Strict comparison and stable rank one

TL;DR

This paper establishes that certain simple, finite $C^*$-algebras with strict comparison and surjective Cuntz semigroup maps have stable rank one and tracial approximate oscillation zero, linking algebraic and functional properties.

Contribution

It proves that surjectivity of the canonical map from the Cuntz semigroup implies stable rank one and oscillation zero in these algebras, extending understanding of their structure.

Findings

Algebras with surjective $ ext{Gamma}$ map have stable rank one.

Such algebras exhibit tracial approximate oscillation zero.

Almost unperforated and divisible Cuntz semigroups imply these properties.

Abstract

Let $A$ be a $\sigma$-unital finite simple $C^*$-algebra which has strict comparison property. We show that if the canonical map $\Gamma$ from the Cuntz semigroup to certain lower semi-continuous affine functions is surjective, then $A$ has tracial approximate oscillation zero and stable rank one. Equivalently, if $A$ has an almost unperforated and almost divisible Cuntz semigroup, then $A$ has stable rank one and tracial approximate oscillation zero.