Optimal extensions of the Banach–Stone theorem

Let C0(K,X)C0(K,X) denote the Banach space of all X-valued continuous functions defined on the locally compact Hausdorff space K which vanish at infinity, provided with the supremum norm. We prove that if X is a real Banach space and T   is an isomorphism from C0(K1,X)C0(K1,X) onto C0(K2,X)C0(K2,X) satisfyingJ(X)‖T‖‖T−1‖<2,J(X)‖T‖‖T−1‖<2, where J(X)J(X) is the James constant of X  , then K1K1 is homeomorphic to K2K2. In the complex case, we provide a similar result for reflexive spaces X  . In other words, we obtain a vector-valued extension of the classical Amir–Cambern theorem (X=RX=R or X=CX=C) which at the same time unifies and strengthens several generalizations of the classical Banach–Stone theorem due to Cambern (1976) and (1985), Behrends–Cambern (1988) and Jarosz (1989). In the case where X=lpX=lp, 2≤p<∞2≤p<∞, our results are optimal.