A dichotomy for groupoid $\text{C}^{\ast }$ -algebras

• Published in: Ergodic theory and dynamical systems 2020-02, Vol.40 (2), p.521-563
• Main Authors: RAINONE, TIMOTHY, SIMS, AIDAN
• Format: Article
• Language: English
• Subjects: Original Article
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2018.52

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.