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SMALL-BOUND ISOMORPHISMS OF FUNCTION SPACES
Published online by Cambridge University Press: 18 March 2020
Abstract
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
${\mathcal{H}}_{2}$.
MSC classification
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 412 - 429
- Copyright
- © 2020 Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by A. Sims
The research was supported by the research grant GAČR 17-00941S.
References
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