**Mathematical Surveys and Monographs**

Volume: 134;
2007;
528 pp;
Hardcover

MSC: Primary 46; 22; 54;

Print ISBN: 978-0-8218-4242-3

Product Code: SURV/134

List Price: $112.00

Individual Member Price: $89.60

**Electronic ISBN: 978-1-4704-1361-3
Product Code: SURV/134.E**

List Price: $112.00

Individual Member Price: $89.60

#### Supplemental Materials

# Crossed Products of $C^{*}$-Algebras

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*Dana P. Williams*

The theory of crossed products is extremely rich and intriguing. There
are applications not only to operator algebras, but to subjects as
varied as noncommutative geometry and mathematical physics. This book
provides a detailed introduction to this vast subject suitable for
graduate students and others whose research has contact with crossed
product \(C^*\)-algebras. In addition to providing the basic
definitions and results, the main focus of this book is the fine ideal
structure of crossed products as revealed by the study of induced
representations via the Green-Mackey-Rieffel machine. In particular,
there is an in-depth analysis of the imprimitivity theorems on which
Rieffel's theory of induced representations and Morita equivalence of
\(C^*\)-algebras are based. There is also a detailed treatment
of the generalized Effros-Hahn conjecture and its proof due to
Gootman, Rosenberg, and Sauvageot.

This book is meant to be self-contained and accessible to any
graduate student coming out of a first course on operator
algebras. There are appendices that deal with ancillary subjects,
which while not central to the subject, are nevertheless crucial for a
complete understanding of the material. Some of the appendices will be
of independent interest.

To view another book by this author, please visit Morita Equivalence and
Continuous-Trace \(C^*\)-Algebras.

#### Table of Contents

# Table of Contents

## Crossed Products of $C^{*}$-Algebras

- Contents vii8 free
- Introduction xi12 free
- 1 Locally Compact Groups 118 free
- 2 Dynamical Systems and Crossed Products 4158
- 3 Special Cases and Basic Constructions 8198
- 4 Imprimitivity Theorems 109126
- 5 Induced Representations and Induced Ideals 151168
- 6 Orbits and Quasi-orbits 171188
- 7 Properties of Crossed Products 189206
- 8 Ideal Structure 227244
- 9 The Proof of the GRS-Theorem 261278
- Appendices 311328
- A: Amenable Groups 311328
- B: The Banach *-Algebra L[sup(1)](G,A) 331348
- C: Bundles of C*-algebras 353370
- D: Groups 369386
- E: Representations of C*-algebras 399416
- F: Direct Integrals 409426
- G: Effros's Ideal Center Decomposition 431448
- H: The Fell Topology 453470
- I: Miscellany 479496
- I.1 The Internal Tensor Product 479496
- I.2 Disintegration of Measures 481498
- I.3 Abelian Von Neumann Algebras 488505
- I.4 The Hilbert Space L[sup(2)](G,H) 490507
- I.5 The Dual of L[sup(1)](G) 495512
- I.6 Point Maps 498515
- I.7 Representations of C[sub(0)](X) 503520
- I.8 Invariants of Morita Equivalent C*-Algebras 507524

- Notation and Symbol Index 509526
- Index 513530
- Bibliography 521538

#### Readership

Graduate students and research mathematicians interested in \(C^*\)-algebras.

#### Reviews

The aim of this nicely written and self-contained book is to provide the tools necessary to begin doing research involving crossed product \(C^*\)-algebras.

-- Zentralblatt MATH

This is the first book devoted exclusively to the theory of \(C^*\)-crossed products, and will be especially useful to graduate students trying to learn something about them. ... A valuable extra feature of this book is the extensive set of appendices (about 200 pages worth!), covering a number of topics of independent interest, such as amenability, Borel structures, direct integrals, and the Fell topology on representations and ideals.

-- Jonathan M. Rosenberg for Mathematical Reviews