Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $K_{i}$ be a locally compact (Hausdorff) topological space and let ${\mathcal{H}}_{i}$ be a closed subspace of ${\mathcal{C}}_{0}(K_{i},\mathbb{F})$ such that each point of the Choquet boundary $\operatorname{Ch}_{{\mathcal{H}}_{i}}K_{i}$ of ${\mathcal{H}}_{i}$ is a weak peak point. We show that if there exists an isomorphism $T:{\mathcal{H}}_{1}\rightarrow {\mathcal{H}}_{2}$ with $\left\Vert T\right\Vert \cdot \left\Vert T^{-1}\right\Vert <2$, then $\operatorname{Ch}_{{\mathcal{H}}_{1}}K_{1}$ is homeomorphic to $\operatorname{Ch}_{{\mathcal{H}}_{2}}K_{2}$. We then provide a one-sided version of this result. Finally we prove that under the assumption on weak peak points the Choquet boundaries have the same cardinality provided ${\mathcal{H}}_{1}$ is isomorphic to ${\mathcal{H}}_{2}$.

中文翻译：

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函数空间的小界同构

让$\mathbb{F}=\mathbb{R}$要么$\mathbb{C}$. 为了$i=1,2$， 让$K_{i}$是一个局部紧致（Hausdorff）拓扑空间，令${\mathcal{H}}_{i}$是一个闭子空间${\mathcal{C}}_{0}(K_{i},\mathbb{F})$使得 Choquet 边界的每个点$\operatorname{Ch}_{{\mathcal{H}}_{i}}K_{i}$的${\mathcal{H}}_{i}$是一个弱峰点。我们证明如果存在同构$T:{\mathcal{H}}_{1}\rightarrow {\mathcal{H}}_{2}$和$\left\Vert T\right\Vert \cdot \left\Vert T^{-1}\right\Vert <2$， 然后$\operatorname{Ch}_{{\mathcal{H}}_{1}}K_{1}$同胚于$\operatorname{Ch}_{{\mathcal{H}}_{2}}K_{2}$. 然后，我们提供此结果的单面版本。最后我们证明了在弱峰值点的假设下，Choquet 边界具有相同的基数${\mathcal{H}}_{1}$同构于${\mathcal{H}}_{2}$.