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

2000 Mathematics Subject Classification: 22A22 (Primary); 46L05, 46L80, 54D35 (Secondary).

Keywords and Phrases: groupoid, $C^*$-algebra, $K$-theory.

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