A dichotomy for groupoid C^* -algebras. (February 2020) - RAINONE, TIMOTHY; SIMS, AIDAN
Abstract : We study the finite versus infinite nature of C ^* -algebras arising from étale groupoids. For an ample groupoid G, we relate infiniteness of the reduced C ^* -algebra C_r^* (G) to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid S(G) which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature of the reduced groupoid C ^* -algebra of G in the sense that if G is ample, minimal, topologically principal, and S(G) is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for C_r^* (G) . A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph C^* -algebras as well.