There's something about the diameter

We study diameter preserving linear bijections from C(X,V) onto C(Y,Z) where X, Y are compact Hausdorff spaces and V, Z are Banach spaces. For instance, we obtain that if X has at least four points, Z is linearly isometric to V and either Z is a C0(L) space or Z∗ is strictly convex or smooth, then there is a diameter preserving linear bijection from C(X,V) onto C(Y,Z) if and only if X is homeomorphic to Y. We also consider the case when X and Y are not compact but locally compact spaces.