Vol. 297, No. 1, 2018

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An Amir–Cambern theorem for quasi-isometries of $C_{0}(K, X)$ spaces

Elói Medina Galego and André Luis Porto da Silva

Vol. 297 (2018), No. 1, 87–100
Abstract

Let $X$ be a finite-dimensional Banach space. We prove that if $K$ and $S$ are locally compact Hausdorff spaces and there exists a bijective map $T:{C}_{0}\left(K,X\right)\to {C}_{0}\left(S,X\right)$ such that

$\frac{1}{M}\parallel f-g\parallel -L\le \parallel T\left(f\right)-T\left(g\right)\parallel \le M\parallel f-g\parallel +L,$

for every $f,g\in {C}_{0}\left(K,X\right)$ then $K$ and $S$ are homeomorphic, whenever $L\ge 0$ and $1\le {M}^{2}, where $S\left(X\right)$ denotes the Schäffer constant of $X$.

This nonlinear vector-valued extension of the Amir–Cambern theorem via quasi-isometries $T$ with large $M$ was previously unknown even for the classical ${\ell }_{p}^{n}$ spaces, $1, $p\ne 2$ and $n\ge 2$.

Keywords
vector-valued Amir–Cambern theorem, $C_{0}(K, X)$ spaces, finite-dimensional uniformly non-square spaces, quasi-isometry, Schäffer constant
Mathematical Subject Classification 2010
Primary: 46B03, 46E15
Secondary: 46B25, 46E40