Non-Hausdorff groupoids, proper actions and $K$-theory

Abstract

Let $G$ be a (not necessarily Hausdorff) locally compact groupoid. We introduce a notion of properness for $G$, which is invariant under Morita-equivalence. We show that any generalized morphism between two locally compact groupoids which satisfies some properness conditions induces a $C^{\ast }$-correspondence from $C_r^{\ast }(G_2)$ to $C_r^{\ast }(G_1)$, and thus two Morita equivalent groupoids have Morita-equivalent $C^{\ast }$-algebras.