Purely infinite partial crossed products

TL;DR

This paper establishes a correspondence between invariant ideals and ideals in partial crossed products for certain dynamical systems, and characterizes pure infiniteness in terms of elements in the algebra, with applications to well-known Cuntz algebras.

Contribution

It provides a new characterization of pure infiniteness for partial crossed products under specific conditions, extending understanding of their ideal structure and applications.

Findings

Bijective correspondence between G-invariant ideals of A and ideals in A xr G

Pure infiniteness characterized by properly infinite positive elements

Applications to Cuntz algebras as partial crossed products

Abstract

Let (A,G,\alpha) be a partial dynamical system. We show that there is a bijective correspondence between G-invariant ideals of A and ideals in the partial crossed product A xr G provided the action is exact and residually topologically free. Assuming, in addition, a technical condition---automatic when A is abelian---we show that A xr G is purely infinite if and only if the positive nonzero elements in A are properly infinite in A xr G. As an application we verify pure infiniteness of various partial crossed products, including realisations of the Cuntz algebras O_n, O_A, O_N, and O_Z as partial crossed products.