Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra

Qingnan An; George Elliott; Zhichao Liu

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arXiv · 2408.16657

Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra

Qingnan An·George Elliott·Zhichao Liu

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Abstract

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.

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BibTeX RIS

Qingnan An, George Elliott, Zhichao Liu. 2024-08-29. Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra. https://arxiv.org/abs/2408.16657

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