Diameter preserving linear bijections and ${\cal C}_0(L)$ spaces

Abstract

We study diameter preserving linear bijections from ${\cal C}(X, V)$ onto ${\cal C}(Y, {\cal C}_0(L))$ where $X, Y$ are compact Hausdorff spaces, $L$ is a locally compact Hausdorff space and $V$ is a Banach space. In the case when $X$ and $Y$ are infinite and ${\cal C}_0(L)^*$ has the Bade property we prove that there is a diameter preserving linear bijection from ${\cal C}(X, V)$ onto ${\cal C}(Y, {\cal C}_0(L))$ if and only if $X$ is homeomorphic to $Y$ and $V$ is linearly isometric to ${\cal C}_0(L)$. Similar results are obtained in the case when $X$ and $Y$ are not compact but locally compact spaces.