Crossed Products With Continuous Trace (Memoirs of the American Mathematical Society) by Siegfried Echterhoff (PDF)

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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. 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.

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