A reflexive admissible topological group must be locally compact

Abstract

Let G be a topological group. The set G of all characters (i.e., continuous homomorphisms from G to the torus T=R/Z) is an abelian group (with pointwise addition). The sets (C,U)={fG : f(C)U}, where C runs over all compact subsets of G and U over all open subsets of T, generate the so-called compact open topology on G. With this topology, G is a topological Hausdorff group. We have the canonical map αG:G→G mapping g to the character χ↦χ(g) of G. If αG is an isomorphism of topological groups, then G is called reflexive. By a result of Pontryagin, every locally compact Hausdorff abelian group is reflexive. There are also examples of reflexive topological groups that are not locally compact (but, necessarily, Hausdorff abelian groups). It seems natural to ask whether the evaluation ω:G×G→T is continuous. Generally, a topology on a subset S of the set C(X,Y) of all continuous mappings from a topological space X to a topological space Y is called admissible iff the evaluation ωS:S×X→Y is continuous. Obviously, the discrete topology on S is admissible. On the other hand, every admissible topology contains the compact open topology. If X is locally compact then the compact open topology is admissible; thus it is the coarsest admissible topology. A result of Arens says that if X is completely regular, the existence of a coarsest admissible topology τ on C(X,R) implies that X is locally compact (and thus τ is the compact open topology). In the paper under review, it is shown that if G is a reflexive topological group, then the compact open topology on G is admissible (if and) only if G is locally compact. This is another indication that the class of locally compact Hausdorff abelian groups is best suited for Pontryagin duality. The proof employs the continuous convergence structure on G; a generalization of the concept of continuous convergence of real functions, as known from classical analysis. In fact, it is shown that if G is a topological Hausdorff abelian group, then the compact open topology on G is admissible if, and only if, the continuous convergence structure coincides with the convergence structure derived from the compact open topology.