Montserrat Bruguera

Completeness properties of locally quasi-convex groups

It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of the Grothendieck theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness

Some properties of locally quasi-convex groups

Abstract We prove in this paper that for a Hausdorff group topology on an Abelian group with sufficiently many continuous characters, there is an associated locally quasi-convex topology which is the strongest among all the locally quasi-convex group topologies weaker than the given one. We also give a result on local quasi-convexity on the line of three-space properties.

Open subgroups, compact subgroups and Binz-Butzmann reflexivity

A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches. The extension to the category of topological abelian groups created the concept of reflexive group. In this paper we deal with the extension of Pontryagin duality to the category of convergence abelian groups. Reflexivity in this category was defined and studied by E. Binz and H. Butzmann. A convergence group is reflexive (subsequently called BB-reflexive by us in our work) if the canonical embedding into the bidual is a convergence isomorphism. Topological abelian groups are, in an obvious way, convergence groups; therefore it is natural to compare reflexivity and BB-reflexivity for them. Chasco and Martin-Peinador (1994) show that these two notions are independent. However some properties of reflexive groups also hold for BB-reflexive groups, and the purpose of this paper is to show two of them. Namely, we prove that if an abelian topological group G contains an open subgroup A, then G is BB-reflexive if and only if A is BB-reflexive. Next, if G has sufficiently many continuous characters and K is a compact subgroup of G, then G is BB-reflexive if and only if GK is BB-reflexive.

EBERLEIN–ŠMULYAN THEOREM FOR ABELIAN TOPOLOGICAL GROUPS

Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological Abelian groups which are strictly angelic when endowed with their Bohr topology. Some extensions are given of the Eberlein–ˇSmulyan theorem for the class of topological Abelian groups, and finally, for a large subclass of the latter, Bohr angelicity is related to the Schur property.