Quasi-isometries of $C_{0}(K, E)$ spaces which determine $K$ for all Euclidean spaces $E$
Volume 243 / 2018
Studia Mathematica 243 (2018), 233-242
MSC: Primary 46B03, 46E15; Secondary 46E40, 46B25.
DOI: 10.4064/sm8747-8-2017
Published online: 26 April 2018
Abstract
We prove that for all Euclidean spaces $E$ and locally compact Hausdorff spaces $K$ and $S$, if there exists a bijective map $T: C_{0}(K,E) \to C_{0}(S, E)$ such that $$ \frac{1}{M} \|f-g\| - L \leq \|T(f)-T(g)\|\leq M \|f-g\|+L $$ for some constants $1 \leq M< \sqrt[4]{2}$ and $L \geq 0$ and for all $f, g \in C_{0}(K, E)$, then $K$ and $S$ are homeomorphic. In other words, by using quasi-isometries we obtain a nonlinear extension of the classical 1976 Hilbert vector-valued Banach–Stone theorem due to Cambern. In the Lipschitz case, that is, when $L=0$, our result improves Jarosz’s 1989 theorem.
