A wider nonlinear extension of Banach-Stone theorem to $C_{0}(K,X)$ spaces which is optimal for $X=\ell _p$, $2 \leq p <\infty$

Abstract:

It is proven that if $X$ is a Banach space, $K$ and $S$ are locally compact Hausdorff spaces and there exists an $(M, L)$-quasi isometry $T$ from $C_{0}(K,X)$ onto $C_{0}(S, X)$, then $K$ and $S$ are homeomorphic whenever $1 \leq M^{2}< S(X)$, where $S(X)$ denotes the Schäffer constant of $X$, and $L \geq 0$.

As a consequence, we show that the first nonlinear extension of Banach-Stone theorem for $C_{0}(K, X)$ spaces obtained by Jarosz in 1989 can be extended to infinite-dimensional spaces $X$, thus reinforcing a 1991 conjecture of Jarosz himself on $\epsilon$-bi-Lipschitz surjective maps between Banach spaces.

Our theorem is optimal when $X$ is the classical space $\ell _p$, $2 \leq p< \infty$.