Pontryagin duality for metrizable groups

Abstract

This paper deals with the validity of the Pontryagin duality theorem in the class of metrizable topological groups. We prove that completeness is a necessary condition for the Pontryagin reflexivity of those groups. We also prove that in order for a metrizable separable topological group to be Pontryagin reflexive it is sufficient that the canonical embedding into its bidual group be an algebraic isomorphism.On the other hand, we consider the notion of reflexivity introduced by E. Binz and H. Butzmann. In [5] it was proved that, for topological abelian groups, the Pontryagin reflexivity and the Binz-Butzmann reflexivity, in general, are independent notions. Here we prove that these notions coincide in the class of metrizable topological abelian groups.

An approach to duality on abelian precompact groups

“…The second assertion comes from the fact that Pontryagin reflexivity and BBreflexivity are equivalent for metrizable groups (see [9]). r Next we draw some consequences for the Bohr topology on a topological group.…”

“…A topological abelian group G will be called a determined group if for any dense subgroup H < G, the restriction mapping from G 5 to H 5 is a topological isomorphism; roughly speaking, if the respective dual groups equipped with the compact open topology coincide algebraically and topologically. Metrizable abelian groups are determined, as proved by the first author in [9], and independently in [1]. The name 'determined group' appeared for the first time in Raczkowski's doctoral thesis [15], where she proves that even compact groups may be non-determined.…”

On Schwartz groups

“…This follows from Corollary 5.5 and the fact that the character group of a metrizable group is a hemicompact k-space (cf. (4.7) in [1] or Theorem 1 in [8]). …”

Strong Reflexivity of Abelian Groups

“…It is a k-group, so the canonical mapping α G is continuous (see [13]). On the other hand, one of the authors proved in [6] that for a metrizable group, the dual group G ∧ is a k-space. The same can be proved for Čech complete groups in a quite analogous way.…”

“…Then, for every closed subgroup H of G, H and G/H are Pontryagin reflexive. So, H and G/H being metrizable, they are also BB-reflexive (see [6]). …”