# Morita Equivalence and Continuous-Trace \(C^*\)-Algebras

Share this page
*Iain Raeburn; Dana P. Williams*

In this text, the authors 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 introduction to recent developments in the
area.

The book is accessible to students who are beginning research in
operator algebras after a standard one-term course in
\(C^*\)-algebras. The authors have included introductions to
necessary but nonstandard background. Thus they have developed the
general theory of Morita equivalence from the Hilbert module,
discussed the spectrum and primitive ideal space of a
\(C^*\)-algebra including many examples, and presented the
necessary facts on tensor products of \(C^*\)-algebras starting
from scratch. Motivational material and comments designed to place the
theory in a more general context are included.

The text is self-contained and would be suitable for an advanced
graduate or an independent study course.

#### Reviews & Endorsements

The exposition is stimulating and well written, and should be regarded as essential reading for any research student in \(C^*\)-algebras. Indeed, the book has a strong claim to be on the shelves of anybody, student or veteran, working in the subject.

-- Bulletin of the London Mathematical Society

A beautiful book …The book provides a very nice introduction to some recent research work of the authors (and others) on the interplay between the theory of group actions on continuous-trace \(C^*\)-algebras and algebraic topology.

-- Zentralblatt MATH

An extremely useful reader's guide is provided in the introduction that summarizes in a clear and precise way what is in each of the seven chapters and four appendices. The authors have indeed made a serious effort to make this volume self-contained and to keep the required background to a minimum. The writing is clear and details are provided throughout. Although this is a volume that will demand the reader's attention to master the material, there is an air of informality to the writing that makes the reading pleasant and enjoyable. The authors are to be commended for writing a beautiful book that will open the way to researchers wishing to learn about a fascinating area of mathematics.

-- Mathematical Reviews Featured Review

#### Table of Contents

# Table of Contents

## Morita Equivalence and Continuous-Trace $C^{*}$-Algebras

- Contents vii8 free
- Introduction ix10 free
- 1 The Algebra of Compact Operators 116 free
- 2 Hilbert C*-Modules 722
- 3 Morita Equivalence 4156
- 4 Sheaves, Cohomology, and Bundles 6782
- 5 Continuous-Trace C*-Algebras 115130
- 6 Applications 155170
- 7 Epilogue: The Brauer Group and Group Actions 173188
- 7.1 Dynamical Systems and Crossed Products 174189
- 7.2 The Equivariant Brauer Group 177192
- 7.3 The Brauer Group of a Point 181196
- 7.4 Group Cohomology and Moore Cohomology 184199
- 7.5 The Brauer Group for Trivial Actions 192207
- 7.6 The Brauer Group for Free and Proper Actions 193208
- 7.7 The Structure of the Brauer Group 196211

- A: The Spectrum 201216
- B: Tensor Products of C*-Algebras 235250
- C: The Imprimitivity Theorem 269284
- D: Miscellany 303318
- Index 309324
- Bibliography 317332