Pureness of Certain Crossed Product C*-Algebras

TL;DR

This paper proves that certain crossed product C*-algebras are pure, have stable rank one, and sometimes real rank zero, by establishing comparison and divisibility properties under various group actions.

Contribution

It introduces new methods to establish purity and rank properties of crossed product C*-algebras beyond previous results, especially in complex cases.

Findings

Crossed products are shown to be pure under various automorphism and group action conditions.

Stable rank one is established for these crossed products.

In some cases, real rank zero is also achieved.

Abstract

We establish comparison and divisibility properties for crossed product C*-algebras arising from automorphisms of algebras C (X, D) which lie over minimal homeomorphisms, from actions of compact groups which have finite Rokhlin dimension with commuting towers, and from actions of compact groups which have the restricted tracial Rokhlin property with comparison. We deduce that these crossed products we consider are pure, and conclude they have stable rank one, and in certain cases have real rank zero. We give examples in which these properties do not follow from previous results, in the case of C (X, D) due to the lack of Z-stability of D, the underlying topological spaces not being finite dimensional, or both.