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*-correspondence from Cr*(G2) to C r*(G1), and thus two Morita equivalent groupoids have Morita-equivalent C*-algebras.