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.
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.
