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arXiv · 1707.04516

A dichotomy for groupoid C*-algebras

Timothy Rainone·Aidan Sims
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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 R{\o}rdam 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.

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Timothy Rainone, Aidan Sims. 2017-07-14. A dichotomy for groupoid C*-algebras. https://arxiv.org/abs/1707.04516

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