A dichotomy for groupoid $\text{C}^{\ast }$-algebras. (February 2020)

• Record Type: Journal Article
• Authors: RAINONE, TIMOTHY; SIMS, AIDAN
• Is Part Of: Ergodic theory and dynamical systems. Volume 40:Number 2(2020)
• Page Start: 521
• Page End: 563
• Publication Date: 2020-02
• Subjects: 46L05; Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2018.52
• Languages: English
• ISSNs: 0143-3857

Abstract

Abstract : 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.