Quasi-isometries on subsets of $C_{0}(K)$ and $C_{0}^{(1)}(K)$ spaces which determine $K$

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.