N-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field K are studied. It is proven that the behaviour of these maps when K is not locally compact is very different fr om the case of real- or complex-valued functions: in general, for N-compact spaces X and Y, the existence of a (weakly) separating additive map T : C* (X) --> C*(Y) implies that X and Y are homeomorphic, whereas when dealing w ith real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactific ations of X and Y. Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.