Stable rank one, tracial local homogeneity and uniform property $\Gamma$

TL;DR

This paper proves that certain stably finite C*-algebras with stable rank one and tracial local homogeneity possess uniform property Γ, confirming the Toms-Winter conjecture for these classes.

Contribution

It establishes uniform property Γ for a broad class of C*-algebras with stable rank one and tracial local homogeneity, providing a new proof of the Toms-Winter conjecture in this context.

Findings

Villadsen algebras of the first type have uniform property Γ

Crossed products of free minimal actions of FC groups have uniform property Γ

These results confirm the Toms-Winter conjecture for the considered classes

Abstract

We prove that separable, simple, unital, non-elementary, stably finite C*-algebras that have stable rank one, and that have locally finite nuclear dimension in a tracial sense, have uniform property $\Gamma$. In particular, Villadsen algebras of the first type and crossed products of free minimal actions of FC (in particular, abelian) groups on compact metric spaces have uniform property $\Gamma$. This implies that all these C*-algebras satisfy the Toms-Winter conjecture, a fact already known for C*-algebras with stable rank one and locally finite nuclear dimension, and here recovered via a different approach.