism ∂0 The homomorphism ∂0 : K0 (A/J) −→ K1 (J) is called the exponential mapping. Of course, one can describe it by using the already known description of the homomorphism ∂1 and of the suspension homomorphism. Explicitly, the formula determining this homomorphism reads )

*

1n 0 = [e2πiz ], ∂0 [w] − 0 0 where w is a projection in M ((A/J)+ ) such that

1 0 w− n ∈ M (A/J) 0 0 and z is a lift of w to M (A+ ).

B.1.6 Stability of K-Groups The following statement holds. Theorem B.15. For an arbitrary C ∗ -algebra A, the following isomorphisms hold : Kj (A) Kj (Mn (A)),

Kj (A) Kj (A⊗K(H)),

j = 1, 2.

These isomorphisms are induced, respectively, by the natural embedding A → Mn (A) (as the left top corner ) and by the embedding A −→ A⊗K(H), a −→ a ⊗ p, where p is an arbitrary given projection of rank 1 in the Hilbert space H.

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B.1.7 K-groups of Local C ∗ -Algebras The theory described in the preceding sections can be transferred to local C ∗ algebras without any essential changes. In this case, we have the following theorem. Theorem B.16. Let A be a C ∗ -algebra, and let B ⊂ A be a dense local C ∗ subalgebra in A. Then Kj (A) = Kj (B),

j = 0, 1.

B.2 K-Homology Just as in the case of covariant K-theory, the K-homology K ∗ (A) of an arbitrary C ∗ -algebra A is a generalization of the K-homology of manifolds to the noncommutative case. Namely, it is deﬁned in such a way that if the algebra A is commutative, then ∗ = 0, 1, K ∗ (A) = K∗ (A), is the maximal ideal space of the algebra A. When passing to algebras, where A the variance of the functor changes to the opposite, so that, despite the name, the K-homology functor for algebras turns out to be contravariant. (Because of this, some authors think that this name is not too apt.) Thus, the reader should be careful here.

B.2.1 K-Homology of a Topological Space First, let us brieﬂy recall the deﬁnition of the K-homology of a compact Hausdorﬀ space X. K-homology is dual to K-theory, and one can give a purely topological deﬁnition of the K-homology groups (for example, if X is a compact manifold, then we embed X in a sphere SN of suﬃciently large odd dimension N and set K∗ (X) = K ∗ (S N \ U ), where U is a tubular neighborhood of the manifold X in SN .) But we are interested in deﬁnitions related to operators and operator algebras. The K-homology classes of the space X can be deﬁned in terms of abstract elliptic operators on X. The notion of abstract elliptic operator was ﬁrst introduced by Atiyah [8]: an abstract elliptic operator is deﬁned to be a triple (D, H0 , H1 ), where H0 and H1 are Hilbert spaces equipped with actions π0 and π1 of the algebra C(X) and D : H0 → H1 is a Fredholm operator such that Dπ0 (ϕ) − π1 (ϕ)D ∈ K(H0 , H1 )

for any ϕ ∈ C(X).

In other words, the operator D intertwines the actions of the algebra C(X) on H0 and H1 modulo compact operators. The K-homology classes of the space X are equivalence classes of abstract elliptic operators with respect to a certain equivalence relation. Let us present Kasparov’s formal deﬁnition.

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Appendix B. K-Theory of Operator Algebras

Deﬁnition B.17. A triple (H, π, F ), where H is a separable Hilbert space, π : C(X) → B(H) is a unital representation of the algebra C(X) in the space H, and F ∈ B(H) is a self-adjoint linear operator such that F 2 − 1 ∈ K(H),

[F, π(ϕ)] ∈ K(H)

for any ϕ ∈ C(X),

(B.9)

is called a Fredholm module over the algebra C(X). A Fredholm module such that the Hilbert space H is Z2 -graded, the action of the algebra C(X) preserves the grading, and the operator F is odd with respect to the grading, is called a graded Fredholm module. A Fredholm module is said to be degenerate if F 2 = 1,

[F, π(ϕ)] = 0 for any ϕ ∈ C(X).

Obviously, a direct sum of Fredholm modules (whose deﬁnition is obvious) is again a Fredholm module. Two Fredholm modules are said to be equivalent if, possibly after the addition of degenerate modules as direct summands and a unitary transformation, they are homotopic to each other. (A homotopy of Fredholm modules is understood as a norm-continuous homotopy of the operator F in the class of operators satisfying the condition (B.9).) For graded Fredholm modules, the deﬁnition of equivalence is similar but uses only graded degenerate modules, unitary transformations that preserve the grading, and homotopies in the class of graded modules. By K1 (X) we denote the set of equivalence classes of Fredholm modules over C(X). By K0 (X) we denote the set of equivalence classes of graded Fredholm modules over C(X). One can readily show that these sets are Abelian groups with respect to the operation induced by the direct sum of modules, and the equivalence class of degenerate modules is the zero element. Deﬁnition B.18. The groups K0 (X) and K1 (X) are called the K-homology groups of the space X. Now let us explain how to assign elements of K-homology groups to abstract elliptic operators on X. If D : H0 → H1 is an abstract elliptic operator, then one can deﬁne a graded Fredholm module by setting

0 D(Pker D + D∗ D)−1/2 , F = H = H0 ⊕ H1 , (Pker D + D∗ D)−1/2 D∗ 0 where Pker D is the projection onto the kernel of the operator D. If D : H → H is a self-adjoint abstract elliptic operator, then one can deﬁne a Fredholm module by setting F = 2P+ (D) − 1,

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191

where P+ (D) is the projection onto the positive spectral subspace of the operator D. Conversely, given a Fredholm module, one can construct an abstract elliptic operator on X. This operator is either F itself or (for graded Fredholm modules) the component of F acting from H0 into H1 . Thus, arbitrary abstract elliptic operators on X generate elements of the group K0 (X), and self-adjoint abstract elliptic operators generate elements of the group K1 (X). Since a continuous mapping f : X → Y determines the algebra homomorphism f ∗ : C(Y ) → C(X) and thus turns any C(X)-module into a C(Y )-module, it is clear that the correspondence X → K∗ (X) is a covariant functor. There are natural pairings between the K-groups and the respective K-homology groups of the space X. Let us describe these pairings. Even Groups Let [E] ∈ K 0 (X) be the equivalence class of some vector bundle E on the manifold X, and let [D] ∈ K0 (X) be the equivalence class of an abstract elliptic operator D : H1 → H2 . The bundle E can be described as the image of some projection P ∈ MatN (C(X)) in the trivial bundle with ﬁber CN over X. The actions (which we denote by the same letters π1 and π2 ) of the algebra MatN (C(X)) in the Hilbert spaces H1 ⊗ CN and H2 ⊗ CN are naturally deﬁned, and we deﬁne the operator D ⊗ E (“the operator D with coeﬃcients in the bundle E”) by setting D ⊗ E = π2 (P ) D ⊗ 1CN : π1 (P )(H1 ⊗ CN ) −→ π2 (P )(H2 ⊗ CN ). Since the operator D “almost commutes” (i.e., commutes modulo compact operators) with the components of the projection P , it readily follows that D ⊗ E is a Fredholm operator. The pairing of the groups K0 (X) and K 0 (X) is given by the formula Z, K0 (X) × K 0 (X) −−−−→ (B.10) ([D], [E]) −→ ind(D ⊗ E). Proposition B.19. The pairing (B.10) is well deﬁned and nondegenerate on the free parts of the groups K0 (X) and K 0 (X). Odd Groups Let [f ] ∈ K 1 (X) be the equivalence class of some invertible n×n matrix function f on the space X, and let [D] ∈ K1 (X) be the equivalence class of some self-adjoint abstract elliptic operator D : H → H. Then the operator P+ (D)f : P+ (D)H n −→ P+ (D)H n

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Appendix B. K-Theory of Operator Algebras

(an abstract Toeplitz operator with symbol f ) is a Fredholm operator (for an almost inverse operator one can take P+ (D)f −1 ), and the pairing of the groups K1 (X) and K 1 (X) is given by the formula K1 (X) × K 1 (X) −−−−→ ([D], [f ])

−→

Z,

(B.11)

ind(P+ (D)f ).

