Non-Hausdorff groupoids, proper actions and K-theory

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*-correspondence from $C{\ast }_r(G_1)$ to $C_r^{\ast }(G_2)$, and thus two Morita equivalent groupoids have Morita-equivalent $C^{\ast }$-algebras.