A dichotomy for groupoid C*-algebras
arXiv:1707.04516
Abstract
We study the finite versus infinite nature of C*-algebras arising from etale groupoids. For an ample groupoid G, we relate infiniteness of the reduced C*-algebra of 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 reflects the finite/infinite nature of the reduced groupoid C*-algebra of G. If G is ample, minimal, and topologically principal, and S(G) is almost unperforated we obtain a dichotomy between stable finiteness and pure infiniteness for the reduced C*-algebra of G.
40 pages. Version 2: Section 9.2 updated to reflect intersection with earlier results of Suzuki; thanks to Suzuki for alerting us. Proofs of Proposition 5.2 and Lemma 9.7 simplified using the refinement property (correcting an oversight in the proof of Proposition 5.2)