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

TL;DR

This paper introduces a topological framework for constructing $C^*$-correspondences between groupoid $C^*$-algebras using bispaces with specific measure and action properties, enriching the theory of groupoid $C^*$-algebras.

Contribution

It defines a new concept of topological correspondence for groupoids and demonstrates how it induces $C^*$-correspondences, with numerous examples illustrating its applicability.

Findings

Topological correspondences induce $C^*$-correspondences between groupoid $C^*$-algebras.

The framework generalizes previous constructions in groupoid $C^*$-algebra theory.

Many explicit examples of topological correspondences are provided.

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 $C^*$-correspondence from $C^*(G,\alpha)$ to $C^*(H,\beta)$. We give many examples of topological correspondences.