Proposition B.20. The pairing (B.11) is well deﬁned and nondegenerate on the free parts of the groups K1 (X) and K 1 (X).

B.2.2 K-Homology of Operator Algebras: Deﬁnitions In this section, we present several diﬀerent versions of the deﬁnition. From now on, we assume as a rule, without stipulating this explicitly, that the C ∗ -algebra A under study is separable and nuclear. Fredholm Modules Kasparov’s deﬁnition, which is apparently the most technically convenient deﬁnition of K-homology in terms of Fredholm modules, provides a direct generalization of the corresponding deﬁnition given above for the algebra C(X). Deﬁnition B.21. Let A be a C ∗ -algebra. A triple (H, π, F ), where H is a separable Hilbert space, π : A → B(H) is a representation of A on H, and F ∈ B(H) is a linear self-adjoint operator such that (F 2 − 1)π(a) ∈ K(H),

[F, π(a)] ∈ K(H)

for any a ∈ A,

(B.12)

is called a Fredholm module over A. A Fredholm module such that the Hilbert space H is Z2 -graded (H = H0 ⊕ H1 ), the action of the algebra A preserves the grading (AHj ⊂ Hj , j = 0, 1), and the operator F is odd with respect to the grading (F Hj ⊂ H1−j , j = 0, 1), is called a graded Fredholm module. The deﬁnition is stated in this form so it can also be used in the case of a nonunital algebra A. Of course, the factor π(a) in the condition (F 2 − 1)π(a) ∈ K(H) can be omitted if the algebra A is unital. Sometimes (if H and π are clear from the context or insigniﬁcant), we denote the Fredholm module simply by F . Two Fredholm modules over A are said to be homotopic if the spaces H and representations π of these modules are the same and the operators F can be related by a norm-continuous homotopy such that all intermediate triples are Fredholm modules. For graded Fredholm modules, it is additionally required that all the intermediate triples be graded Fredholm modules. Direct sums and unitary equivalence of Fredholm modules are naturally deﬁned in both the graded and ungraded cases.

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Deﬁnition B.22. We deﬁne K-homology groups K ∗ (A), ∗ = 0, 1, to be the Abelian group generated by the following generators and relations. The generators of this group are the classes of unitary equivalence of graded Fredholm modules (for ∗ = 0) or of Fredholm modules (for ∗ = 1). The relations have the form [F1 ] = [F2 ] if F1 and F2 are homotopic, and [F1 ⊕ F2 ] = [F1 ] + [F2 ] for any F1 and F2 . A Fredholm module is said to be degenerate if (F 2 − 1)π(a) = 0,

[F, π(a)] = 0

for any a ∈ A.

Proposition B.23. The degenerate Fredholm modules determine the zero class in the K-homology of the algebra A. If f : A → B is a C ∗ -algebra homomorphism, then to each representation π of the algebra B we can assign the representation f ∗ (π) = π ◦ f of the algebra A, and hence each Fredholm module over B becomes a Fredholm module over A. One can readily see that this correspondence gives rise to a group homomorphism f ∗ : K ∗ (B) −→ K ∗ (A), and the correspondence A → K ∗ (A) turns out to be a contravariant functor from the category of C ∗ -algebras to the category of Abelian groups. Dual Algebras Another deﬁnition of K-homology can be given in terms of dual algebras. Let A be a C ∗ -algebra. Deﬁnition B.24. A representation π : A → B(H) of the algebra A in a Hilbert space H is said to be ample if 1. The representation π is nondegenerate. 2. The operator π(a) is compact only if a = 0. Suppose that an ample representation π : A → B(H) of the algebra A is given. Deﬁnition B.25. The subalgebra D(A) = {T ∈ B(H) : [T, π(a)] ∈ K(H) is called the dual algebra of the algebra A.

for any a ∈ A}

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Proposition B.26. The algebra D(A) is well deﬁned. Namely, Deﬁnition B.25 gives the same result (up to an isomorphism) for any ample representation of A. Now we can give the desired deﬁnition. Deﬁnition B.27. The K-homology groups of a C ∗ -algebra A are determined by the formulas K j (A) = K1−j (D(A+ )),

j = 0, 1.

(B.13)

Recall that here A+ is the algebra obtained from A by adjoining a unit. We present one more useful expression for the K-homology groups in terms of the dual algebra. Deﬁnition B.28. Let J be an ideal in the C ∗ -algebra A. By D(A//J) we denote the ideal D(A//J) = {T ∈ D(A) : T π(a) ∈ K(H)

for any a ∈ J}

of the algebra D(A) consisting of the elements that become compact after multiplication by any element of J. In particular, the ideal D(A//A) is called the ideal of locally compact operators. This terminology becomes clear if we consider the algebra A = C0 (X) of functions vanishing at inﬁnity on a locally compact Hausdorﬀ space X. In this case, the elements of the ideal D(C0 (X)//C0 (X)) are the operators that become compact after multiplication by any compactly supported function. Proposition B.29. The following formula holds: K j (A) = K1−j D(A)/D(A//A) ,

j = 0, 1.

(B.14)

This formula does not appeal to unitization (which is often important in practice if an ample representation precisely of the algebra A itself is originally given) and holds both for unital and nonunital algebras. Remark B.30. Let M be a compact smooth manifold, and consider the natural representation C(M ) → BL2 (M ) as operators of multiplication. Since C(M ) is unital, D(C(M )//C(M )) = K and the K-homology group is by deﬁnition K 0 (C(M )) = K1 (D(C(M ))/K). Since K1 is generated by homotopy classes of unitary elements, the latter formula recovers Atiyah’s realization of K-homology classes by abstract elliptic operators.

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Extensions of C ∗ -Algebras The third (and the last) deﬁnition of K-homology, which we present here, is related to extensions of C ∗ -algebras. We consider only the deﬁnition of the group K 1 (A); the deﬁnition of the group K 0 (A) can be obtained by using the suspension (for details, see below). Deﬁnition B.31. A short exact sequence of C ∗ -algebras of the form 0 −→ K(H) −−−−→ B −−−−→ A −→ 0

(B.15)

is called an extension of the C ∗ -algebra A (by the algebra of compact operators in a separable inﬁnite-dimensional Hilbert space H). Two extensions of the algebra A are said to be isomorphic if there exists an isomorphism of the corresponding exact sequences identical in the term A. An extension is said to split if there exists a C ∗ -algebra homomorphism A → B splitting the sequence (B.15). Deﬁnition B.32. A homomorphism of a C ∗ -algebra A into the Calkin algebra Q(H) = B(H)/K(H) of a separable Hilbert space H is called a quantization of the algebra A. Two quantizations μj : A → Q(Hj ), j = 1, 2, are said to be unitarily equivalent if there exists a unitary operator U : H1 → H2 such that μ2 (a) = AdU (μ1 (a)),

a ∈ A.

