Non-surjective coarse version of the Banach–Stone theorem

Abstract

Let X and Y be compact Hausdorff spaces and let $F:C(X)\rightarrow C(Y)$ be a (non-surjective) coarse $(M,\varepsilon )$-quasi-isometry with $M<\sqrt{8/7}$. Assume also that the range of F “approximates” C(Y) well, then X and Y are homeomorphic. Moreover, if $U:C(X)\rightarrow C(Y)$ is the canonical isometry induced by the homeomorphism, then

$$\begin{aligned} \Vert F(f)-MU(f)\Vert \le 8\frac{M^2-1}{M}\Vert f\Vert +\varDelta \varepsilon \end{aligned}$$

for every $f\in C(K)$, where $\varDelta$ depends only on M and the degree of approximation of C(Y) by the range of F. (See the Introduction for the precise technical approximation condition and the estimate of $\varDelta$). The approximation result is new even for non-surjective isometries.