Composition of topological correspondences

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.