Tracial oscillation zero and stable rank one
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866909541099110400 |
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| author | Fu, Xuanlong Lin, Huaxin |
| author_facet | Fu, Xuanlong Lin, Huaxin |
| contents | 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 $Γ$ 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 $Γ$ is surjective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14007 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Tracial oscillation zero and stable rank one Fu, Xuanlong Lin, Huaxin Operator Algebras 46L35, 46L05 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 $Γ$ 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 $Γ$ is surjective. |
| title | Tracial oscillation zero and stable rank one |
| topic | Operator Algebras 46L35, 46L05 |
| url | https://arxiv.org/abs/2112.14007 |