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Tracial oscillation zero and stable rank one (2112.14007v4)
Published 28 Dec 2021 in math.OA
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.