Operator Algebras and Geometry

168 Pages·2008·11.397 MB·English

by  Hitoshi Moriyoshi and Toshikazu Natsume

Translations of
MATHEMATICAL
MONOGRAPHS
Volume 237
Operator Algebras
and Geometry
Hitoshi Moriyoshi
Toshikazu Natsume
American Mathematical Society
Operator Algebras
and Geometry

Translations of
MATHEMATICAL
MONOGRAPHS
Volume 237
Operator Algebras
and Geometry
Hitoshi Moriyoshi
Toshikazu Natsume
Translated by Hitoshi Moriyoshi
and Toshikazu Natsume
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 ©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
www.ams.org/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
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10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08
Contents
Preface vii
Chapter 1. C*- Algebras 1
1.1. C*-algebras i
1.2. C*-algebras on Hilbert spaces 3
1.3. Spectral theory 4
1.4. GePfand’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 (7*-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. AT-Theory 41
2.1. RT-groups 41
2.2. RTi-group 45
2.3. Basic properties of AT-groups 51
2.4. Applications of AT-theory 54
2.5. Twisted AT-theory 56
Chapter 3. KK-Theory 59
3.1. A" AT-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 If-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 MjT 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. PurselPs the­
orem says that two compact smooth manifolds are diffeomorphic if and only if the
E-algebras of smooth functions are isomorphic.
It may be natural to consider Ck(M) (k = 0,1,2, ••• ,oo) for a given C°°-
manifold. 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 Frechet 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 de­
valued Ck-class functions. Complex conjugation of C-valued functions defines a
C*-algebra structure on C(M) = C°(M), a Banach *-algebra structure (§1) on
Ck(M) (0 < k < oo) and a Frechet *-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 GePfand corre­
spondence 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 geome­
try 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
vii
viii , PREFACE
concepts, e.g. spin geometry, geometry of fractals, geometry of discrete groups,
pseudo-differential calculus, and so on.
In the early 1980’s topologists and geometers for the first time came across
unfamiliar words like C*-algebras and von Neumann algebras through the discovery
of new knot polynomials by V.F.R. Jones or through S. Hurder’s remarkable result
on the relationship between the vanishing of the Godbillon-Vey classes for foliations
and the types of foliation von Neumann algebras. During the following decade, a
great deal of progress in the area of interaction between geometry and analysis was
achieved. We list just a few developments: cyclic cohomology theory, if if-theory,
applications of operator algebras to the Novikov conjecture on homotopy invariance
of higher signature in topology.
Geometers in Japan organized a workshop in 1998 to learn operator alge­
bras and its applications to other fields. At that time when topologists/geometers
wanted to study the theory of operator algebras, not much suitable material was
available. Of course, there were many good books, but they were mostly aimed
at those who had a thorough knowledge of functional analysis. This book was ini­
tially prepared for the workshop “Surveys in Geometry” held in the fall of 1998
at the University of Tokyo, and it is aimed at topologists and geometers, with less
background in analysis. We shall provide an overview of operator algebra theory
and explain basic tools used in noncommutative geometry and finally applications
to Atiyah-Singer type index theorems. Our purpose here is to convey an outline
and general idea of the theory of operator algebras, to some extent focusing on
examples. To that end some details and proofs will be omitted. Hence, we give
a list of references that can be easily obtained. For those who do not care about
details of proofs, [46] is easy reading. For those who care about some detail, [80]
may be suitable. Finally, for those who want to thoroughly understand the proofs,
[67] and [109] are excellent. Additional reading material will be referred to in each
chapter.
We would like to express our gratitude to both the American Mathematical
Society and the Mathemtical Society of Japan for giving us this opportunity to
publish this book in English. Finally, we would also like to thank our former
colleague Catherine L. Olsen of SUNY at Buffalo for her considerable assistance in
preparing the manuscript.
Hitoshi Moriyoshi and Toshikazu Natsume
CHAPTER 1
C*-Algebras
1.1. C*-algebras
A (complex) Banach space is a C-vector space with a norm || • || which is
complete in the metric topology given by the norm. A C-algebra B is a Banach
algebra, if B is a Banach space, and for any a>b € B the following inequality holds:
When a Banach algebra B has a (multiplicative) unit 1, we say that B is a unital
Banach algebra. When a Banach algebra has a unit, we do not assume ||1|| = 1.
However, we can always replace the norm with an equivalent norm 111 • 111 satisfying
1111111 = !•
Example 1.1.1. Let P cC b e the open unit disk in the complex plane with
closure D. Then
A = { f G (7(D) ; /|p : holomorphic } (the disk algebra)
is a Banach algebra with respect to the norm ||/||oo = sup{|/(x)| ; x G D}.
Definition 1.1.2. When a Banach algebra B has an antilinear map * : B —> B
such that (i) a** = a, (ii) (ab)* = i>*a*, (iii) ||a*|| = ||a|| for arbitrary a, 6 G £, we
call B a Banach *-algebra.
For a unital B we have
1* = 11* = 1**1* = (11*)* = 1** = 1.
Example 1.1.3. The Banach algebra A of Example 1.1.1 does not have a
Banach *-algebra structure.
Example 1.1.4. Let \i be a left-invariant measure on a locally compact group
(7, and let A be the modular function of /¿. Denote by Ll(G) the space of (equiva­
lence classes of) integrable functions on G. The space Ll(G) is a Banach *-algebra
with the following structure:
product : (<t>*4>)(g) = [ <l>(h)íp(h~1g)dfji,(h), <j>, tp € Ll(G),
Jg
* ■ 4>*(g) = A
norm : ||<^||i = / \<p(g)\dfi(g).
Jg
1