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On a duality for crossed products of \(C^*\)-algebras. (English) Zbl 0295.46088

Summary: Let \(\mathfrak A\) be a \(C^*\)-algebra, and \(G\) be a locally compact abelian group. Suppose \(\alpha\) is a continuous action of \(G\) on \(\mathfrak A\). Then there exists a continuous action \(\hat{\alpha}\) of the dual group of \(\hat G\) on the \(C^*\)-crossed product \(C^*(\mathfrak A; \alpha)\) of \(\mathfrak A\) by \(\alpha\) such that the \(C^*\)-crossed product \(C^*(C^*(\mathfrak A; \alpha); \hat\alpha)\) is isomorphic to the tensor product \(\mathfrak A\hat\otimes_* \mathfrak C(L^2(G))\) of \(\mathfrak A\) and the \(C^*\)-algebra \(\mathfrak C(L^2(G)\) of all compact operators on \(L^2(G)\).

MSC:

46L05 General theory of \(C^*\)-algebras
46M05 Tensor products in functional analysis
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
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References:

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