Table Of Contenthttp://dx.doi.org/10.1090/surv/060
Selected Titles in This Series
60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace
C*-algebras, 1998
59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998
58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on
representation theory and Knizhnik-Zamolodchikov equations, 1998
57 Marc Levine, Mixed motives, 1998
56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum
groups: Part I, 1998
55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998
54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in
analysis, 1997
53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997
52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in
domains with point singularities, 1997
51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial
differential equations, 1997
50 Jon Aaronson, An introduction to infinite ergodic theory, 1997
49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential
equations, 1997
48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997
47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by
M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997
46 Stephen Lipscomb, Symmetric inverse semigroups, 1996
45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in
categories of associative rings, 1996
44 J. Amoros, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental
groups of compact Kahler manifolds, 1996
43 James E. Humphreys, Conjugacy classes in semisimple algebraic groups, 1995
42 Ralph Freese, Jaroslav Jezek, and J. B. Nation, Free lattices, 1995
41 Hal L. Smith, Monotone dynamical systems: an introduction to the theory of
competitive and cooperative systems, 1995
40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the
finite simple groups, number 3, 1998
40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the
finite simple groups, number 2, 1995
40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the
finite simple groups, number 1, 1994
39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994
38 Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, 1993
37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991
36 John B. Conway, The theory of subnormal operators, 1991
35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990
34 Victor Isakov, Inverse source problems, 1990
33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean
fields, 1990
32 Howard Jacobowitz, An introduction to CR structures, 1990
31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and
harmonic analysis on semisimple Lie groups, 1989
30 Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989
(Continued in the back of this publication)
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Mathematical
Surveys
and
Monographs
Volume 60
Morita Equivalence
and Continuous-Trace
C*-Algebras
lain Raeburn
Dana P. Williams
American Mathematical Society
^DEO
Editorial Board
Georgia M. Benkart Tudor Stefan Ratiu, Chair
Peter Landweber Michael Renardy
1991 Mathematics Subject Classification. Primary 46L05;
Secondary 46L35, 46L40, 43A65.
ABSTRACT. We give a modern treatment of the classification of continuous-trace C*-algebras up
to Morita equivalence. This includes a detailed discussion of Morita equivalence of C*-algebras,
a review of the necessary sheaf cohomology, and an overview of recent developments in the area.
The book should be accessible to anyone familiar with the basics of C*-algebras up to the GNS-
construction.
The authors were supported by grants from the Australian Research Council, the Uni
versity of Newcastle, and the Edward Shapiro Fund at Dartmouth College.
Library of Congress Cataloging-in-Publication Data
Raeburn, Iain, 1949-
Morita equivalence and continuous-trace C*-algebras / Iain Raeburn, Dana P. Williams,
p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 60)
Includes bibliographical references and index.
ISBN 0-8218-0860-5 (alk. paper)
1. C*-algebras. I. Williams, Dana P., 1952- . II. Title. III. Series: Mathematical surveys
and monographs ; no. 60.
QA326.R34 1998
512/.55-dc21 98-25838
CIP
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
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Republication, systematic copying, or multiple reproduction of any material in this publication
(including abstracts) is permitted only under license from the American Mathematical Society.
Requests for such permission should be addressed to the Assistant to the Publisher, American
Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also
be made by e-mail to [email protected].