The mapping AdU : B(H1 ) → B(H2 ) takes K(H1 ) to K(H2 ) and hence can be factorized to a mapping of Calkin algebras. Proposition B.33. There exists a one-to-one correspondence between classes of isomorphic extensions and classes of unitarily equivalent quantizations of the algebra A. This correspondence can be described by the commutative diagram 0 −−−−→ K(H) −−−−→

B ⏐ ⏐

−−−−→

A −−−−→ 0 ⏐ ⏐μ

0 −−−−→ K(H) −−−−→ B(H) −−−−→ Q(H) −−−−→ 0, where the upper row represents an arbitrary extension in a given equivalence class and the last vertical arrow μ represents some quantization (depending on the choice of the extension) in the class of equivalent quantizations corresponding to it. Deﬁnition B.34. The equivalence classes of extensions of the algebra A form a semigroup with respect to the operation induced by the direct sum of representatives of these classes. This semigroup is denoted by Ext(A).

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Here we slightly digress from the traditional deﬁnition, where only injective extensions (extensions for which the corresponding quantization is injective) are considered. The point is that we deal only with separable algebras, for which the requirement to be injective is superﬂuous: any extension can be transformed into an injective extension by adding a direct summand that is a trivial injective extension, and by the following theorem, this addition does not change the equivalence class of the extension. Proposition B.35 (Voiculescu theorem). If A is a separable C ∗ -algebra, then the equivalence class of any split extension is the zero element of the semigroup Ext(A). In particular, all split extensions are equivalent. Of all possible quantizations, the special class of so-called Toeplitz quantizations can be distinguished. Let π : A → B(H) be a representation of a C ∗ -algebra A in a Hilbert space H, and let P : H → H be a projection such that [P, π(a)] ∈ K(H),

a ∈ A.

Then the mapping a −→ P a : P H −→ P H gives a well-deﬁned homomorphism μP : A −→ Q(P H), which is called the Toeplitz quantization corresponding to the projection P . Theorem B.36. Let A be a separable nuclear C ∗ -algebra. Then the following statements hold : 1. The semigroup Ext(A) is a group, whose zero element is the equivalence class of split extensions. 2. In each class μ ∈ Ext(A), there is a Toeplitz quantization. 3. The inverse element of the class [μP ] of a Toeplitz quantization μP is the class [μ1−P ] of the Toeplitz quantization μ1−P acting in the complementary subspace (P H)⊥ = (1 − P )H. Now we can give the desired deﬁnition. Deﬁnition B.37. The K 1 -homology group of a separable nuclear C ∗ -algebra A is given by the formula K 1 (A) = Ext(A) . Equivalence of Diﬀerent Deﬁnitions The above three versions of the deﬁnition of K-homology of separable nuclear C ∗ -algebras are equivalent to each other. Let us show how the correspondence between the elements of the K-homology groups described in diﬀerent deﬁnitions can be established.

B.2. K-Homology

197

Fredholm modules and duality. Let [p] ∈ K0 (D(A+ )) be the equivalence class of some projection over D(A+ ). Let us write out the Fredholm module determining the corresponding element in K 1 (A). Let p ∈ Mn (D(A+ )). One can readily see that if the algebra D(A+ ) is realized via an ample representation π of the algebra A+ in the space H, then Mn (D(A+ )) = {T ∈ B(H ⊗ Cn ) : [T, π(a) ⊗ 1n ] ∈ K(H ⊗ Cn ),

a ∈ A+ }.

But the representation π( · ) ⊗ 1n of the algebra A+ is also ample in H ⊗ Cn (recall that A+ is unital), and hence the algebra Mn (D(A+ )) is isomorphic to the algebra D(A+ ) by Proposition B.26. Thus, without loss of generality, we can assume that p ∈ D(A+ ). Then the corresponding Fredholm module has the form (H, π, 2p − 1). (For brevity, we simply write π, although, strictly speaking, we mean the restriction of the representation π to the subalgebra A.) Now assume that [u] ∈ K1 (D(A+ )) is the equivalence class of some unitary element over D(A+ ). Let us write out the graded Fredholm module determining the corresponding element in K 0 (A). Just as above, without loss of generality, we can assume that u ∈ D(A+ ). Then the corresponding Fredholm module has the form

0 u∗ . H, π, u 0 Fredholm modules and the theory of extensions. Let us show how to assign an element of the group K 1 (A) described in terms of the Fredholm modules to an element of the group Ext(A). Here the construction is quite simple. By Theorem B.36, each element of the group Ext(A) is the equivalence class of some Toeplitz quantization μP , where P is a projection in the space H of the representation π of the algebra A and compactly commutes with the representation operators. This readily gives the Fredholm module (H, π, 2P − 1).

B.2.3 Suspension and Bott Periodicity Just as in the case of covariant K-theory, the following statements hold in Ktheory of operator algebras. Proposition B.38. Let A be a separable nuclear C ∗ -algebra. Then 1. (Suspension.) K 0 (SA) = K 1 (A). 2. (Bott periodicity.) K ∗ (S 2 A) K(A),

∗ = 0, 1.

We do not explicitly write out the corresponding isomorphisms.

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Appendix B. K-Theory of Operator Algebras

B.2.4 Long Exact Sequence in K-Homology Theorem B.39 (Six-term sequence in K-homology of algebras). Let 0 −→ J −−−−→ A −−−−→ A/J −→ 0 be a short exact sequence of C ∗ -algebras. Then there is a long exact sequence K 0 (J) ⏐ ⏐ δ0

←−−−− K 0 (A) ←−−−− K 0 (A/J) 8 ⏐δ ⏐1

K 1 (A/J) −−−−→ K 1 (A) −−−−→

(B.16)

K 1 (J)

of K-homology groups. Remark B.40. Here we do not present any explicit formulas for the connecting homomorphisms δ0 and δ1 . (For example, they can be found in [17] and [41].) We only note that for the special (commutative) case in which A = C(X) is the algebra of continuous functions on a smooth compact manifold X with boundary ∂X, J is the ideal of functions vanishing on ∂X, and A/J = C(∂X) is the algebra of continuous functions on ∂X, the homomorphism δ0 can be interpreted as follows. Let D be an elliptic ﬁrst-order diﬀerential operator on X. Then a pseudodiﬀerential operator T on X with symbol σ(T ) = σ(D)(1 + σ(D)∗ σ(D))−1/2 can be deﬁned. This operator compactly commutes with the action of the algebra J and is elliptic in the interior of X and hence determines an element [T ] in K 0 (J). The element δ0 [T ] ∈ K 1 (A/J) corresponds to the Calder´on projection (e.g., see [42]) of the operator D, and its triviality is equivalent to the fact that the operator D admits (possibly, after stabilization) classical elliptic boundary value problems (Atiyah–Bott condition). More details about this can be found, e.g., in [16]. Remark B.41. Strictly speaking, in the exact sequence (B.16) the K-homology of the ideal J stands in place of the relative K-homology K j (A, A/J) = K1−j D(A+ )/D(A+ //J) . The two are equal by the following theorem. Theorem B.42 (Excision in K-homology). There is a natural isomorphism K j (A, A/J) K j (J).

