Purely infinite C*-algebras arising from crossed products

TL;DR

This paper establishes conditions under which crossed products of C*-algebras by discrete exact groups are purely infinite, with applications to actions on the Cantor set resulting in Kirchberg algebras.

Contribution

It provides new criteria for pure infiniteness in crossed products involving non-amenable groups and commutative C*-algebras, linking paradoxicality to algebraic properties.

Findings

Conditions for pure infiniteness in crossed products by exact groups.

Existence of free amenable minimal actions on the Cantor set.

Crossed products can be Kirchberg algebras in the UCT class.

Abstract

We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.