A weak vector-valued Banach-Stone theorem

Abstract:

For a locally compact Hausdorff space $X$ and a Banach space $E$, we denote by $C_0(X, E)$ the space of $E$-valued continuous functions on $X$ which vanish at infinity, endowed with the supremum norm. In the spirit of the classical Banach-Stone theorem, we prove that if $C_0(X, E)$ is isomorphic to $C_{0}(Y, E)$, where $E$ has non-trivial cotype and such that $E$ is separable or $E^*$ has the Radon-Nikodým property, then either $X$ and $Y$ are finite or $X$ and $Y$ have the same cardinality. In other words, we obtain a vector-valued extension of a 1978 B. Cengiz result, the scalar case $E=\mathbb R$ or $E=\mathbb C.$