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199

B.2.5 Stability Just as the K-groups, the K-homology of the C ∗ -algebra A does not change if the K(H). Indeed, for example, in the second algebra A is replaced by Mn (A) or A ⊗ case the desired relation follows from the deﬁnition of K-homology with the use of duality theory and the following easy-to-verify relation: K(H)) = D(A) ⊗ C D(A). D(A ⊗

B.2.6 Duality between K-Homology and K-Theory of Operator Algebras There are natural pairings between the K-groups and the K-homology groups of the same parity of the algebra A. Let us describe these pairings. Even Groups Let [p] ∈ K0 (A) be the equivalence class of a projection p in the algebra A, and let [F ] ∈ K 0 (A) be the equivalence class of a graded Fredholm module (H, π, F ). In the decomposition of the space H into graded components, the operator F has the form

0 T∗ . F = T 0 The pairing of the groups K0 (A) and K 0 (A) is given by the formula K 0 (A) × K0 (A) −−−−→ ([F ], [p])

−→

Z, ind(pF : pH −→ pH).

(B.17)

(The fact that the operator pF is Fredholm is obvious.) The case in which p is the projection not in the algebra A itself but in some matrix algebra Mn (A) can be considered in a similar way. Proposition B.43. The pairing (B.17) is well deﬁned. Odd Groups Let [u] ∈ K1 (A) be the equivalence class of some unitary matrix u over the algebra A, and let [F ] ∈ K1 (A) be the equivalence class of some graded Fredholm module (H, π, F ). For simplicity assume that u is a unitary element of the algebra A itself. Then the Toeplitz operator P+ (F )u : P+ (F )H → P+ (F )H is a Fredholm operator. The pairing of the groups K1 (A) and K 1 (A) is given by the formula K 1 (A) × K1 (A) −−−−→ ([F ], [u])

−→

Z, ind(P+ (F )u).

Proposition B.44. The pairing (B.18) is well deﬁned.

(B.18)

Appendix C

Cyclic Homology and Cohomology C.1 Cyclic Cohomology We start from the deﬁnition of cyclic cohomology of algebras, which is simpler than the deﬁnition of cyclic homology.

C.1.1 Enveloping Diﬀerential Algebra Let A be a local C ∗ -algebra. For this algebra, we construct a universal enveloping diﬀerential algebra (Ω, d), i.e., a graded topological diﬀerential algebra (Ω, d), Ω=

∞

Ωn ,

d(Ωn ) ⊂ Ωn+1 ,

n=0

along with a homomorphism j : A → Ω0 such that for each homomorphism

ρ: A → Ω 0 into the degree zero term of a topological diﬀerential graded algebra (Ω , d ) there exists a unique homomorphism ρ : (Ω, d) → (Ω , d )

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of topological diﬀerential graded algebras for which the diagram j

A −−−−→

Ω0 ⏐ ⏐ρ

A −−−−→ Ω 0 ρ

commutes. It turns out that the universal enveloping diﬀerential algebra exists, is unique up to an isomorphism, and can be described by the following explicit construction. We set Ωn ≡ Ωn (A) = A+ ⊗ A · · ⊗ A< . 9 ⊗ ·:; n factors

(This is the projective tensor product of Fr´echet spaces. Recall that A+ is obtained from A by adding a unit even if A already has one.) Thus, the decomposable elements in Ωn (A) become

a0 ⊗ a1 ⊗ · · · ⊗ an ≡ (a0 + λ01 ) ⊗ a1 ⊗ · · · ⊗ an , where aj ∈ A, j = 0, 1 . . . , n, λ ∈ C, and 1 is the adjoined unity and, from now on, to preserve the notation, we write a = a + λ11 . The space Ωn , whose elements will be called noncommutative n-forms, is a right A-module with respect to the action deﬁned on the decomposable elements by the formula n−1 (−1)n−j a0 ⊗ · · · ⊗ aj aj+1 ⊗ · · · ⊗ a

a0 ⊗ a1 ⊗ · · · ⊗ an a = j=0

+ a0 ⊗ a1 ⊗ · · · ⊗ an a. (C.1) We equip the direct sum Ω=

∞

Ωn

(C.2)

n=0

with the multiplication (inductively) described on the decomposable elements by the formula def ω· a0 ⊗ a1 ⊗ · · · ⊗ an = ωa0 ⊗ a1 ⊗ · · · ⊗ an . (C.3) This multiplication is associative and makes Ω a graded topological algebra. We deﬁne the diﬀerential (of degree +1) on Ω by the formula d a0 ⊗ a1 ⊗ · · · ⊗ an = 1 ⊗ a 0 ⊗ a1 ⊗ · · · ⊗ an . (C.4) In particular, d(11 ⊗ a1 ⊗ a2 ⊗ · · · ⊗ an ) = 0.

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203

The above description of the universal enveloping diﬀerential algebra looks rather cumbersome, and the formulas deﬁning the product seem to be artiﬁcial. Therefore, we give an alternative description of this algebra. To this end, to each element a ∈ A we assign the element da = 1 ⊗ a ∈ Ω1 . It turns out that each element ω ∈ Ωn can be expressed as a linear combination of elements of the form

a0 da1 da2 · · · dan ; for such elements, the formal rules of multiplication and diﬀerentiation are the same as for the usual de Rham diﬀerential forms (with the noncommutativity of the algebra taken into account). In particular, d a0 da1 · · · dan = da0 da1 · · · dan , da · b = d(ab) − a db, etc. Remark C.1. A similar universal algebra can be constructed in the category of unital algebras and unital maps. In this case, one deﬁnes the space of n-forms as Ωn = A ⊗ (A/C1)n , where 1 ∈ A is the unit. In this case, we have d1 = 0 (unlike the previous construction for general algebras, where d1 = 0.)

C.1.2 Graded Traces and Cyclic Functionals Let τ : Ωn −→ C be a continuous linear functional possessing the cyclic invariance property τ(ω1 ω2 ) = (−1)kl τ(ω2 ω1 ),

ω 1 ∈ Ωk ,

ω 2 ∈ Ωl ,

k + l = n.

(C.5)

Each functional of this form will be called a graded trace of degree n on Ω. To the trace τ we can assign a continuous (n + 1)-linear functional on the algebra A by the formula (C.6) τ (a0 , a1 , . . . , an ) = τ(a0 da1 · · · dan ). We also assume that the trace τ is closed, i.e., satisﬁes τ(dω) = 0,

ω ∈ Ωn−1 .

(C.7)

If this is the case, then the functional τ is cyclic, i.e., satisﬁes τ (an , a0 , a1 , . . . , an−1 ) = (−1)n τ (a0 , a1 , . . . , an ).

