Elena Martín-Peinador

Open subgroups and Pontryagin duality

For an abelian topological group G, let G∧ denote the character group of G. The group G is called reflexive if the evaluation map is a topological isomorphism of G onto G∧∧, and G is called strongly reflexive if all closed subgroups and quotient groups of G and G∧ are reflexive. In this paper the authors study the relationship of reflexivity (and strong reflexivity) among G, A, and G/K, where A is an open subgroup and K a compact subgroup of G. Strong reflexivity is closely connected with the notion of strong duality introduced by R. Brown, P. J. Higgins and S. A. Morris [Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32]. In fact, G is strongly reflexive if and only if the natural homomorphism G∧×G→T is a strong duality. R. Venkataraman [Math. Z. 143 (1975), no. 2, 105–112] originally claimed that if G is reflexive, then so is A. However, his proof includes inaccuracies. The present paper includes a new proof in this regard. In all, the following theorems are proved. Theorem 1: G is reflexive [resp. strongly reflexive] if and only if A is reflexive [resp. strongly reflexive]. Theorem 2: If G admits sufficiently many continuous characters and G/K is reflexive [resp. strongly reflexive], then G is reflexive [resp. strongly reflexive]. Conversely, if G is reflexive and K is dually closed in G, then G/K is reflexive. Theorem 3: Every closed subgroup H and the quotient group G/H of a strongly reflexive group G are strongly reflexive

A property of Dunford-Pettis type in topological groups

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 Dunford-Pettis 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.

A Reflexive Admissible Topological Group must be Locally Compact

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

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

Binz-Butzmann duality versus Pontryagin duality

A convergence structure Ξ on a set G is a set of pairs (F,x) consisting of a filter F on X and an element x∈X satisfying a few simple axioms expressing the idea that F converges to x. A set with a convergence structure is a convergence space. It should be clear what continuous functions between convergence spaces are. A subset of a convergence space is compact if every ultrafilter on it converges to an element in it. If G is a group, then (G,Ξ) is called a convergence group if (x,y)↦xy−1:G×G→G is continuous. For an abelian topological group G let ΓG denote the group of continuous characters G→R/Z. Then ΓG is a convergence group with respect to a convergence structure for which a filter F converges to χ if for any filter H on G converging to g the filter basis F(H) converges to χ(g). The convergence group arising in this fashion is denoted by Γc (G). The topological character group endowed with the compact-open topology is written Gˆ. The authors establish the following theorem: Let G be a topological abelian group such that the canonical evaluation morphism G→Gˆˆ is continuous. Then ΓcG is locally compact (in the sense that every convergent filter contains a compact member), and the bidual Gˆˆ may be identified with a topological subgroup of ΓcΓc (G). Examples show that equality does not prevail in general as it does when G is locally compact. The examples are mostly taken from topological vector spaces. For a countable family of locally compact abelian groups Gn one knows from Kaplans theorem that the character group of ∑Gn with the box topology is ∏Gnˆ. Then the continuous convergence structure on ∏Gnˆ is finer than the convergence structure of the product topology and coarser than the convergence structure of the box topology—in general properly so in both instances. It is shown that ∑Gn≅ΓcΓc (∑Gn) canonically and, similarly, for ∏Gn.

An approach to duality on abelian precompact groups

Abstract We prove that every dense subgroup of a topological abelian group has the same ‘convergence dual’ as the whole group. By the ‘convergence dual’ we mean the character group endowed with the continuous convergence structure. We draw as a corollary that the continuous convergence structure on the character group of a precompact group is discrete and therefore a non-compact precompact group is never reflexive in the sense of convergence. We do not know if the same statement holds also for reflexivity in the sense of Pontryagin; at least in the category of metrizable abelian groups it does.

Weakly pseudocompact subsets of nuclear groups

Let G be an Abelian topological group and G(+) the group G endowed with the weak topology induced by continuous characters. We say that G respects compactness (pseudocompactness, countable compactness, functional boundedness) if G and G+ have the same compact (pseudocompact, countably compact, functionally bounded) sets. The well-known theorem of Glicksberg that LCA groups respect compactness was extended by Trigos-Arrieta to pseudocompactness and functional boundedness. In this paper we generalize these results to arbitrary nuclear groups, a class of Abelian topological groups which contains LCA groups and nuclear locally convex spaces and is closed with respect to subgroups, separated quotients and arbitrary products.

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.

“Varopoulos paradigm”: Mackey property versus metrizability in topological groups

The class of all locally quasi-convex (lqc) abelian groups contains all locally convex vector spaces (lcs) considered as topological groups. Therefore it is natural to extend classical properties of locally convex spaces to this larger class of abelian topological groups. In the present paper we consider the following well known property of lcs: “A metrizable locally convex space carries its Mackey topology ”. This claim cannot be extended to lqc-groups in the natural way, as we have recently proved with other coauthors (Außenhofer and de la Barrera Mayoral in J Pure Appl Algebra 216(6):1340–1347, 2012; Díaz Nieto and Martín Peinador in Descriptive Topology and Functional Analysis, Springer Proceedings in Mathematics and Statistics, Vol 80 doi:10.1007/978-3-319-05224-3_7, 2014; Dikranjan et al. in Forum Math 26:723–757, 2014). We say that an abelian group G satisfies the Varopoulos paradigm (VP) if any metrizable locally quasi-convex topology on G is the Mackey topology. In the present paper we prove that in any unbounded group there exists a lqc metrizable topology that is not Mackey. This statement (Theorem C) allows us to show that the class of groups satisfying VP coincides with the class of finite exponent groups. Thus, a property of topological nature characterizes an algebraic feature of abelian groups.

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.