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.