(C.8)

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Appendix C. Cyclic Homology and Cohomology

Indeed, τ(an da0 · · · dan−1 ) = τ(d(an a0 ) da1 · · · dan−1 ) − τ(dan · a0 da1 · · · dan−1 ) = − τ (dan · a0 da1 · · · dan−1 ) (because of the closedness) = (−1)n τ(a0 da1 · · · dan ) (because of the cyclic invariance).

C.1.3 Cochains, Cyclic Cochains, and the Hochschild Diﬀerential The set of all cyclic continuous (n + 1)-linear functionals (cyclic n-cochains) on A is a linear space, which we denote by Cλn = Cλn (A). (The subscript λ stands for graded invariance with respect to the cyclic permutation operator, which we denote by the same letter.) There is a theorem stating that all such functionals can be obtained from closed graded traces. We deﬁne the Hochschild diﬀerential b : Cλn −→ Cλn+1 by the formula (bτ )(a0 , a1 , . . . , an+1 ) =

n

τ (a0 , a1 , . . . , aj aj+1 , . . . , an+1 )

j=0

(C.9)

+ (−1)n+1 τ (an+1 a0 , a1 , . . . , an ). The Hochschild diﬀerential on the space C n = C n (A) of all (not only cyclic) cochains can be deﬁned by the same formula. Generally speaking, the relation bλ + λb = 0 does not hold on C n (A), but it already holds on the subspace Cλn (A) ⊂ C n (A) of cyclic cochains, so that b(Cλn ) ⊂ Cλn+1 and Deﬁnition (C.9) makes sense.

C.1.4 Cyclic Cohomology and Hochschild Cohomology A straightforward veriﬁcation shows that b2 = 0. Thus, the Hochschild diﬀerential deﬁnes the two complexes b

b

b

(C.10)

b

b

b

(C.11)

0 −−−−→ Cλ0 (A) −−−−→ Cλ1 (A) −−−−→ Cλ1 (A) −−−−→ · · · , 0 −−−−→ C 0 (A) −−−−→ C 1 (A) −−−−→ C 1 (A) −−−−→ · · · .

The cohomology groups of the complexes (C.10) and (C.11) are called the cyclic cohomology and the Hochschild cohomology, respectively, of the algebra A and are denoted by HC n (A) = Zλn (A)/Bλn (A),

HH n (A) = Z n (A)/B n (A),

with the usual notation for the corresponding spaces of cocycles and coboundaries.

C.1. Cyclic Cohomology

205

C.1.5 Example Now let us consider a simple example. Let M be a smooth oriented n-dimensional manifold, and consider the algebra A = MatN (C ∞ (M )) of matrix C ∞ -functions. What multilinear functionals on A can be indicated? Suppose that we wish to describe a k-cochain, i.e., a (k + 1)-linear functional. The following simple method suggests itself: to a diﬀerential form ω of degree n − k on M , we assign the functional (C.12) tr a0 da1 ∧ · · · ∧ dak ∧ ω, τω (a0 , a1 , . . . , ak ) = M

where d is the usual de Rham diﬀerential and tr is the matrix trace. For an arbitrary form ω, the functional (C.12) need not be cyclic, but it is necessarily Hochschild closed: bτω = 0. But if dω = 0 (the form ω is closed), then the functional τω is already cyclic, which can be shown by the following straightforward computations: tr dak ∧ a0 da1 ∧ · · · ∧ dak−1 ∧ ω τω (a0 , a1 , . . . , ak ) = (−1)k−1 M k−1 tr d(ak a0 ) ∧ da1 ∧ · · · ∧ dak−1 ∧ ω = (−1) M + (−1)k tr ak da0 ∧ da1 ∧ · · · ∧ dak−1 ∧ ω M

= (−1)k τω (ak , a0 , . . . , ak−1 ), since the ﬁrst term in the sum is an integral of an exact form and hence is zero. Thus, the functional τω assigns a number to each diﬀerential form a0 da1 ∧ . . . dak , and hence the above argument suggests that HC k (C ∞ (M )) ∼ Hk (M ). (And, in a similar way, HH k (C ∞ (M )) ∼ Currentsk (M ), where Currentsk (M ) is the space of de Rham currents of degree k on M .) But this is not true. The point is that there also exist other cyclic multilinear functionals noncohomological to the above-described functionals. Indeed, take a form ω of degree n − m, where m = k − s < k, and deﬁne a k-linear functional on A by the formula ± tr a0 da1 ∧ · · · ∧ da τω (a0 , . . . , ak ) = j1 ∧ · · · ∧ dajs ∧ · · · ∧ dak ∧ ω, 1≤j1 <j2 <···<js ≤k

M

(C.13)

206

Appendix C. Cyclic Homology and Cohomology

where the “hat” over a factor daj means that this factor is replaced by aj and the signs ± are chosen in a speciﬁc way. (Here we do not describe the choice of the signs.) It turns out that if s = k − m is even and ω is closed, then, under an appropriate choice of the signs, we obtain a cyclic cocycle. The right answer has the form [26] HC k (C ∞ (M )) ker b ⊕ Hk−2 (M, C) ⊕ Hk−4 (M, C) ⊕ · · · ,

(C.14)

where b is the de Rham diﬀerential on currents of degree k and Hl (M, C) is the usual homology of manifold M .

C.1.6 Cup Product in Cyclic Cohomology Let A and B be two algebras, and let τ and τ be closed graded traces of degrees n and n on Ωn (A) and Ωn (B), respectively. We wish to deﬁne the cup product τ # τ , which should be a closed graded trace on Ω(A ⊗ B) (more precisely, on n +n (A ⊗ B), since on the other components this is 0). Although Ω(A ⊗ B) = Ω Ω(A) ⊗ Ω(B) in the general case, this does not prevent us from presenting a wellposed deﬁnition. Namely, we set

τ (ω ); τ(ω ⊗ ω ) = (−1)n n τ (ω )

(C.15)

def

this determines the trace τ = τ # τ on the graded tensor product Ω(A) ⊗ Ω(B). Further, we use the universal property of the diﬀerential enveloping algebra, which implies that there is a unique homomorphism π of diﬀerential algebras making the diagram below commute: A⊗B ⏐ ⏐j ⊗j A B

A⊗B ⏐ ⏐ jA⊗B

Ω(A ⊗ B) −−−−→ Ω(A) ⊗ Ω(B). π

Using this homomorphism, we lift the trace τ to Ω(A ⊗ B). The same can be stated in the language of cyclic functionals: if ϕ ∈ Zλm (A),

ψ ∈ Zλn (B),

then we determine ϕ#ψ ∈ Zλm+n (A ⊗ B) by the formula ∧ ϕ#ψ = π ∗ ϕ ⊗ ψ ≡ ϕ ⊗ ψ ◦ π. One can show that this product gives a well-posed deﬁnition of cup product # : HC ∗ (A) × HC ∗ (B) → HC ∗ (A ⊗ B) in cyclic cohomology.

