An optimal nonlinear extension of Banach–Stone theorem

We prove that if K and S   are locally compact Hausdorff spaces and there exists a bijective coarse (M,L)(M,L)-quasi-isometry T   between the Banach spaces of real continuous functions C0(K)C0(K) and C0(S)C0(S) with M<2, then K and S are homeomorphic. This nonlinear extension of Banach–Stone theorem (1933/1937) is in some sense optimal and improves some results of Amir (1965), Cambern (1967), Jarosz (1989), Dutrieux and Kalton (2005) and Górak (2011).In the Lipschitz case, that is when L=0L=0, we also improve the estimations of the distance of the map T   from the isometries between the spaces C0(K)C0(K) and C0(S)C0(S) obtained by Górak when K and S are compact spaces or not. As a consequence, we get a linear sharp refinement of the Amir–Cambern theorem.