A generalized Banach-Stone theorem for \(C_{0}(K,X)\) spaces via the modulus of convexity of \(X^\ast\)
Let \(X\) be a real or complex Banach space and let \(K\) be a locally compact Hausdorff space. Let \(C_0(K,X)\) denote the space of \(X\)-valued continuous functions on \(K\) that vanish at infinity, equipped with the supremum norm. Let \(T: C_0(K,X) \rightarrow C_0(S,X)\) be a surjective isomorphism such that \(\|T\|\|T^{-1}\| \leq \lambda\). Suppose that \( \sup \{ \|x_1^\ast-x_2^\ast\|: \|x_1^\ast + x_2^\ast\| =2,~ x_1^\ast,x_2^\ast \in X^\ast,~\|x_1^\ast\|,\|x_2^\ast\| \leq \lambda \} < \frac{4}{3}\). A classical result of \textit{K. Jarosz} [Stud. Math. 73, 33--39 (1982; Zbl 0501.46037)] is that in this case \(K\) and \(S\) are homeomorphic. In the present paper, the authors show that the same conclusion holds with a larger number which is approximately \(4\times 0.41883\).
scientific article; zbMATH DE number 6684623
scientific article