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SMALL-BOUND ISOMORPHISMS OF FUNCTION SPACES
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 111 / Issue 3 / December 2021
- Published online by Cambridge University Press:
- 18 March 2020, pp. 412-429
- Print publication:
- December 2021
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- Article
- Export citation
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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 
${\mathcal{H}}_{2}$.
