Tracial oscillation zero and stable rank one

Abstract

Let $A$ be a separable (not necessarily unital) simple $C^{\ast }$-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 $C^{\ast }$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma$ is surjective.

This paper will appear in Canadian Journal of Math

References in corpus (3)

• Completely positive maps of order zero
• Nuclear dimension, Z-stability, and algebraic simplicity for stably projectionless C*-algebras
• The C*-algebra of a minimal homeomorphism of zero mean dimension