CORRESPONDENCES, HOMOLOGY AND K-THEORY FOR ÉTALE GROUPOIDS

One approach to understanding the K-theory of an étale groupoid C*-algebra is to approximate it with the groupoid homology. I will describe the functoriality of the K-theory, the homology and the approximation with respect to a class of groupoid morphisms called étale correspondences. These are a broad class of morphisms encompassing both Morita equivalences and groupoid homomorphisms that are local homeomorphisms, and I will discuss more examples. The main result of the talk states that, under some reasonably tame conditions, an étale correspondence that induces an isomorphism in homology will also induce an isomorphism in K-theory. This can be used to compute the K-theory of some Toeplitz-like C*-algebras.