Characteristics of the Mackey Topology for Abelian Topological Groups

Nieto, José Manuel Díaz

doi:10.1007/978-3-319-05224-3_7

Cited by 10 publications

(11 citation statements)

References 14 publications

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“…More precisely, ν is the topology of uniform convergence on the sets of E, and {L ⊳ , L ∈ E} is a basis of zero neighborhoods for ν. A thorough study of this topic is done in [11]. Definition 4.1.…”

Section: On the Existence Of G-barrelled Topologies In A Group Dualitymentioning

confidence: 99%

“…The proof of (i) follows from the fact that the equicontinuous subsets with respect to τ and Qτ coincide ( [11,Proposition 7.1]). The statement (ii) has an straightforward proof.…”

Section: On the Existence Of G-barrelled Topologies In A Group Dualitymentioning

confidence: 99%

“…Thus, whenever the existence of a g-barrelled locally quasi-convex topology in a fixed duality (G, G ∧ ) can be guaranteed, it is unique and it is the Mackey topology. Consequently, if the Mackey topology for a topological group (G, τ ) does not exist, the dual pair (G, G ∧ ) does not contain either a g-barrelled topology on G. However, a Mackey topology may not be g-barrelled: these relationships, together with a grading of the Mackey property, are deeply analyzed in [11]. In Theorem 4.6 we give a necessary and sufficient condition for the existence of a g-barrelled topology in a group duality.…”

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confidence: 99%

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On the existence of topologies compatible with a group duality with predetermined properties

Borsich¹,

Domínguez²,

Martín-Peinador³

2021

Preprint

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The paper deals with group dualities. A group duality is simply a pair (G, H) where G is an abstract abelian group and H a subgroup of characters defined on G. A group topology τ defined on G is compatible with the group duality (also called dual pair) (G, H) if G equipped with τ has dual group H.A topological group (G, τ ) gives rise to the natural duality (G, G ∧ ), where G ∧ stands for the group of continuous characters on G. We prove that the existence of a g-barrelled topology on G compatible with the dual pair (G, G ∧ ) is equivalent to the semireflexivity in Pontryagin's sense of the group G ∧ endowed with the pointwise convergence topology σ(G ∧ , G). We also deal with k-group topologies. We prove that the existence of k-group topologies on G compatible with the duality (G, G ∧ ) is determined by a sort of completeness property of its Bohr topology σ(G, G ∧ ) (Theorem 3.3).For a topological abelian group (G, τ ), denote by G ∧ := CHom(G, T) the group of all continuous characters on G. The weak topology associated to G ∧ is defined as the weakest topology on G for which all the elements of G ∧ are continuous. It is a group topology which will be denoted by τ + (or by σ(G, G ∧ ) if the duality (G, G ∧ ) is the prevailing point of view). Clearly, τ + ≤ τ and it is the bottom element in the duality (G, G ∧ ). By its relationship with the Bohr compactification of (G, τ ), τ + is called the Bohr topology of (G, τ ). It is precompact and Hausdorff provided (G, τ ) has sufficiently many continuous characters. The question of when a precompact and Hausdorff group topology on an abelian group is the Bohr topology corresponding to a locally compact group has been considered in [10], in [15] and recently in [17]. The present paper was originated by a thorough reading of [17].More explicitly, the main question in [17] was: If (G, w) denotes a totally bounded abelian topological group (that is, precompact and Hausdorff), is there a locally compact topology on G, say τ , such that τ + = w? If such τ exists, it can be said in categorical language that (G, w) is the Bohr reflection of (G, τ ). The authors of [17] denote by B the class of all totally bounded abelian groups which are the Bohr reflection of a locally compact group. In the present paper we consider the question from another point of view. As a matter of fact a precompact Hausdorff topological group (G, w) is in B if there is a locally compact topology in the duality (G, G ∧ ), where G ∧ denotes the character group of (G, w). Since in particular, every locally compact abelian group is g-barrelled, the question can be generalized to the following one: Question 1. Let (G, τ ) be an abelian topological group. Under which conditions on G or in G ∧ is there a g-barrelled topology in the duality (G, G ∧ )?The g-barrelled groups were introduced in [7]. In Section 4 we formulate their definition, and we obtain a necessary and sufficient condition for a duality (G, G ∧ ) to contain g-barrelled group topologies (Theorem 4.6).

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Section: On the Existence Of G-barrelled Topologies In A Group Dualitymentioning

confidence: 99%

“…The proof of (i) follows from the fact that the equicontinuous subsets with respect to τ and Qτ coincide ( [11,Proposition 7.1]). The statement (ii) has an straightforward proof.…”

Section: On the Existence Of G-barrelled Topologies In A Group Dualitymentioning

confidence: 99%

mentioning

confidence: 99%

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On the existence of topologies compatible with a group duality with predetermined properties

Borsich¹,

Domínguez²,

Martín-Peinador³

2021

Preprint

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“…This problem is called the Mackey Problem. During recent years several mathematicians have tried to solve the Mackey Problem: de Leo [9], Aussenhofer, Dikranjan, Martín-Peinador [4], Díaz-Nieto [10], Gabriyelyan [11]. A survey including similarities and differences between the Mackey topology in spaces and the Mackey Problem for groups has recently appeared [14].…”

Section: Definition 112 ([12]mentioning

confidence: 99%

Non metrizable topologies on Z with countable dual group.

Mayoral¹

2017

Appl. Gen. Topol.

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In this paper we give two families of non-metrizable topologies on the group of the integers having a countable dual group which is isomorphic to a infinite torsion subgroup of the unit circle in the complex plane. Both families are related to D-sequences, which are sequences of natural numbers such that each term divides the following. The first family consists of locally quasi-convex group topologies. The second consists of complete topologies which are not locally quasi-convex. In order to study the dual groups for both families we need to make numerical considerations of independent interest. 2010 MSC: Primary: 22A05; 55M05; Secondary: 20K45.

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“…(b) The Prüfer groups Z( p ∞ ) endowed with the topology inherited from T are metrizable locally-quasi-convex groups which are not Mackey [8]. (c) The group of rationals Q endowed with the topology inherited from the real line is a non precompact metrizable locally quasi-convex group which is not Mackey [8,16].…”

Section: Example 17mentioning

confidence: 99%

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

et al. 2016

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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 ( -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.

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Characteristics of the Mackey Topology for Abelian Topological Groups

Cited by 10 publications

References 14 publications

On the existence of topologies compatible with a group duality with predetermined properties

On the existence of topologies compatible with a group duality with predetermined properties

Non metrizable topologies on Z with countable dual group.

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

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