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An equivariant Brauer semigroup and the symmetric imprimitivity theorem

Authors:	Astrid an Huef Iain Raeburn Dana P Williams

Institution:	Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551 ; Department of Mathematics, University of Newcastle, Callaghan, New South Wales 2308, Australia ; Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551

Abstract:	Suppose that 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 is a $C_{0}(X)$ -algebra and $\alpha$ is compatible with the given -action on . 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 on $C_{0}(X)$ -algebras. If and act freely and properly on the left and right of a space , 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 $B\rtimes_{\beta}H$ .

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