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

DOI: 10.2140/pjm.2018.297.87

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 $1\le M^2<S\left(X\right)$, 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, $1<p<\infty$, $p\ne 2$ 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

Milestones

Received: 6 January 2018

Revised: 12 April 2018

Accepted: 20 April 2018

Published: 7 October 2018