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 $C^{\ast }$-correspondence $\mathcal{H}(X):C^{\ast }(G,\alpha )\to C^{\ast }(H,\beta )$. We give sufficient topological conditions which when satisfied the $C^{\ast }$-correspondence $\mathcal{H}(X)$ is proper, that is, the $C^{\ast }$-algebra $C^{\ast }(G,\alpha )$ acts on the Hilbert $C^{\ast }(H,\beta )$-module $H(X)$ via the comapct operators. Thus a proper topological correspondence produces an element in $KK(C^{\ast }(G,\alpha ),C^{\ast }(H,\beta ))$.

43 pages, 3 figures. Comments are welcome