Coarse version of the Banach-Stone theorem

Rafal Gorak

doi:10.1016/j.jmaa.2010.11.012

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arXiv · 1005.0937

Coarse version of the Banach-Stone theorem

Rafal Gorak

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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 $C(Y)$.

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Rafal Gorak. 2010-05-06. Coarse version of the Banach-Stone theorem. https://doi.org/10.1016/j.jmaa.2010.11.012

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