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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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