Isomorphisms of C(K, E) Spaces and Height of K

Let K-1, K-2 be compact Hausdorff spaces and E-1, E-2 be Ba-nach spaces not containing a copy of c(0). We establish lower estimates of the Banach-Mazur distance between the spaces of continuous functions C(K-1, E-1) and C(K-2, E-2) based on the ordinals ht(K-1), ht(K-2), which are new even for the case of spaces of real-valued functions on ordinal intervals. As a corollary we deduce that C(K-1, E-1) and C(K-2, E-2) are not isomorphic if ht(K-1) is substantially different from ht(K-2).