Journal of Operator Theory

Volume 60, Issue 1, Summer 2008 pp. 85-112.

Duality for crossed products of Hilbert

$C^*$ -modules

Authors: Masaharu Kusuda
Author institution: Department of Mathematics, Faculty of Engineering Science, Kansai University, Yamate-cho 3-3-35, Suita, Osaka 564-8680, Japan

Summary: Let

$(A, G, \alpha)$ be a

$C^*$ -dynamical system and let

$X$ be an

$A$ -Hilbert module with an

$\alpha$ -compatible action

$\eta$ of

$G$ . Then it is shown that there exist a coaction

$\delta{\scriptscriptstyle _A}$ of

$G$ on the reduced crossed product

$A \times_{\alpha, r} G$ and a coaction

$\delta{\scriptscriptstyle _X}$ of

$G$ on the reduced crossed product

$X \times_{\eta, r} G$ such that

$(X \times_{\eta, r} G) \times_{\delta{\scriptscriptstyle _X}} G \cong X \otimes \mathcal{C}(L^2(G))$ , where

$\mathcal{ C}(L^2(G))$ denotes the

$C^*$ -algebra of all compact operators on

$L^2(G)$ . Furthermore, when

$A$ has a nondegenerate coaction

$\delta{\scriptscriptstyle _A}$ of

$G$ on

$A$ and

$X$ is an

$A$ -Hilbert module with a nondegenerate

$\delta{\scriptscriptstyle _A}$ -compatible coaction

$\delta{\scriptscriptstyle _X}$ of

$G$ , it is shown that there exists a dual action

$\widehat{\delta}{\scriptscriptstyle _X}$ of

$G$ on the crossed product

$X \times_{\delta{\scriptscriptstyle _X}} G$ such that

$(X \times_{\delta{\scriptscriptstyle _X}} G) \times_{\widehat{\delta}{\scriptscriptstyle _X}, r} G \cong X \otimes \mathcal{C}(L^2(G))$ .

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