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Let $A$ be a separable (not necessarily unital) simple $C^*$-algebra withstrict comparison. We show that if $A$ has tracial approximate oscillation zerothen $A$ has stable rank one and the canonical map $\Gamma$ from the Cuntzsemigroup of $A$ to the corresponding affine function space is surjective. Theconverse 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.

Tracial oscillation zero and stable rank one