Realcompactness and spaces of vector-valued functions

It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X, E) and A(Y, F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T : C* (X, E) --> C* (Y, F) is a biseparating map, then the realcompactifications of X and Y are homeomorphic.