An equivariant Brauer semigroup and the symmetric imprimitivity theorem

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 $\operatorname{S}_{H}(G\backslash X)$ 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 $B\rtimes_{\beta}H$.