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Crossed Products with Continuous Trace
Siegfried Echterhoff, University of Paderborn, Germany
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Memoirs of the American Mathematical Society
1996; 134 pp; softcover
Volume: 123
ISBN-10: 0-8218-0563-0
ISBN-13: 978-0-8218-0563-3
List Price: US$48 Individual Members: US$28.80
Institutional Members: US\$38.40
Order Code: MEMO/123/586

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.