Crossed products of C*-algebras / Dana P. Williams.

Format:

Book

Author/Creator:

Williams, Dana P., 1952- author.

Series:

Mathematical surveys and monographs ; v. 134.

Mathematical surveys and monographs, 0076-5376 ; volume 134

Language:

English

Subjects (All):

C*-algebras.

Operator algebras.

Physical Description:

1 online resource (546 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2007]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Introduction""; ""1 Locally Compact Groups""; ""1.1 Preliminaries on Topological Groups""; ""1.2 Preliminaries on Locally Compact Groups""; ""1.3 Haar Measure""; ""1.4 An Interlude: Abelian Harmonic Analysis""; ""1.5 Integration on Groups""; ""2 Dynamical Systems and Crossed Products""; ""2.1 Dynamical Systems""; ""2.2 Covariant Representations""; ""2.3 The Crossed Product""; ""2.4 Representations of the Crossed Product""; ""2.5 Comments on Examples""; ""2.6 Universal Property""; ""3 Special Cases and Basic Constructions""

""3.1 Another Interlude: The C*-Algebra of an Abelian group""""3.2 Third Interlude: The C*-Algebra of a Compact group""; ""3.3 Semidirect Products""; ""3.4 Invariant Ideals""; ""3.5 The Orbit Space""; ""3.6 Proper Actions and Induced Algebras""; ""4 Imprimitivity Theorems""; ""4.1 The Symmetric Imprimitivity Theorem""; ""4.2 Some Special Cases""; ""4.3 Green's Imprimitivity Theorem""; ""4.4 The Stone-von Neumann Theorem""; ""4.5 Transitive Transformation Groups""; ""5 Induced Representations and Induced Ideals""; ""5.1 Induced Representations of Crossed Products""; ""5.2 An Example""

""5.3 Inducing Ideals""""5.4 Invariant Ideals and the Induction Process""; ""6 Orbits and Quasi-orbits""; ""6.1 The Mackey-Glimm Dichotomy""; ""6.2 The Res Map and Quasi-Orbits""; ""7 Properties of Crossed Products""; ""7.1 The Takai Duality Theorem""; ""7.2 The Reduced Crossed Product""; ""7.3 Crossed Products Involving the Compacts""; ""7.4 Aside on Twisted Crossed Products""; ""7.5 The Type Structure of Regular Crossed Products""; ""8 Ideal Structure""; ""8.1 Fibering Over the Orbit Space""; ""8.2 EH-Regularity and the GRS-Theorem""

""8.3 The Ideal Structure of C[sub(0)](X) [omitted][sub(lt)] G""""8.4 The Fell Subgroup Crossed Product""; ""9 The Proof of the GRS-Theorem""; ""9.1 Step I: Induced Primitive Ideals""; ""9.2 Step II: Restricting to the Stability Groups""; ""9.3 Step III: Sauvageot's Induced Representation""; ""9.4 Step IV: G Amenable""; ""9.5 Step V: Gootman-Rosenberg a la Renault""; ""Appendices""; ""A: Amenable Groups""; ""A.1 States and Positive Definite Functions""; ""A.2 Amenability""; ""A.3 Another GNS Construction""; ""B: The Banach *-Algebra L[sup(1)](G,A)""; ""B.1 Vector-Valued Integration""

""B.2 Product Measures""""C: Bundles of C*-algebras""; ""C.1 C[sub(0)](X)-algebras""; ""C.2 Upper Semicontinuous C*-bundles""; ""D: Groups""; ""D.1 Group homomorphisms""; ""D.2 Borel Groups and Invariant Measures""; ""D.3 Projective Representations""; ""D.4 Quasi-invariant Measures""; ""D.5 Technical Results""; ""E: Representations of C*-algebras""; ""E.1 Multiplicity""; ""E.2 Decomposable Operators""; ""F: Direct Integrals""; ""F.1 Borel Hilbert Bundles""; ""F.2 The Direct Integral of Hilbert Spaces""; ""F.3 Decomposable Operators""; ""F.4 τ-isomorphisms""

""F.5 Uniqueness of Direct Integrals""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 521-528) and indexes.

Description based on print version record.

ISBN:

1-4704-1361-2