Topological construction of $C^{\ast }$-correspondences for groupoid $C^{\ast }$-algebras

Abstract

Let $(G,\alpha )$ and $(H,\beta )$ be locally compact groupoids with Haar systems. We define a topological correspondence from $(G,\alpha )$ to $(H,\beta )$ to be a $G$-$H$-bispace $X$ on which $H$ acts properly and $X$ carries a continuous family of measures which is $H$-invariant and each measure in the family is $G$-quasi invariant. We show that a topological correspondence produces a $C^{\ast }$-correspondence from $C^{\ast }(G,\alpha )$ to $C^{\ast }(H,\beta )$. We give many examples of topological correspondences.

This version has 24 pages; the case of groupoid equivalence is removed from Section 2.1. A modified version of this article is accepted for publication in Journal of Operator Theory

Cited by in corpus (1)

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