Separating maps and linear isometries between some spaces of continuous functions (Q1268735)

For a given locally compact Hausdorff space \(X\), a Banach space \(E\) and a function \(\sigma: X\to (0,\infty)\) satisfying certain conditions, the author defines the Banach space \(C^\sigma_0(X,E)\) of continuous functions from \(X\) into \(E\). An additive map \(T: C^\sigma_0(X, E)\to C^\tau_0(Y, F)\) between two such Banach spaces is said to be separating if whenever \(f,g\in C^\sigma_0(X,E)\) satisfy \(\| f(x)\| \| g(x)\|= 0\) for every \(x\in X\), then \(\|(Tf)(y)\| \|(Tg)(y)\|= 0\) for every \(y\in Y\). \(T\) is said to be biseparating if it is bijective and both \(T\) and \(T^{-1}\) are separating. The author proves that the existence of a biseparating map \(T: C^\sigma(X, E)\to C^\tau_0(Y, F)\) implies that the spaces \(X\) and \(Y\) are homeomorphic.