Coarse version of the Banach-Stone theorem

TL;DR

This paper establishes that under certain Lipschitz conditions, the topological structure of compact spaces can be recovered from their function spaces, improving previous bounds on the Lipschitz constants needed.

Contribution

It proves a refined version of the Banach-Stone theorem with a tighter Lipschitz constant threshold for homeomorphism of underlying spaces.

Findings

Spaces are homeomorphic if Lipschitz constants satisfy l(T) * l(T^{-1}) < 6/5

Improves previous constant bounds from 17/16 to 6/5

Provides estimates on the distance of T from an isometry

Abstract

We show that if there exists a Lipschitz homeomorphism $T$ between the nets in the Banach spaces $C(X)$ and $C(Y)$ of continuous real valued functions on compact spaces $X$ and $Y$, then the spaces $X$ and $Y$ are homeomorphic provided $l(T) \times l(T^{-1})< 6/5$. By $l(T)$ and $l(T^{-1})$ we denote the Lipschitz constants of the maps $T$ and $T^{-1}$. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is 17/16. We also estimate the distance of the map $T$ from the isometry of the spaces $C(X)$ and $C(Y)$.