A dichotomy for groupoid -algebras

We study the finite versus infinite nature of C$^{\ast }$-algebras arising from étale groupoids. For an ample groupoid$G$, we relate infiniteness of the reduced C$^{\ast }$-algebra$\text{C}_{r}^{\ast }(G)$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$^{\ast }$-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$\text{C}_{r}^{\ast }(G)$. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph$\text{C}^{\ast }$-algebras as well.