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.

arXiv commentpaper shortened based on reports of referees. To appear in IMRN

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Cited by in corpus (2)

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