Crossed Products of C-algebras

Erscheinungsjahr 2007
ISBN: 978-0-8218-4242-3
Verlag: American Mathematical Society

Buch, Englisch, Band 134, 528 Seiten, Gewicht: 1106 g

Reihe: Mathematical Surveys and Monographs

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.

Zielgruppe

Graduate students and research mathematicians interested in $C^*$-algebras.

Autoren/Hrsg.

Williams, Dana P.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Elementare Algebra
• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis
• Mathematik | Informatik Mathematik Topologie Algebraische Topologie

Inhaltsverzeichnis

Introduction xi
1 Locally Compact Groups 1
1.1 Preliminaries on Topological Groups. 2
1.2 Preliminaries on Locally Compact Groups. 5
1.3 HaarMeasure. 16
1.4 An Interlude: Abelian Harmonic Analysis. 22
1.5 Integration on Groups. 28
2 Dynamical Systems and Crossed Products 41
2.1 Dynamical Systems. 41
2.2 Covariant Representations. 44
2.3 The Crossed Product. 47
2.4 Representations of the Crossed Product. 54
2.5 Comments on Examples. 64
2.6 Universal Property. 72
3 Special Cases and Basic Constructions 81
3.1 Another Interlude: The C*-Algebra of an Abelian group. 81
3.2 Third Interlude: The C*-Algebra of a Compact group. 84
3.3 Semidirect Products. 86
3.4 Invariant Ideals. 92
3.5 The Orbit Space. 94
3.6 Proper Actions and Induced Algebras. 99
4 Imprimitivity Theorems 109
4.1 The Symmetric Imprimitivity Theorem. 110
4.2 Some Special Cases. 125
4.3 Green’s Imprimitivity Theorem. 130
4.4 The Stone-von Neumann Theorem. 133
4.5 Transitive Transformation Groups. 137
5 Induced Representations and Induced Ideals 151
5.1 Induced Representations of Crossed Products. 151
5.2 An Example. 161
5.3 Inducing Ideals. 164
5.4 Invariant Ideals and the Induction Process. 166
6 Orbits and Quasi-orbits 171
6.1 TheMackey-Glimm Dichotomy. 171
6.2 The ResMap and Quasi-Orbits. 179
7 Properties of Crossed Products 189
7.1 The Takai Duality Theorem. 189
7.2 The Reduced Crossed Product. 197
7.3 Crossed Products Involving the Compacts. 203
7.4 Aside on Twisted Crossed Products. 208
7.5 The Type Structure of Regular Crossed Products. 221
8 Ideal Structure 227
8.1 Fibering Over the Orbit Space. 227
8.2 EH-Regularity and the GRS-Theorem. 239
8.3 The Ideal Structure of C0(X) lt G. 242
8.4 The Fell Subgroup Crossed Product. 256
9 The Proof of the GRS-Theorem 261
9.1 Step I: Induced Primitive Ideals. 268
9.2 Step II: Restricting to the Stability Groups. 271
9.3 Step III: Sauvageot’s Induced Representation. 283
9.4 Step IV: G Amenable. 292
9.5 Step V: Gootman-Rosenberg a la Renault. 298
Appendices
A Amenable Groups 311
A.1 States and Positive Definite Functions. 311
A.2 Amenability. 318
A.3 Another GNS Construction. 327
B The Ban

Über Autor(innen)

Dana P. Williams, Dartmouth College, Hanover, NH