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SMALL-BOUND ISOMORPHISMS OF FUNCTION SPACES

Part of: Special classes of linear operators Topological linear spaces and related structures

Published online by Cambridge University Press:  18 March 2020

JAKUB RONDOŠ Open the ORCID record for JAKUB RONDOŠ [Opens in a new window]  and
JIŘÍ SPURNÝ
JAKUB RONDOŠ*
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic
JIŘÍ SPURNÝ
Affiliation:
Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic e-mail: spurny@karlin.mff.cuni.cz
*
e-mail: jakub.rondos@gmail.com
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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 ${\mathcal{H}}_{2}$.

Keywords

function spaceBanach–Stone theoremCambern theorem

MSC classification

Primary: 47B38: Operators on function spaces (general) 46A55: Convex sets in topological linear spaces; Choquet theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 412 - 429
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Sims

The research was supported by the research grant GAČR 17-00941S.

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