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Crossed Products with Continuous Trace

Available Formats:
Electronic ISBN: 978-1-4704-0171-9
Product Code: MEMO/123/586.E
List Price: $48.00 MAA Member Price:$43.20
AMS Member Price: $28.80 Click above image for expanded view Crossed Products with Continuous Trace Siegfried Echterhoff University of Paderborn, Paderborn, Germany Available Formats:  Electronic ISBN: 978-1-4704-0171-9 Product Code: MEMO/123/586.E  List Price:$48.00 MAA Member Price: $43.20 AMS Member Price:$28.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 1231996; 134 pp
MSC: 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, continuous-trace $C^*$-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 two-step nilpotent groups and solvable Lie groups.

For readers' convenience, expositions of the Mackey-Green-Rieffel 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
• Requests

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Volume: 1231996; 134 pp
MSC: 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, continuous-trace $C^*$-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 two-step nilpotent groups and solvable Lie groups.

For readers' convenience, expositions of the Mackey-Green-Rieffel 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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