Composition of topological correspondences

TL;DR

This paper explores how to compose topological correspondences between locally compact groupoids with Haar systems, focusing on the case where bispaces are Hausdorff and second countable, to form new $C^*$-correspondences.

Contribution

It extends previous work by detailing the composition process of topological correspondences under additional topological conditions.

Findings

Provides a method to compose topological correspondences.

Shows the composition results in a new $C^*$-correspondence.

Clarifies conditions under which composition is well-defined.

Abstract

In the previous article, we proved that a topological correspondence $(X,\lambda)$ from a locally compact groupoid with a Haar system $(G,\alpha)$ to another one, $(H,\beta)$, produces a $C^*$-correspondence $\mathcal{H}(X)$ from $C^*(G,\alpha)$ to $C^*(H,\beta)$. In the present article, we describe how to form a composite of two topological correspondences when the bispaces are Hausdorff and second countable in addition to being locally compact.