Strict comparison and stable rank one

Abstract

Let A be a σ-unital finite simple $C^{⁎}$-algebra which has strict comparison property. We show that if the rank map Γ from the Cuntz semigroup to certain lower semicontinuous affine functions is surjective, then A has tracial approximate oscillation zero and stable rank one. Equivalently, if A is a σ-unital finite simple $C^{⁎}$-algebra which has an almost unperforated and almost divisible Cuntz semigroup, then A has stable rank one and tracial approximate oscillation zero.