Separating maps and linear isometries between some spaces of continuous functions

this paper we deal with some spaces of vector-valued continuous functions C-0(sigma)(X, E) and C-0(tau)(Y, F) on locally compact spaces X and Y (among them the spaces of continuous functions vanishing at infinity). After giving some properties of these spaces, it is shown that the existence of a biseparating map T between these spaces always implies the existence of a homeomorphism between X and Y. Some results on automatic continuity are given when T is just separating, linear and has closed range, as well as a description of T. Finally these results are used to characterize the linear isometries between some of these spaces, (C) 1998 Academic Press.