Diameter preserving linear bijections and C0(L) spaces

We study diameter preserving linear bijections from C(X, V) onto C(Y,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 C-0(L)* has the Bade property we prove that there is a diameter preserving linear bijection from C(X, V) onto C(Y, C-0 (L)) if and only if X is homeomorphic to Y and V is linearly isometric to C-0(L). Similar results are obtained in the case when X and Y are not compact but locally compact spaces.