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# 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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