**Memoirs of the American Mathematical Society**

1996;
134 pp;
Softcover

MSC: Primary 46;
Secondary 22

Print ISBN: 978-0-8218-0563-3

Product Code: MEMO/123/586

List Price: $48.00

Individual Member Price: $28.80

**Electronic ISBN: 978-1-4704-0171-9
Product Code: MEMO/123/586.E**

List Price: $48.00

Individual Member Price: $28.80

# Crossed Products with Continuous Trace

Share this page
*Siegfried Echterhoff*

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.

#### Table of Contents

# Table of Contents

## Crossed Products with Continuous Trace

- Contents v6 free
- Abstract vii8 free
- Introduction 110 free
- Chapter 1. Preliminaries and basic definitions 1120 free
- Chapter 2. Morita equivalent twisted actions and duality 2938
- Chapter 3. Representations of type I abelian twisted systems 3746
- Chapter 4. Subgroup crossed products 5564
- Chapter 5. Crossed products with continuous trace 7786
- 5.1. Subgroup crossed products and σ-proper G-spaces 7786
- 5.2. Pointwise unitary subgroup actions 8291
- 5.3. Systems with continuous choices of maximally pointwise unitary subgroups 91100
- 5.4. Continuous trace for systems with Hausdorff quasi-orbit space 95104
- 5.5. Systems with continuously varying stabilizers 100109

- Chapter 6. Applications and Examples 115124
- Some concluding remarks 129138
- References 131140

#### Readership

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.