Locally free actions of groupoids and proper topological correspondences

Abstract

Let $(G,\alpha)$ and $(H,\beta)$ be locally compact Hausdorff groupoids with Haar systems, and let $(X,\lambda)$ be a topological correspondence from $(G,\alpha)$ to $(H,\beta)$ which induce the ${C}^*$-correspondence $\mathcal{H}(X)\colon {C}^*(G,\alpha)\to {C}^*(H,\beta)$. We give sufficient topological conditions which when satisfied the ${C}^*$-correspondence $\mathcal{H}(X)$ is proper, that is, the ${C}^*$-algebra ${C}^*(G,\alpha)$ acts on the Hilbert ${C}^*(H,\beta)$-module ${H}(X)$ via the comapct operators. Thus a proper topological correspondence produces an element in ${KK}({C}^*(G,\alpha),{C}^*(H,\beta))$.