Stable rank one in nonnuclear crossed products

TL;DR

This paper studies the local structure of certain nonnuclear C*-crossed products, demonstrating that stable rank one is common in specific minimal group actions on the Cantor set, with broader implications for free product groups.

Contribution

It introduces the first results showing stable rank one is generic in nonnuclear crossed products for minimal free group actions, extending previous amenable case results.

Findings

Stable rank one is generic in minimal free group actions on the Cantor set.

The approach generalizes to some other free product groups.

Provides a streamlined proof and extends stable rank one results to product actions.

Abstract

We initiate an investigation into the local structure of simple nonnuclear C$^*$-crossed products by showing that stable rank one is generic within two natural classes of minimal actions of free groups on the Cantor set. The arguments also apply to some other free product groups. Our approach is inspired by Li and Niu's stable rank one theorem in the amenable setting and also yields a streamlined argument in that case, along with a generalization to product actions.