C.1. Cyclic Cohomology

207

C.1.7 Periodicity in Cyclic Cohomology Periodicity is a key operation connecting the cyclic cohomology groups of diﬀerent dimensions, which serves as an analog of the Bott isomorphism in topological Ktheory. This operation is related to the tensor multiplication of an algebra by C; therefore, we ﬁrst compute the cyclic cohomology of the algebra C of complex numbers. We denote the unit in C by e ∈ C, and by 1, as usual, we denote the adjoined unit in the algebra C+ obtained from C by unitization. For each n, the space Ωn (C) is the two-dimensional complex space C2 with the basis de 9 de:;· · · de<,

e 9de de:;· · · de< .

n factors

n factors

(C.16)

Let us ﬁnd out what closed graded traces of degree n exist on Ω(C). If τ is such a trace, then τ(de de · · · de) = 0 owing to closedness, and hence the trace is completely determined by its value on the second basis element in (C.16). One can readily verify the following identities: e2 = e e de + de e = de e de e = 0 e de de = de de e

(the former unit is an idempotent),

(C.17)

(diﬀerentiate (C.17)), (C.18) (multiply (C.18) by e on the right and on the left), (C.19) (diﬀerentiate (C.19)).

(C.20)

Using these formulas, we readily compute τ(e de de · · · de) = τ(e2 de de · · · de) = (−1)n τ(e de de · · · de e) 0 τ(e de e de · · · de) = 0 for n = 2k + 1, = τ(e de de · · · de) for n = 2k

by (C.17) (by cyclicity) by (C.20) and (C.19).

One can show that for even n, the value attained by τ can indeed be arbitrary, so that the space of closed graded traces of degree n is C for even n and {0} for odd n. Then the same statement also holds for the spaces of cyclic cochains. This implies that the Hochschild diﬀerential is zero in this case, and ﬁnally, 0 C, n = 2k, n HC (C) = (C.21) {0}, n = 2k + 1. Since C ⊗ C = C, we see that HC ∗ (C) is equipped with a multiplication making the cyclic cohomology space of the algebra of complex numbers a ring with a single

208

Appendix C. Cyclic Homology and Cohomology

multiplicative generator σ ∈ HC 2 (C). We eliminate the ambiguity in the choice of this generator by using the normalization condition σ(e de de) = 1. Theorem C.2. For any local C ∗ -algebra A, the space HC ∗ (A) is an HC ∗ (C)module. (The module structure is induced by the tensor multiplication A ⊗ C = A, as was explained above.) We now deﬁne the periodicity operator S : HC n (A) −→ HC n+2 (A), τ −→ τ #σ, where σ ∈ HC 2 (C) is the element constructed earlier. This operator enables us to deﬁne the periodic cyclic cohomology groups HP ∗ (A) = lim HC n+∗ (A), n→∞

where ∗ = 0, 1,

as the direct limit with respect to the mappings given by operator S. Example C.3. Cyclic cohomology of the algebra of C ∞ functions on a smooth compact manifold is (see Eq. (C.14)) HC k (C ∞ (M )) ker b ⊕ Hk−2 (M, C) ⊕ Hk−4 (M, C) ⊕ · · · . It can be shown that S preserves this decomposition and, moreover, periodic cyclic cohomology in this case is HP 0 (C ∞ (M )) H2k (M, C), k≥0

HP 1 (C ∞ (M ))

H2k+1 (M, C).

k≥0

Let us write out a formula for the periodicity operator at the level of representatives. Let τ be a cyclic n-cocycle on A corresponding to a closed graded trace τ of degree n. We act on the class [τ ] ∈ HC n (A) by the operator S. The operator S is given by the tensor multiplication by σ; therefore, to write out a representative Sτ of the class S[τ ], we should compute the corresponding tensor product. We have τ ⊗ σ = τ ⊗ σ . (In this case, the preimage under π ∗ is not required, since the isomorphism Ω(A ⊗ C) ≡ Ω(A) ∼ Ω(A) ⊗ Ω(C) occurs under the multiplication by C.) Next, Sτ (a0 , a1 , . . . , an+2 ) = τ ⊗ σ (a0 ⊗ e) d(a1 ⊗ e) ∧ · · · ∧ d(an+2 ⊗ e) . (C.22)

C.2. Cyclic Homology

209

Using the identity d(a ⊗ e) = da ⊗ e + a ⊗ de and the linearity of the graded trace τ ⊗ σ , we multiply out on the right-hand side in (C.22). Then the sum contains only terms with precisely two occurrences of de. Moreover, it follows from identities (C.17) and (C.18) that only the terms containing these two occurrences of de in neighboring positions can “survive.” In these positions, the corresponding aj are not diﬀerentiated, and, as a result, for Sτ we obtain the formula (Sτ )(a0 , a1 , . . . , an , an+1 , an+2 ) = τ (a0 a1 a2 , a3 , . . . , an+2 ) − τ (a0 , a1 a2 a3 , a4 , . . . , an+2 ) + τ (a0 , a1 , a2 a3 a4 , . . . , an+2 ) − · · · + (−1)n τ (a0 , a1 , . . . , an an+1 an+2 ) + (−1)n+1 τ (an+1 an+2 a0 , a1 , . . . , an ). (C.23)

C.1.8 Example Now let us describe how the operator S acts in the case of algebra of C ∞ functions in terms of diﬀerential forms. Thus, let M be an n-dimensional manifold, and let A = MatL (C ∞ (M )). Let ω be a closed diﬀerential (n − k)-form on M determining the functional τ = τω by formula (C.12): τ (a0 , a1 , . . . , ak ) = tr a0 da1 ∧ · · · ∧ dak ∧ ω, M ∞

so that [τ ] ∈ HC (C (M )). We apply the operator S to the class [τ ]. The representative Sτ of the class S[τ ] is given using (C.23) by the formula k

Sτ (a0 , a1 , . . . , ak+2 ) n+1 = tr a0 da1 ∧ · · · ∧ daj−1 aj aj+1 daj+2 ∧ · · · dak+2 ∧ ω. (C.24) j=1

M

C.2 Cyclic Homology C.2.1 Deﬁnition of Cyclic Homology Again let A be a local C ∗ -algebra. (We assume that it is unital, i.e., adjoin the unit if necessary.) Set Ωn (A) = A⊗(n+1) ≡ A · · ⊗ A< . 9 ⊗ ·:; n+1 factors

The elements of this space are called Hochschild chains. A chain is said to be cyclic if it changes or does not change the sign (according to whether the number n is

210

Appendix C. Cyclic Homology and Cohomology

even) under a cyclic permutation of the factors: j a0 ⊗ aj1 ⊗ · · · ⊗ ajn = (−1)n ajn ⊗ aj0 ⊗ · · · ⊗ ajn−1 . ω= j

j

We denote the set of cyclic chains by Cn (A) ⊂ Ωn (A) and introduce the boundary operator b : Ωn (A) −→ Ωn−1 (A) by the formula b(a0 ⊗ · · · ⊗ an ) = a0 a1 ⊗ a2 ⊗ · · · ⊗ an − a0 ⊗ a1 a2 ⊗ a3 ⊗ · · · ⊗ an + · · · + (−1)n an a0 ⊗ a1 ⊗ · · · ⊗ an−1 . Under this deﬁnition, we have b(Cn (A)) ⊂ Cn−1 (A), and the cyclic homology of the algebra A is determined as usual as the homology of the complex b

b

b

b

· · · −−−−→ Cn (A) · · · −−−−→ C2 (A) −−−−→ C1 (A) −−−−→ C0 (A) −−−−→ 0. (C.25) Namely, HCn (A) = Zn (A)/Bn (A), n = 0, 1, 2, . . . . (C.26) In particular, b(a0 ⊗ a1 ) = a0 a1 − a1 a0 ≡ [a0 , a1 ] is the commutator of the elements a0 and a1 , so that the homology at the ﬁrst step measures the noncommutativity of the algebra A. Namely, HC0 (A) = A/[A, A].

