A dichotomy for groupoid C*-algebras

TL;DR

This paper establishes a dichotomy for ample groupoid C*-algebras, linking their finite or infinite nature to K-theoretic paradoxicality and introducing a new monoid S(G) that generalizes existing concepts.

Contribution

It introduces the monoid S(G) for ample groupoids, connecting algebraic properties to the finite/infinite classification of their C*-algebras.

Findings

S(G) generalizes the type semigroup for transformation groups.

A dichotomy between stable finiteness and pure infiniteness is proven.

Conditions for the dichotomy include minimality, topological principality, and almost unperforation of S(G).

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.