© 1998 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at URL: http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 03 02 01 00 99 98
To our parents:
Margaret and Fraser Raeburn
on their golden wedding anniversary,
and in memory of
Ruth and James Williams
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Contents
Introduction ix
1 The Algebra of Compact Operators 1
2 Hilbert C*-Modules 7
2.1 Hilbert Modules 8
2.2 Bounded Maps on Hilbert Modules 16
2.3 Multiplier Algebras 23
2.4 Induced Representations 30
3 Morita Equivalence 41
3.1 Imprimitivity Bimodules 42
3.2 Morita Equivalence 47
3.3 The Rieffel Correspondence 52
3.4 The External Tensor Product 62
4 Sheaves, Cohomology, and Bundles 67
4.1 Sheaf Cohomology 68
4.2 Fibre Bundles 92
4.3 The Dixmier-Douady Classification of Locally Trivial Bundles . . . . 102
5 Continuous-Trace C*-Algebras 115
5.1 C*-Algebras with Hausdorff Spectrum 116
5.2 Continuous-Trace C*-Algebras 121
5.3 The Dixmier-Douady Classification of
Continuous-Trace C*-Algebras 126
5.4 Automorphisms of Continuous-Trace C*-Algebras 140
5.5 Classification up to Stable Isomorphism 143
6 Applications 155
6.1 The Brauer Group 155
6.2 Pull-back C*-Algebras 160
6.3 Induced C*-Algebras 163
vii
viii Contents
7 Epilogue: The Brauer Group and Group Actions 173
7.1 Dynamical Systems and Crossed Products 174
7.2 The Equivariant Brauer Group 177
7.3 The Brauer Group of a Point 181
7.4 Group Cohomology and Moore Cohomology 184
7.5 The Brauer Group for Trivial Actions 192
7.6 The Brauer Group for Free and Proper Actions 193
7.7 The Structure of the Brauer Group 196
A The Spectrum 201
A.l States and Representations 201
A.2 The Spectrum of a C*-Algebra 210
A.3 The Dauns-Hofmann Theorem 223
A.4 The State Space of a C*-Algebra 226
B Tensor Products of C*-Algebras 235
B.l The Spatial Tensor Product 236
B.2 Fundamental Examples 244
B.3 Other C*-Norms 246
B.4 C*-Algebras with Hausdorff Spectrum 254
B.5 Tensor Products of General C*-Algebras 257
C The Imprimitivity Theorem 269
C.l Haar Measure and Measures on Homogeneous Spaces 269
C.2 Vector-Valued Integration on Groups 274
C.3 The Group C*-Algebra 280
C.4 The Imprimitivity Theorem 285
C.5 Induced Representations of Groups 293
C.6 The Stone-von Neumann Theorem 298
D Miscellany 303
D.l Direct Limits 303
D.2 The Inductive Limit Topology 304
Index 309
Bibliography 317
Introduction
The Gelfand-Naimark theorem says that a commutative C*-algebra A with identity
is determined up to isomorphism by its spectrum T, in the very strong sense that A
is isomorphic to the algebra C(T) of continuous functions on T. There are also non-
commutative C*-algebras with spectrum T: for example, the algebra C(T, M (C))
n
of continuous functions from T into the matrix algebra M (C). In this book we
n
study theorems which classify algebras with spectrum T and automorphisms of
these algebras in terms of topological invariants of the space T.
One of the goals of algebraic topology is to associate to a space T algebraic
objects whose properties reflect the topological structure of T. The ones of interest
to us are cohomology groups Hn(T;7j): very roughly speaking, iJn(T;Z) counts
n-dimensional holes in the space T by chalking up a copy of Z for each such hole.
Our C*-algebras with spectrum T are classified by classes in the 3-dimensional
cohomology group H3(T; Z), and their automorphisms by classes in H2(T; Z); these
theorems were first proved, respectively, by Dixmier and Douady in 1963 and by
J. Phillips and Raeburn in 1980. They have been used extensively in recent years:
in the analysis of C*-dynamical systems and their crossed products, in the K-theory
of C*-algebras [150], in differential geometry [14], and in mathematical physics [16].
We shall discuss all the background material needed to prove these classification
theorems and apply them to C*-dynamical systems, including the definitions and
properties of the cohomology groups themselves.
There are several versions of the Dixmier-Douady Classification Theorem, in
volving different families of C*-algebras and classifying up to different equivalence
relations. The simplest concerns locally trivial bundles over T with fibres isomor
phic to the algebra of compact operators on an infinite-dimensional Hilbert space,
and identifies them up to bundle isomorphism; we shall prove this one first. For
applications to operator algebra, it is more satisfactory to work with continuous-
trace C*-algebras, as Dixmier and Douady did. They used continuous fields of
C*-algebras, which had been developed by Fell to prove structure theorems for
important classes of C*-algebras. However, Fell's theory never became standard
C*-algebraic equipment, so anybody wanting to use the Dixmier-Douady classifi
cation faced a heavy start-up cost. We have used instead Rieffel's theory of Morita
equivalence for C*-algebras, which is now recognized to be a fundamental tool for
C*-algebraists. It has been folklore for the past 20 years that the Dixmier-Douady
classification can be couched in terms of Morita equivalence (see, for example, [5] or
[140, 20]), but this version has remained logically dependent on the special structure
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