C.2.2 Periodicity Just as in the case of cyclic cohomology, there is a (dual) periodicity operator S : HCn (A) −→ HCn−2 (A), which is an epimorphism. The projective limits 0 HP0 (A) limprojS HCn (A) = HP1 (A)

for even n, for odd n

(C.27)

(C.28)

are called the periodic cyclic homology of the algebra A. In the case of the algebra A = C ∞ (M ), there are natural isomorphisms HP0 (A) = ⊕H 2k (M, C),

HP1 (A) = ⊕H 2k+1 (M, C).

Here the formulas for the periodicity operator S are more complicated than those in the case of the cohomology, and we do not write them out because of their cumbersomeness.

C.2. Cyclic Homology

211

C.2.3 Pairing between Cyclic Homology and Cohomology There is a natural pairing between cyclic homology and cohomology, HCn (A) × HC n (A) −→ C,

(C.29)

generated on the level of representatives by the mapping (a0 ⊗ a1 ⊗ · · · ⊗ an ) × τ (·, . . . , ·) −→ τ (a0 , a1 , . . . , an ). (We consider the case of a unital algebra A.)

C.2.4 Morita Invariance There exist natural isomorphisms HC n (A) = HC n (MN (A)), HCn (A) = HCn (MN (A)),

(C.30) (C.31)

N = 1, 2, . . . , where MN (A) is the algebra of N ×N matrices over A. For example, the isomorphism (C.30) can be given by the formula def

τ (a0 ⊗ m0 , a1 ⊗ m1 , . . . , an ⊗ mn ) = τ (a0 , a1 , . . . , an ) tr(m0 m1 · · · mn ), where mj ∈ MN (C) and tr(· · · ) is the usual matrix trace.

C.2.5 Chern Characters in Homology Let us present mappings of K-theory into cyclic homology. These mappings are called Chern characters. First, we describe the Chern character on the odd K-group chn1 : K1 (A) −→ HC2n−1 (A). The descriptions look somewhat diﬀerent in the case of unital algebras and in the case of algebras without unity. 1. Let A be unital. Then for elements a ∈ GL(A) we deﬁne the Chern character by the formula K1 (A) [a] −→ [cn a−1 ⊗ a ⊗ · · · ⊗ a−1 ⊗ a] ∈ HC2n−1 (A), :; < 9

(C.32)

2n factors

where cn = 2−2n+1 Γ(n + 1/2) is the normalization constant. In this formula we also used stability of cyclic homology HC2n−1 (M (A)) HC2n−1 (A). 2. Let A be nonunital. Then, by deﬁnition, K1 (A) consists of the equivalence classes of invertible matrices a with elements in A+ such that a − 1 ∈ M (A). We set 2 1 (C.33) chn1 [a] = cn (a−1 − 1) ⊗ (a − 1) ⊗ · · · ⊗ (a − 1 ) .

212

Appendix C. Cyclic Homology and Cohomology Now we describe the Chern character on the even K-group, chn0 : K0 (A) −→ HC2n (A).

For simplicity, we consider only the case of unital algebras. The elements of the group K0 (A) are the equivalence classes [P ] of matrix projections P with entries in A. We specify the mapping chn0 : K0 (A) −→ HC2n (A)

(C.34)

by the formula [P ] −→ cn P ⊗ · · · ⊗ P , :; < 9

(C.35)

2n+1 factors

where cn = (−1)n (2n)!/(n!) is the normalization constant. Theorem C.4. The mapping (C.34), (C.35) is well deﬁned and is a group homomorphism. Remark C.5. The normalization constants are chosen in such a way that the Chern characters deﬁned for diﬀerent values of n are compatible with the periodicity operator S in cyclic homology: chn−1 = S ◦ chn∗ , ∗

∗ = 0, 1.

Thus, these Chern characters induce Chern character with values in periodic cyclic homology.

Concise Bibliographical Remarks Chapter 1 A detailed presentation of the theory of crossed products, as well as further references, can be found, for example, in the monographs [62,77]. An important role in the theory of crossed products is played by the isomorphism theorem (see [1, 3]). Smooth subalgebras in group algebras of polynomial growth and in the corresponding crossed products were studied in [67, 68]. We note that there are also known local subalgebras for other classes of groups, for example, for hyperbolic groups (e.g., see [43, 45] and the recent survey in [44]).

Chapter 2 The notion of symbol for pseudodiﬀerential operators with shifts was introduced in [3]. The Fredholm property of elliptic operators generated by an amenable group of shifts was also proved there. The use of the isomorphism theorem is a key point in this proof. Note that in our book we do not consider problems related to studying the ellipticity of the symbol (i.e., its invertibility). This is a very interesting and meaningful ﬁeld of studies closely related to the theory of dynamical systems. For example, in the case of an action of the group Z (i.e., for nonlocal operators generated by iterations of a given diﬀeomorphism of a manifold), the symbol invertibility condition is equivalent to the hyperbolicity condition for a linear extension of the diﬀeomorphism. This class of problems was studied in [1], where the reader can ﬁnd further references.

Chapter 3 The theory of (local) elliptic operators over C ∗ -algebras was constructed in [57]. Further development of this theory can be found in [48] and in the monographs [53, 72]. As far as we know, nonlocal operators over C ∗ -algebras have not been studied earlier.

Chapter 4 Atiyah and Singer [10] showed that the group of stable homotopy classes of elliptic operators on a smooth manifold is isomorphic to the K-group of the cotangent bundle of the manifold. In fact, the isomorphism is induced by the symbol mapping. In this chapter, this result is generalized to the case of nonlocal operators.

Chapter 5 In the case of a nonlocal operator corresponding to a ﬁnite group Γ of shifts, Antonevich [5] proved that such an operator determines a new Γ-invariant operator. Passing to the Γ-index of the latter operator, we see that nonlocal operators have

214

Concise Bibliographical Remarks

an invariant ranging in the virtual representation ring of the group Γ. In this chapter, we deﬁne the corresponding invariant in the case of inﬁnite groups. In this case, this invariant ranges in the K-group of the C ∗ -algebra of the group Γ. In fact, the Fredholm index is contained in this invariant.

Chapter 6 In this chapter, we establish the Bott periodicity theorem for isometric actions of polynomial growth groups. This theorem is new. By the way, we note that there is a close generalization of the periodicity theorem presented in [40]. These authors state the periodicity theorem in terms of noncommutative algebras including functions of anticommuting variables. The statement and the proof of the periodicity theorem presented in our book are closer to the classical version. In particular, we construct the mapping providing the inverse of the Bott homomorphism in terms of indices of some operators.

