Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916374641639424 |
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| author | An, Qingnan Elliott, George Liu, Zhichao |
| author_facet | An, Qingnan Elliott, George Liu, Zhichao |
| contents | Let $Ω$ be a compact subset of $\mathbb{C}$ and let $A$ be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero and strict comparison. We show that, given a Cu-morphism $α:{\rm Cu}(C(Ω))\to {\rm Cu}(A)$ with $α(\langle \mathds{1}_Ω\rangle)\leq \langle 1_A\rangle$, there exists a homomorphism $ϕ: C(Ω)\to A$ such that ${\rm Cu}(ϕ)=α$ and $ϕ$ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^*$-algebras to $A$ in terms of the Cuntz semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16657 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra An, Qingnan Elliott, George Liu, Zhichao Operator Algebras Let $Ω$ be a compact subset of $\mathbb{C}$ and let $A$ be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero and strict comparison. We show that, given a Cu-morphism $α:{\rm Cu}(C(Ω))\to {\rm Cu}(A)$ with $α(\langle \mathds{1}_Ω\rangle)\leq \langle 1_A\rangle$, there exists a homomorphism $ϕ: C(Ω)\to A$ such that ${\rm Cu}(ϕ)=α$ and $ϕ$ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^*$-algebras to $A$ in terms of the Cuntz semigroup. |
| title | Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2408.16657 |