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

### Summary

Summary: 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 $C^*_r(G_2)$ to $C^*_r(G_1)$, and thus two Morita equivalent groupoids have Morita-equivalent $C^*$-algebras.

### Mathematics Subject Classification

22A22, 46L05, 46L80, 54D35

### Keywords/Phrases

groupoid, $C^*$-algebra, $K$-theory