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)-quasi-isometry T between the Banach spaces of real continuous functions C-0(K) and C-0(S) with M < root 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 Gorak (2011). In the Lipschitz case, that is when L = 0, we also improve the estimations of the distance of the map T from the isometries between the spaces C-0(K) and C-0(S) obtained by Gorak when K and S are compact spaces or not. As a consequence, we get a linear sharp refinement of the Amir-Cambern theorem. (C) 2016 Elsevier Inc. All rights reserved.