A dichotomy for groupoid $\text{C}^{* }$-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 $\text{C}_{r}^{* }(G)$ 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 $\text{C}_{r}^{* }(G)$ . A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph $\text{C}^{* }$ -algebras as well.