Reflexivity of Topological Groups

It is shown that under mild conditions the path-component of the identity in the dual group G* of an Abelian topological group G is precisely the union of the one-parameter subgroups of CT. This yields several corollaries, including a necessary condition for certain groups to be reflexive (to satisfy the Pontrjagin duality theorem), and a negative answer to a question of N. Noble. The main result of this paper is a theorem on the structure of the dual group G" of an Abelian topological group G; we show (under mild restrictions) that the path-component of the identity in G~ is precisely the union of the one-parameter subgroups of G~. Several authors have studied the class of Abelian topological groups which are reflexive (that is, which satisfy the Pontrjagin duality theorem): it is known, for example, that the class contains many nonlocally compact groups ([1], [7], [8], [14], [16], [17]). It appears, however, that a few general necessary conditions for reflexivity are known. We derive such a condition here as a consequence of the main theorem, and then explore some of its applications. In particular, we show that many free Abelian topological groups fail to be reflexive, and we settle a question of N. Noble [14] in the negative. The author wishes to thank Dr. S. A. R. Disney and Dr. S. A. Morris for helpful comments. We shall use additive notation for all our groups except the circle group T, which we regard as the (compact) multiplicative group of complex numbers of modulus one. As usual, the dual group G" of an Abelian topological group G is the group of continuous characters of G, with the compact-open topology. All topological groups and spaces considered here will be assumed Hausdorff. Theorem. // the Abelian topological group G is a k-space, then the pathcomponent of the identity in G~ is the union of all the one-parameter subgroups ofG*. Proof. The union of the one-parameter subgroups of G' is clearly Presented to the Society, June 16, 1975; received by the editors August 5, 1975 and, in revised form, December 13, 1976. AMS (A/05) subject classifications (1970). Primary 22A05; Secondary 54D30.