Let C 0 (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 isomorphismwhere J(X) is the James constant of X, then K 1 is homeomorphic to K 2 . 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 = R or X = C) which at the same time unifies and strengthens several generalizations of the classical Banach-Stone theorem due to Cambern (1976) and (1985), BehrendsCambern (1988) andJarosz (1989). In the case where X = l p , 2 ≤ p < ∞, our results are optimal.

No abstract

No abstract

No abstract

For a compact Hausdorff space, we denote by C(K) the Banach space of continuous functions defined in K with values in R or C. A well known result in Banach spaces of continuous functions is the Holsztyński theorem which establishes that if C(K) is isometric to a subspace of C(S), then K is a continuous image of S. The aim of this paper is to give an alternative proof of this result for extremely regular subspaces of C(K).