C*-ALGEBRAS DEFINED BY GROUPOID CORRESPONDENCES

In this talk, I define correspondences between étale groupoids, and show that they contain topological graphs and self-similarities of groups and graphs as special cases. A correspondence between two groupoids induces a C*-correspondence between the groupoid C*-algebras, which yields a Cuntz-Pimsner algebra. We also define the groupoid model, a groupoid naturally attached to the groupoid correspondence. Under some assumptions, the groupoid C*-algebra of the groupoid model is naturally isomorphic to the Cuntz-Pimsner algebra of the C*-correspondence of the groupoid correspondence. More generally, we may consider diagrams of groupoid correspondences. These contain higher-rank, self-similar graphs and discrete Conduché fibrations as special cases. A diagram of groupoid correspondences induces a diagram of C*-correspondences, which is a variant of a product system over a category. This also yields a Cuntz-Pimsner algebra. In addition, a diagram of groupoid correspondences has a groupoid model. In some cases, the C*-algebra of the groupoid model is naturally isomorphic to the Cuntz-Pimsner algebra of the product system of the diagram.