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arXiv · 2112.14007

Tracial oscillation zero and stable rank one

Xuanlong Fu·Huaxin Lin
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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.

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Xuanlong Fu, Huaxin Lin. 2021-12-28. Tracial oscillation zero and stable rank one. https://doi.org/10.4153/s0008414x24000099

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