从 $C(Ω)$ 到 $C^*$ 代数的同态分类

Qingnan An, George Elliott, Zhichao Liu
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引用次数: 0

摘要

让 $\Omega$ 是 $\mathbb{C}$ 的一个紧凑子集,让 $A$ 是一个具有稳定秩一、实秩零和严格比较的单简单、可分离的 $C^*$ 代数。我们证明,给定一个 Cu-morphism $\alpha:{\rm Cu}(C(\Omega))\to{rm Cu}(A)$ 带有 $\alpha(\langle \mathds{1}_{\Omega}\rangle)leq \langle1_A\rangle$, 存在一个同态性 $\phi:C(\Omega)\to A$,使得 ${\rmCu}(\phi)=\alpha$ 并且 $\phi$ 在近似单元等价性上是唯一的。我们还给出了从一大类$C^*$-代数到$A$的映射在 Cuntz 半群方面的分类结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Classification of homomorphisms from $C(Ω)$ to a $C^*$-algebra
Let $\Omega$ 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 $\alpha:{\rm Cu}(C(\Omega))\to {\rm Cu}(A)$ with $\alpha(\langle \mathds{1}_{\Omega}\rangle)\leq \langle 1_A\rangle$, there exists a homomorphism $\phi: C(\Omega)\to A$ such that ${\rm Cu}(\phi)=\alpha$ and $\phi$ 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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