• DOI:10.1017/etds.2018.52
• Corpus ID: 119151366

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

@article{Rainone2017ADF,
  title={A dichotomy for groupoid \$\text\{C\}^\{\ast \}\$ -algebras},
  author={Timothy Rainone and Aidan Sims},
  journal={Ergodic Theory and Dynamical Systems},
  year={2017},
  volume={40},
  pages={521 - 563}
}

• T. Rainone, A. Sims
• Published 14 July 2017
• Mathematics
• Ergodic Theory and Dynamical Systems

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.