An equivariant Brauer semigroup and the symmetric imprimitivity theorem

an Huef, Astrid; Raeburn, Iain; Williams, Dana

doi:10.1090/S0002-9947-00-02618-0

An equivariant Brauer semigroup and the symmetric imprimitivity theorem
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by Astrid an Huef, Iain Raeburn and Dana P. Williams PDF

Trans. Amer. Math. Soc. 352 (2000), 4759-4787 Request permission

Abstract:

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 on $C_{0}(X)$-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 $\operatorname {S}_{H}(G\setminus X)$ 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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