A property of Dunford-Pettis type in topological groups

Abstract

Abstract. The property of Dunford-Pettis for a locally convex space was introduced by Grothendieck in 1953. Since then it has been intensively studied, with especial emphasis in the framework of Banach space theory.In this paper we define the Bohr sequential continuity property (BSCP) for a topological Abelian group. This notion could be the analogue to the DunfordPettis property in the context of groups. We have picked this name because the Bohr topology of the group and of the dual group plays an important role in the definition. We relate the BSCP with the Schur property, which also admits a natural formulation for Abelian topological groups, and we prove that they are equivalent within the class of separable metrizable locally quasi-convex groups.For Banach spaces (or for metrizable locally convex spaces), considered in their additive structure, we show that the BSCP lies between the Schur and the Dunford-Pettis properties. PreliminariesThe aim of this paper is to give an analogue to the Dunford-Pettis property in the context of topological Abelian groups, as explained in Section 3. In this spirit we define in Section 2 the Bohr sequential continuity property, which holds for a class of groups larger than that of Hausdorff locally compact Abelian (LCA) groups. The most satisfactory situation would be, if for any locally convex space E, the BSCP were equivalent to the Dunford-Pettis property. The question does not go so smoothly; we have obtained some related results and an insight to properties which so far had only been treated for topological vector spaces and, in fact, they can be extended to topological Abelian groups.We now introduce notation and relevant facts. Let G be an Abelian topological group; we denote by N e (G) the set of all neighborhoods of the neutral element e. The group of all continuous homomorphisms (also named characters) from G into the torus T = {z ∈ C : |z| = 1}, considered with pointwise multiplication,

Duality of locally quasi-convex convergence groups

“…This result explains the role of local compactness in the Pontryagin duality theory. This result is further generalized in [11,Theorem 1.1] where the same statement with the condition "reflexive" replaced by the "quasi-convex compactness property" is proved.…”

Topological properties of the group of the null sequences valued in an Abelian topological group

“…Note that the Glicksberg property has been studied beyond LCA groups by many authors, see [2,8,9,38,57,71]. The Schur property in Abelian topological groups was intensively studied by Martín-Peinador and Tarieladze in [51] (see also [36,46]). Many other results concerning preservation and respectness of topological properties under the Bohr functor see [11].…”

“…I wish to thank Professor D. Dikranjan for the comment [22] and suggestions. It is a pleasure to thank Professors S. Hernández, E. Martín-Peinador and V. Tarieladze for pointing out the articles [44,23,51].…”

On the continuity of separately continuous bihomomorphisms

“…The following notion was defined for topological groups by E. Martin-Peinador and V. Tarieladze in [15] and [11].…”

“…Also separable Baire or metrizable hereditarily Baire groups are g-barrelled ( [16], [22], [11]). Finally, the additive group of a barrelled topological vector space is g-barrelled ( [15]). To obtain an example of a non-topological g-barrelled convergence group we recall that a topological group G is said to respect compactness if each σ(G, ΓG)-compact subset of G is compact.…”