Separating Maps and Linear Isometries between Some Spaces of Continuous Functions

In this paper we deal with some spaces of vector-valued continuous functions C0s X, E.and Ct0 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