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Crossed Products with Continuous Trace
Siegfried Echterhoff, University of Paderborn, Germany

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
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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, 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.


  • 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.

Table of Contents

  • Abstract
  • Introduction
  • Preliminaries and basic definitions
  • Morita equivalent twisted actions and duality
  • Representations of type I abelian twisted systems
  • Subgroup crossed products
  • Crossed products with continuous trace
  • Applications and examples
  • Some concluding remarks
  • References
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