Coarse version of the Banach–Stone theorem

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 ) × 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 ) .