Purely Infinite Locally Compact Hausdorff étale Groupoids and Their C*-algebras.

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 |$C^*$| -algebras. As an application, we show a dichotomy of stably finiteness against pure infiniteness for reduced groupoid |$C^*$| -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 that appeared in the literature for locally compact Hausdorff étale groupoids. [ABSTRACT FROM AUTHOR]