Chapter 7 Here we consider the direct image and the index formula in K-theory. We express the index of an elliptic operator as the direct image (under the projection of the manifold into a point) of an element in K-theory corresponding to the symbol of the operator. But, in contrast to the classical proof of the index theorem due to Atiyah and Singer [10], we do not use the axiomatic approach but successively transform the elliptic operator and simultaneously prove that its analytical index is preserved under these transformations. Note that a proof, close in spirit, of an index theorem was given in [54]. Let us also mention another approach to index formulas based on K-theory [48]. In this approach, an arbitrary elliptic operator is ﬁrst reduced to the Dolbeault operator on the cotangent bundle of the manifold twisted by the bundle determined by the symbol of the original operator. In our situation, there would be no obstacle to such a realization and establishing the corresponding reduction to the Dolbeault operator, but then, to obtain a cohomology formula for the index, we would have to calculate the equivariant Chern–Connes character of the Dolbeault operator. Here the calculations by means of the local Connes–Moscovici index theorem [31] encounter some technical diﬃculties related to small denominators, and one has to impose unnecessary additional conditions; see [59]. In the approach based on the K-theory index formula in terms of the direct image, these additional conditions do not arise.

Chapter 8 The deﬁnition of the Chern character in noncommutative geometry goes back to Connes and Karoubi (see the monographs [27,46] and the references therein). Here the Chern character ranges in the cyclic homology group or in the noncommutative

Concise Bibliographical Remarks

215

de Rham cohomology, respectively. In the case of actions of discrete groups, the Chern character was also constructed in [15]. We use Karoubi’s approach and construct a Chern character ranging in the cohomology of ﬁxed point sets of the action. (To deﬁne such a Chern character, we have to choose some closed graded trace over a smooth group algebra.)

Chapter 9 The Todd class contained in our index formula arises already in the classical Lefschetz formula for the index of equivariant operators [9]; see also [15]. The index formulas on noncommutative toric manifolds [28, 29], in particular, on the noncommutative sphere [50] are speciﬁc cases of the index formula established in this chapter.

Chapter 10 To obtain numerical invariants from the index of an operator over a C ∗ -algebra, it was proposed in [30] to couple the index with closed graded traces over a local subalgebra. Formulas expressing such a coupling in topological terms are called “higher index formulas.”

Chapter 11 The index of (local) elliptic operators over C ∗ -algebras was calculated in [57] modulo ﬁnite-order elements. In [73], the so-called “exact” index formula was obtained in terms of direct image in K-theory. Other approaches to this theory, in particular, to the cohomology formulas, were given in [18, 19, 65, 74, 76]. As far as we know, the index of nonlocal operators over C ∗ -algebras is considered in this book for the ﬁrst time.

Chapter 12 Elliptic theory on the noncommutative torus constructed in [24, 25, 27] has served as the starting point for our studies of nonlocal equations. These papers present ellipticity conditions for the corresponding nonlocal operators, and an index formula was obtained there. A beautiful example is given by a ﬁrst-order operator deﬁned in terms of the Rieﬀel projection [64].

Chapter 13 In the situation of local elliptic operators, the theory of the index of almost projection operators (or self-adjoint operators), in which the index is an element of an odd K-group, was constructed in [12]. In fact, this theory can be reduced to the usual elliptic theory by using the suspension operation.

216

Concise Bibliographical Remarks

Chapter 14 As was already noted, the index of nonlocal operators with a ﬁnite group of shifts was calculated in [5]. For Dirac operators, the calculation of the index in this and similar situations was also performed in [61], [14], and [63].

Appendices Appendix A. There are numerous good textbooks concerned with the theory of C ∗ -algebras. Details of the proofs and further information can be found, for example, in the monographs [6, 35, 37, 58] Appendix B. The reader can ﬁnd a more detailed presentation of K-theory of C ∗ -algebras in the monographs [22, 33, 41]. Appendix C. The reader can ﬁnd a more detailed description of the cyclic (co)homology theory in the monographs [27, 46, 51] and the survey papers [20, 32, 75].

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Index action tempered, 16 topologically free, 13 Atiyah–Bott–Patodi compactiﬁcation, 97

crossed product, 7 reduced, 8 smooth, 16 curvature form, 97 cyclic homology, 209

Bott element, 70 Bott periodicity in K-theory, 181, 186 for inﬁnite groups, 71

diﬀerence construction of elliptic symbol, 61 of Λ-elliptic symbol, 128 direct image mapping, 78

C ∗ -algebras, 162 enveloping, 5 extension of, 164 local, 167 nuclear, 177 projection in, 168 representations of, 169 tensor product of, 177 unital, 162 Chern character in cyclic homology, 211 localized, 99 multiplicativity of, 108 of Λ-elliptic symbol, 130 of elliptic symbol, 94 of projections, 92, 97, 122 on the group K0 (Λ), 102 Chern–Simons character, 148 cohomology cyclic, 204 cup product in, 206 periodic, 208 periodicity in, 207 Hochschild, 204

EllΓ (M ) group, 59 EllΓ (M, Λ) group, 128 enveloping diﬀerential algebra, 201 exterior product of operators, 80, 84 of symbols, 69, 76 ﬁniteness theorem for elliptic operators, 28, 30, 33 for Λ-elliptic operators, 50, 54 Fredholm modules, 192 graded, 192 Gelfand–Naimark theorem, 162 for commutative algebras, 165 group amenable, 6 Heisenberg, 15 of polynomial growth, 15 group C ∗ -algebras, 6 reduced, 6 Hilbert Λ-modules, 39 countably generated, 40 standard, 40

Index Hilbert Λ-modules tensor product of, 65 holomorphic functional calculus, 166 homotopy classiﬁcation of elliptic operators, 61 of Λ-elliptic operators, 128 index Fredholm, 63 Λ-Fredholm, indΛ , 64 topological, 101 index theorem for elliptic operators in cohomology, 102, 123 for ﬁnite group, 155 for group Zn , 104 odd case, 147 in K-theory, 79 index theorem for Λ-elliptic operators, 131 isomorphism theorem, 13 K-homology of operator algebra, 193, 194, 196 of topological space, 189 K-theory of operator algebra, 183, 184 of topological space, 179 relative, 180 Kasparov stabilization theorem, 41 K¨ unneth formula, 129 Λ-bundle, 44 Λ-pseudodiﬀerential operators local, 49 calculus, 49 elliptic, 50 nonlocal, 51, 52 elliptic, 54 Morita invariance, 211 noncommutative cylinder, 10

223 noncommutative diﬀerential forms, 89, 116, 117, 202 torus, 9 diﬀerential operators on, 31, 140 nonlocal bundles, 22 nonlocal functions continuous, C(X)Γ , 18, 23 rapidly decaying, 17 smooth, C ∞ (X)Γ , 21 nonlocal pseudodiﬀerential operators, 28, 30, 32, 35 calculus, 28 elliptic, 33 operators classical Euler, 69 equivariant, 68 Euler in Rn , 70 exterior product of, 68 in Hilbert modules, 41 Λ-Fredholm, 42 Prim A, 175 quantization of C ∗ -algebra, 195 of smooth symbols, 29 representation ample, 193 covariant, 8 Rieﬀel projections, 9 Schweitzer theorem, 17 symbols homogeneous elliptic, 36 multiplicatively trivial at inﬁnity, 35 Todd class, 101 trace closed, 90 graded, 90, 117, 203

224 trajectory representation, 18 symbol, 154 vanishing theorem, 103

Index