ALGEBRAIC CHARACTERIZATIONS OF LOCALLY COMPACT-GROUPS

Let G(1), G(2) be locally compact real-compact spaces. A linear map T defined from C(G(1)) into C(G(2)) is said to be separating or disjoint ness presenting if f.g = 0 implies T f.T g = 0 for all f, g is an elem ent of C(G(1)). In this paper we prove that both a separating map whic h preserves non-vanishing functions and a separating bijection which s atisfies condition (M) (see Definition 4) are automatically continuous and can be written as weighted composition maps. We also study the ef fect of separating surjections (respectively injections) on the underl ying spaces G(1) and G(2). Next we apply the above results to give an algebraic characterization of locally compact Abelian groups, similar to the one given in [7] for compact Abelian groups in the presence of ring isomorphisms. Finally, locally compact (not necessarily Abelian) groups are considered. We provide a sharpening of a result of Edwards and study the effect of onto (respectively injective) weighted composi tion maps on the groups G(1) and G(2).