Tracial oscillation zero and stable rank one

TL;DR

This paper proves that for certain simple separable C*-algebras with strict comparison, having tracial approximate oscillation zero is equivalent to having stable rank one, linking algebraic and functional properties.

Contribution

It establishes the equivalence between tracial approximate oscillation zero and stable rank one in simple C*-algebras with strict comparison, and explores implications for the canonical map mma.

Findings

Tracial approximate oscillation zero implies stable rank one.

The canonical map mma is surjective under these conditions.

Almost stable rank one implies stable rank one with strict comparison.

Abstract

Let $A$ be a separable (not necessarily unital) simple $C^*$-algebra with strict comparison. We show that if $A$ has tracial approximate oscillation zero then $A$ has stable rank one and the canonical map $\Gamma$ from the Cuntz semigroup of $A$ to the corresponding affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma$ is surjective.