Journal of Operator Theory

Volume 43, Issue 2, Spring 2000 pp. 295-327.

Pointwise unitary coactions on $C^*$-algebras with continuous trace

Authors: Klaus Deicke
Author institution: Universitaet-Gesamthochschule Paderborn, Fachbereich Mathematik-Informatik, Warburger Strasse 100, D-33095 Paderborn, Germany

Summary: Let $G$ be a second countable locally compact group, $A$ a separable continuous trace $C^*$-algebra and $\delta$ a pointwise unitary coaction of $G$ on $A$. It is shown that the crossed product $A\times_\delta G$ of $(A,G,\delta)$ has continuous trace and that the restriction map $\opn{Res}:(A\times_\delta G)^{\wedge}\to\widehat{A}$ is a proper $G$-bundle via the dual action of $G$ on $(A\times_\delta G)^{\wedge}$. Further, $A\times_\delta G$ is isomorphic to the pull-back $\opn{Res}^*A$. We obtain a characterization of continuous trace crossed products $A\times_{\alpha,{\rm r}} G$ by an action $\alpha$ of $G$ on $A$: when $\alpha$ acts freely on $\widehat{A}$, the crossed product has continuous trace if and only if the actio n of $G$ on $\widehat{A}$ is proper.

Keywords: Continuous trace $C^*$-algebra, coaction, crossed product, pointwise unitary

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