Topological construction of $C^*$-correspondences for groupoid $C^*$-algebras

Abstract

Let G and H be locally compact, Hausdor groupoids with Haar systems. We de fine a topological correspondence from G to H to be a G-H bispace X carrying a G-quasi invariant and H-invariant family of measures. We show that such a correspondence gives a C*-correspondence from C *(G) to C* (H). If the groupoids and the spaces are second countable, then this construction is functorial. We show that under a certain amenability assumption, similar results hold for the reduced C *-algebras. We apply this theory of correspondences to study induction techniques for groupoid representations, construct morphisms of Brauer groups and produce some odd unbounded KK-cycles.