Electronic ISBN:  9781470401719 
Product Code:  MEMO/123/586.E 
List Price:  $48.00 
MAA Member Price:  $43.20 
AMS Member Price:  $28.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 123; 1996; 134 ppMSC: Primary 46; Secondary 22;
The importance of separable continuous trace \(C^*\)algebras arises from the following facts: Firstly, their stable isomorphism classes are completely classifiable by topological data and, secondly, continuoustrace \(C^*\)algebras form the building blocks of the more general type I \(C^*\)algebras. This memoir presents an extensive study of strongly continuous actions of abelian locally compact groups on \(C^*\)algebras with continuous trace. Under some natural assumptions on the underlying system \((A,G,\alpha )\), necessary and sufficient conditions are given for the crossed product \(A{\times }_{\alpha }G\) to have continuous trace, and some relations between the topological data of \(A\) and \(A{\times }_{\alpha }G\) are obtained. The results are applied to investigate the structure of group \(C^*\)algebras of some twostep nilpotent groups and solvable Lie groups.
For readers' convenience, expositions of the MackeyGreenRieffel machine of induced representations and the theory of Morita equivalent \(C^*\)dynamical systems are included. There is also an extensive elaboration of the representation theory of crossed products by actions of abelian groups on type I \(C^*\)algebras, resulting in a new description of actions leading to type I crossed products.
Features: The most recent results on the theory of crossed products with continuous trace.
 Applications to the representation theory of locally compact groups and structure of group \(C^*\)algebras.
 An exposition on the modern theory of induced representations.
 New results on type I crossed products.
ReadershipGraduate students and research mathematicians working in operator algebras or representation theory of locally compact groups. Theoretical physicists interested in operator algebras and \(C^*\)dynamical systems.

Table of Contents

Chapters

Introduction

1. Preliminaries and basic definitions

2. Morita equivalent twisted actions and duality

3. Representations of type I abelian twisted systems

4. Subgroup crossed products

5. Crossed products with continuous trace

6. Applications and examples


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The importance of separable continuous trace \(C^*\)algebras arises from the following facts: Firstly, their stable isomorphism classes are completely classifiable by topological data and, secondly, continuoustrace \(C^*\)algebras form the building blocks of the more general type I \(C^*\)algebras. This memoir presents an extensive study of strongly continuous actions of abelian locally compact groups on \(C^*\)algebras with continuous trace. Under some natural assumptions on the underlying system \((A,G,\alpha )\), necessary and sufficient conditions are given for the crossed product \(A{\times }_{\alpha }G\) to have continuous trace, and some relations between the topological data of \(A\) and \(A{\times }_{\alpha }G\) are obtained. The results are applied to investigate the structure of group \(C^*\)algebras of some twostep nilpotent groups and solvable Lie groups.
For readers' convenience, expositions of the MackeyGreenRieffel machine of induced representations and the theory of Morita equivalent \(C^*\)dynamical systems are included. There is also an extensive elaboration of the representation theory of crossed products by actions of abelian groups on type I \(C^*\)algebras, resulting in a new description of actions leading to type I crossed products.
Features:
 The most recent results on the theory of crossed products with continuous trace.
 Applications to the representation theory of locally compact groups and structure of group \(C^*\)algebras.
 An exposition on the modern theory of induced representations.
 New results on type I crossed products.
Graduate students and research mathematicians working in operator algebras or representation theory of locally compact groups. Theoretical physicists interested in operator algebras and \(C^*\)dynamical systems.

Chapters

Introduction

1. Preliminaries and basic definitions

2. Morita equivalent twisted actions and duality

3. Representations of type I abelian twisted systems

4. Subgroup crossed products

5. Crossed products with continuous trace

6. Applications and examples