This is an announcement for the paper "Coarse version of the Banach Stone theorem" by Rafal Gorak.

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 $l(T) \times l(T^{-1})<\frac{6}{5}$. 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 $\frac{17}{16}$. We also estimate the distance of the map $T$ from the isometry of the spaces $C(X)$ and $C(Y)$.

Archive classification: math.FA math.GN

Mathematics Subject Classification: 46E15, 46B26, 46T99

Submitted from: R.Gorak@mini.pw.edu.pl

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1005.0937

or

http://arXiv.org/abs/1005.0937