Purely infinite locally compact Hausdorff étale groupoids and their $C^{\ast }$-algebras

Abstract

In this paper, we introduce properties including groupoid comparison, pure infiniteness and paradoxical comparison as well as a new algebraic tool called groupoid semigroup for locally compact Hausdorff étale groupoids. We show these new tools help establishing pure infiniteness of reduced groupoid $C^{\ast }$-algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid $C^{\ast }$-algebras arising from locally compact Hausdorff étale minimal topological principal groupoids. This generalizes the dichotomy obtained by Bönicke-Li and Rainone-Sims. We also study the relation among our paradoxical comparison, $n$-filling property and locally contracting property appeared in the literature for locally compact Hausdorff étale groupoids.

paper shortened based on reports of referees. To appear in IMRN

References in corpus (2)

• The type semigroup, comparison and almost finiteness for ample groupoids
• Graph-Based Models for Kirchberg Algebras

Cited by in corpus (2)

• Almost elementary étale groupoids
• Comparison and Simplicity of Commutator Subgroups of Full Groups