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Quasi-isometries on subsets of $ C_{0}(K)$ and $ C_{0}^{(1)}(K)$ spaces which determine $ K$


Authors: Elói Medina Galego and André Luis Porto da Silva
Journal: Proc. Amer. Math. Soc. 147 (2019), 3455-3470
MSC (2010): Primary 46B03, 46E15; Secondary 46B25, 47H99
DOI: https://doi.org/10.1090/proc/14498
Published electronically: April 8, 2019
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Abstract: We introduce the concept of Banach-Stone subsets of $ C_{0}(K)$
spaces. This allows us to unify and improve several extensions of the classical theorem due to Banach (1933) and Stone (1937). More precisely, we prove that if $ K$ and $ S$ are locally compact Hausdorff spaces, $ A$ and $ B$ are Banach-Stone subsets of $ C_{0}(K)$ and $ C_{0}(S)$, respectively, and there exists a map $ T$ from $ A$ to $ B$ (not necessarily injective) with image $ \theta $-dense in $ B$ for some $ \theta >0$ such that

$\displaystyle \frac {1}{M} \Vert f-g\Vert-L \leq \Vert T(f)-T(g)\Vert\leq M \Vert f-g\Vert+L,$    

for every $ f, g \in A$, then $ K$ and $ S$ are homeomorphic whenever $ L \geq 0$ and $ M< \sqrt {2}$. As an application of this more general theorem concerning the quasi-isometries $ T$ on subsets of $ C_{0}(K)$ spaces, we show that certain quasi-isometries on $ C_0^{(1)}(K)$ spaces also determine the locally compact subspaces $ K$ of the real line $ \mathbb{R}$ with no isolated points. In turn, this result enables us to prove a unification and improvement of some theorems of Cambern, Pathak, and Vasavada for the first time to the nonlinear case.

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Additional Information

Elói Medina Galego
Affiliation: Department of Mathematics, University of São Paulo, IME, Rua do Matão 1010, São Paulo, Brazil
Email: eloi@ime.usp.br

André Luis Porto da Silva
Affiliation: Department of Mathematics, University of São Paulo, IME, Rua do Matão 1010, São Paulo, Brazil
Email: porto@ime.usp.br

DOI: https://doi.org/10.1090/proc/14498
Keywords: Banach-Stone theorem, quasi-isometry, $C_{0}(K)$ and $C_{0}^{(1)}(K)$ spaces
Received by editor(s): May 6, 2018
Received by editor(s) in revised form: November 13, 2018, and November 21, 2018
Published electronically: April 8, 2019
Additional Notes: This work was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
Communicated by: Stephen Dilworth
Article copyright: © Copyright 2019 American Mathematical Society