Comparison and pure infiniteness of crossed products

Abstract

Let $\alpha: G\curvearrowright X$ be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action $\alpha$ is topologically free and has no $G$-invariant regular Borel probability measure on $X$, dynamical comparison implies that the reduced crossed product of $\alpha$ is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action $\alpha$ is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of $\alpha$ is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero-dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds a light to a question arising in the paper of R{\o}rdam and Sierakowski.