Tracial approximate divisibility and stable rank one

TL;DR

This paper proves that separable simple tracially approximately divisible C*-algebras have strict comparison and stable rank one, with implications for finite simple Z-stable C*-algebras, advancing the understanding of their structure.

Contribution

It establishes that such algebras are either purely infinite or have stable rank one, and shows finite simple Z-stable C*-algebras have stable rank one, extending classification results.

Findings

Separable simple tracially approximately divisible C*-algebras have strict comparison.

Such algebras are either purely infinite or have stable rank one.

Finite simple Z-stable C*-algebras have stable rank one.

Abstract

We show that every separable simple tracially approximately divisible $C^*$-algebra has strict comparison, is either purely infinite, or has stable rank one. As a consequence, we show that every (non-unital) finite simple ${\cal Z}$-stable $C^*$-algebra has stable rank one.