Linear isometries between certain subspaces of continuous vector-valued functions (Q1129754)

Let \(E\), \(F\) be Banach spaces and \(X\), \(Y\) locally compact spaces. Let \(A\) be a linear subspace of \(C_0(X)\). We will denote by \({\mathcal A}[A]\) any linear subspace of \(C_0(X,E)\) which contains the set \(\{\xi\cdot e:\xi\in A, e\in S_E\}\), where \(S_E\) stands for the unit sphere of \(E\). In this paper, we prove that a linear isometry \(T\) of \({\mathcal A}[A]\) into \(C_0(Y, F)\), where \(F\) is strictly convex, can be written as a weighted composition map on a subspace \(Y_0\) of \(Y\). Furthermore, \(X\) is shown to be the continuous image of \(Y_0\). We also prove that \(X\) and \(Y\) are homeomorphic if \(T\) is a linear isometry of \({\mathcal A}[A]\) onto such a subspace \({\mathcal B}[B]\) of \(C_0(Y, F)\), where both \(E\) and \(F\) are strictly convex.