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An equivariant Brauer semigroup and the symmetric imprimitivity theorem. (English) Zbl 1001.46044
Summary: Suppose that $$(X,G)$$ is a second countable locally compact transformation group. We let $$\operatorname{S}_G(X)$$ denote the set of Morita equivalence classes of separable dynamical systems $$(A,G,\alpha)$$ where $$A$$ is a $$C_{0}(X)$$-algebra and $$\alpha$$ is compatible with the given $$G$$-action on $$X$$. We prove that $$\operatorname{S}_{G}(X)$$ is a commutative semigroup with identity with respect to the binary operation $$[A,G,\alpha][B,G,\beta]=[A\otimes_{X}B,G,\alpha\otimes_{X}\beta]$$ for an appropriately defined balanced tensor product of $$C_{0}(X)$$-algebras. If $$G$$and $$H$$ act freely and properly on the left and right of a space $$X$$, then we prove that $$\operatorname{S}_{G}(X/H)$$ and $$\operatorname{S}_{H}(G\backslash X)$$ are isomorphic as semigroups. If the isomorphism maps the class of $$(A,G,\alpha)$$ to the class of $$(B,H,\beta)$$, then $$A\rtimes_{\alpha}G$$ is Morita equivalent to $$B\rtimes_{\beta}H$$.

##### MSC:
 46L55 Noncommutative dynamical systems 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 22D30 Induced representations for locally compact groups
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