Translations of MATHEMATICAL MONOGRAPHS - American Mathematical Society

[Pages:22]Translations of

MATHEMATICAL MONOGRAPHS

Volume 237

Operator Algebras and Geometry

Hitoshi Moriyoshi Toshikazu Natsume

American Mathematical Society

Operator Algebras and Geometry

10.1090/mmono/237

Translations of

MATHEMATICAL MONOGRAPHS

Volume 237

Operator Algebras and Geometry

Hitoshi Moriyoshi Toshikazu Natsume

Translated by Hitoshi Moriyoshi and Toshikazu Natsume

AMERICAN !'%7-%

MATHEMATICA

42(4/3 -(

FOUNDED 1888

SOCIETY L

%)3)47

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE

Shoshichi Kobayashi (Chair) Masamichi Takesaki

(OPERATOR ALGEBRAS AND GEOMETRY) Hitoshi Moriyoshi and Toshikazu Natsume

This work was originally published in Japanese by the Mathematical Society of Japan

under the title "

" c 2001. The present translation was created under license

for the American Mathematical Society and is published by permission.

Translated from the Japanese by Hitoshi Moriyoshi and Toshikazu Natsume.

2000 Mathematics Subject Classification. Primary 46L87, 46L80; Secondary 46L05, 46L10.

For additional information and updates on this book, visit bookpages/mmono-237

Library of Congress Cataloging-in-Publication Data

Moriyoshi, Hitoshi, 1961? Operator algebras and geometry / Hitoshi Moriyoshi, Toshikazu Natsume ; translated by

Hitoshi Moriyoshi, Toshikazu Natsume. p. cm. -- (Translation of mathematical monographs ; v. 237)

Includes bibliographical references and index. ISBN 978-0-8218-3947-8 (alk. paper) 1. Operator algebras. 2. Geometry. I. Natsume, Toshikazu. II. Title.

QA326.M66 2008 512 .556--dc22

2008029381

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@.

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The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

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Contents

Preface

vii

Chapter 1. C-Algebras

1

1.1. C-algebras

1

1.2. C-algebras on Hilbert spaces

3

1.3. Spectral theory

4

1.4. Gel'fand's theorem

8

1.5. Compact operators

13

1.6. Fredholm operators and index

18

1.7. Multiplier algebras

21

1.8. Nuclearity

23

1.9. Representations of C-algebras

27

1.10. C-dynamical systems and crossed products

32

1.11. Fields of Hilbert spaces and C-algebras

35

1.12. Appendix: Bounded linear operators on Hilbert spaces

37

Chapter 2. K-Theory

41

2.1. K-groups

41

2.2. K1-group

45

2.3. Basic properties of K-groups

51

2.4. Applications of K-theory

54

2.5. Twisted K-theory

56

Chapter 3. KK-Theory

59

3.1. KK-group

59

3.2. Construction of the Kasparov product

66

3.3. Extensions

80

Chapter 4. Von Neumann Algebras

87

4.1. Definitions and examples

87

4.2. Factors

89

4.3. Classification

91

4.4. Modular theory (Tomita-Takesaki theory)

97

4.5. Dixmier traces

99

v

vi

CONTENTS

4.6. Further reading

102

Chapter 5. Cyclic Cohomology

105

5.1. Definitions and examples

105

5.2. Relationship with K-theory

109

Chapter 6. Quantizations and Index Theory

115

6.1. Quantizations

115

6.2. Proof of an index theorem

117

Chapter 7. Foliation Index Theorems

121

7.1. The Atiyah-Singer index theorem

121

7.2. Topology of the leaf space M/F

125

7.3. Geometric aspects of the leaf space

133

7.4. Toward noncommutative geometry

142

References

147

Index

153

Preface

In this book we describe the elementary theory of operator algebras and basic tools in noncommutative geometry.

In his early work on noncommutative geometry, A. Connes proposed to treat a general noncommutative C-algebra as the C-algebra of "continuous functions on a noncommutative space". The study of interactions between topology/geometry and analysis via algebraic objects originates with I.M. Gel'fand. According to Gel'fand's theory, the topological structure of a compact topological space X is completely determined by the algebra C(X) of continuous functions. Even a smooth structure of a differentiable manifold can be captured by an algebraic object. Pursell's theorem says that two compact smooth manifolds are diffeomorphic if and only if the R-algebras of smooth functions are isomorphic.

It may be natural to consider Ck(M ) (k = 0, 1, 2, ? ? ? , ) for a given Cmanifold. These are infinite-dimensional vector spaces. In order to control "infinite dimension", we need to study these spaces with topology. For a finite k the space Ck(M ) has the structure of a Banach space, and C(M ) is a Fr?echet space. Since we study linear algebras (e.g. eigenvalue problems) of infinite dimension, it would be suitable to consider algebras over C. Hence from now on Ck(M ) denotes Cvalued Ck-class functions. Complex conjugation of C-valued functions defines a C-algebra structure on C(M ) = C0(M ), a Banach -algebra structure (?1) on Ck(M ) (0 < k < ) and a Fr?echet -algebra structure on C(M ).

Generally speaking, C-algebras are more well behaved than Banach -algebras, and beautiful theories have been established. Once we accept the Gel'fand correspondence of topological spaces and abelian C-algebras as natural, nothing keeps us from investigating noncommutative C-algebras which correspond to "singular" spaces such as the leaf spaces of foliations. Accordingly, it is important to learn not only about commutative C-algebras but also about noncommutative C-algebras.

In a sense noncommutative geometry is a geometry of "virtual spaces" or "pointless spaces". However, that may be misleading. Noncommutative geometry should be thought of rather as a paradigm than as a theory. The core idea is to express geometry as an operator on the representation space of an algebra. As it turns out, noncommutative geometry provides unification of various mathematical

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