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.

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Banach-Stone 定理的非满射粗略版本

令 X 和 Y 是紧致的豪斯多夫空间，并令 $$F:C(X)\rightarrow C(Y)$$ 是（非满射的）粗糙 $$(M,\varepsilon )$$ -quasi-isometry with $ $M<\sqrt{8/7}$$ 。还假设 F 的范围很好地“近似”了 C(Y)，那么 X 和 Y 是同胚的。此外，如果 $$U:C(X)\rightarrow C(Y)$$ 是由同胚引起的规范等距，则 $$\begin{aligned} \Vert F(f)-MU(f)\Vert \ le 8\frac{M^2-1}{M}\Vert f\Vert +\varDelta \varepsilon \end{aligned}$$ 对于每个 $$f\in C(K)$$ ，其中 $$\varDelta $$ 仅取决于 M 和 C(Y) 在 F 范围内的近似程度。（有关精确的技术近似条件和 $$\varDelta $$ 的估计，请参阅简介）。即使对于非投影等距，近似结果也是新的。