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

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.