Quasi-isometries on subsets of $C_{0}(K)$ and $C_{0}^{(1)}(K)$ spaces which determine $K$
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- by Elói Medina Galego and André Luis Porto da Silva PDF
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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 \begin{equation*} \frac {1}{M} \|f-g\|-L \leq \|T(f)-T(g)\|\leq M \|f-g\|+L, \end{equation*} 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.References
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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
- MR Author ID: 647154
- 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
- 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
- © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 3981124