Purely infinite locally compact Hausdorff \'etale groupoids and their C*-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 \'etale 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 \'etale minimal topological principal groupoids. This generalizes the dichotomy obtained by B\"onicke-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 \'etale groupoids.