Abelian Topological Group Research Articles

Local compactness in free topological groups

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

Banach-Dieudonné Theorem Revisited

AbstractWe prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have calledthe quasi-convex compactness property, or brieflyqcp(Section 3).

Topologies on the direct sum of topological Abelian groups

We prove that the asterisk topologies on the direct sum of topological Abelian groups, used by Kaplan and Banaszczyk in duality theory, are different. However, in the category of locally quasi-convex groups they do not differ, and coincide with the coproduct topology.

A criterion for right continuity of filtrations generated by group-valued additive processes

Let X={ X( t), t∈ T} be a stochastic process with independent increments indexed by the multidimensional set of parameters T=(R +) q, q⩾1 , taking values in a T 0 topological Abelian group G. In this note, we give conditions under which the filtration F t=σ{X(s),s⩽t} , t∈ T, generated by the process X is right continuous, i.e. F t=⋂ v>t F v for all t∈ T.

The free abelian topological group as a subgroup of the free locally convex topological vector space

Article The free abelian topological group as a subgroup of the free locally convex topological vector space was published on March 27, 2003 in the journal Journal of Group Theory (volume 6, issue 3).

Universal countable-dimensional topological groups

We construct a separable metrizable countable-dimensional (respectively strongly countable-dimensional) abelian topological group that contains all countable-dimensional separable metrizable abelian topological absolute G δ -groups (respectively all finite-dimensional separable metrizable abelian topological absolute G δσ -groups). A similar result is proved for separable metrizable abelian topological group that are countable-dimensional with respect to integral cohomological dimension.

Modular functions on multilattices

We prove that every modular function on a multilattice L with values in a topological Abelian group generates a uniformity on L which makes the multilattice operations uniformly continuous with respect to the exponential uniformity on the power set of L.

A characterization of Pontryagin–van Kampen duality for locally convex spaces

Topological vector spaces (TVSs) are topological Abelian groups when considered under the operation of addition. It is therefore natural to ask when they satisfy Pontryagin–van Kampen (P–vK) duality. In 1984 S. Kye published a characterization of P–vK duality for real locally convex spaces (LCSs). His proof however is incorrect. In this paper we offer an alternative characterization of P–vK duality for real LCSs. We also compare our results with some other contributions and state a number of questions.

A CHARACTERIZATION OF PONTRYAGIN-VAN KAMPEN DUALITY FOR COMPLEX LOCALLY CONVEX SPACES

ABSTRACT Topological vector spaces (TVS) are topological Abelian groups when considered under the operation of addition. It is therefore natural to ask when they satisfy Pontryagin-van Kampen (P-vK) duality. A number of attempts have been considered for real LCS. In particular, in a previous paper we showed a characterization of (P-vK) duality for real LCS. In this article we show that this characterization also holds for complex spaces.

Coproducts of abelian topological groups

This paper is a study of certain topological group topologies on the weak or restricted direct sum of a family of abelian topological groups. Amongst the topologies studied are the topology of the coproduct in the category of abelian topological groups and Kaplan's asterisk topology. A simple, explicit description of the coproduct topology is given, and a detailed investigation is made of the relations between the coproduct topology, the asterisk topology and several other topologies of interest. The reflexivity properties of the topological coproduct are also analysed. In particular, it is shown that the coproduct of a family of locally compact Hausdorff abelian topological groups is reflexive if and only if all but countably many of the groups are discrete.

Fréchet-Urysohn spaces in free topological groups

Let F(X) and A(X) be respectively the free topological group and the free Abelian topological group on a Tychonoff space X. For every natural number n we denote by F n (X) (A n (X)) the subset of F(X) (A(X)) consisting of all words of reduced length 3 except for n = 4. In addition, however, there is a first countable, and hence Frechet-Urysohn subspace Y of F(X) (A(X)) which is not contained in any F n (X) (A n (X)). We shall show that if such a space Y is first countable, then it has a special form in F(X) (A(X)). On the other hand, we give an example showing that if the space Y is Frechet-Urysohn, then it need not have the form.

STRONGLY SEQUENTIALLY CONTINUOUS FUNCTIONS

Given a topological abelian group G, we study the class of strongly sequentially continuous functions on G. Strong sequential continuity is a property intermediate between sequential continuity and uniform sequential continuity, which appeared naturally in the study of smooth functions on Banach spaces. In this paper, we shall mainly concentrate on the gap between strong sequential continuity and uniform sequential continuity. It turns out that if G has some completeness property—for example, if it is completely metrizable—then all strongly sequentially continuous functions on G are uniformly sequentially continuous. On the other hand, we exhibit a large and natural class of groups for which the two notions differ. This class is defined by a property reminiscent of the classical Dirichlet theorem; it includes all dense sugroups of R generated by an increasing sequence of Dirichlet sets, and groups of the form (X, w), where X is a separable Banach space failing the Schur property. Finally, we show that the family of bounded, real-valued strongly sequentially continuous functions on G is a closed subalgebra of l∞(G).

Decay measures on locally compact abelian topological groups

We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces MD and MND, where MD is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and MND is the set of measures μ ∈ M such that ν ∉ MD for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μD + μND (μD ∈ MD and μND ∈ MND) there exist a Borel set A = A(μ) such that μD is the restriction of μ to A, therefore the measures μD and μND are singular with respect to each other. The measures μD and μND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from MD and MND.

Pontryagin duality for topological Abelian groups

A topological Abelian group G is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism. We look at the question, set by Kaplan in 1948, of characterizing the topological Abelian groups that are P-reflexive. Thus, we find some conditions on an arbitrary group G that are equivalent to the P-reflexivity of G and give an example that corrects a wrong statement appearing in previously existent characterizations of P-reflexive groups.

On strongly reflexive topological groups

<p>An Abelian topological group G is strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of its dual group G<sup>⋀</sup>, is reflexive.</p> <p>In this paper we prove the following: the annihilator of a closed subgroup of an almost metrizable group is topologically isomorphic to the dual of the corresponding Hausdorff quotient, and an analogous statement holds for the character group of the starting group. As a consequence of this perfect duality, an almost metrizable group is strongly reflexive just if its Hausdorff quotients, as well as the Hausdorff quotients of its dual, are reflexive. The simplification obtained may be significant from an operative point of view.</p>

Topological groups and [formula omitted]-embeddings

The notion of a Moscow space is applied to the study of some problems of topological algebra, following an approach introduced by A.V. Arhangel'skii [Comment. Math. Univ. Carolin. 41 (2000) 585–595]. In particular, many new, and, it seems, unexpected, solutions to the equation νX×νY=ν(X×Y) are identified. We also find new large classes of topological groups G, for which the operations in G can be extended to the Dieudonné completion of the space G in such a way that G becomes a topological subgroup of the topological group μG. On the other hand, it was shown by A.V. Arhangel'skii [Comment. Math. Univ. Carolin. 41 (2000) 585–595] that there exists an Abelian topological group G for which such an extension is impossible (this provided an answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985). Some new open questions are formulated.

Covering group theory for compact groups

We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism σ : G ̄ →G in this category. The abelian topological group π 1 K (G)≔ ker σ is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems.

Weakly complete free topological groups

A topological group G is sequentially complete if it is sequentially closed in any other topological group. We show that for a Tychonoff space X, the free topological group F( X) is sequentially complete iff the free Abelian topological group A( X) is sequentially complete iff X is sequentially closed in βX. Furthermore, the free precompact Abelian group F(X, PA ) is sequentially complete iff the space X is sequentially closed in βX. We consider also other forms of weak completeness, namely ω-completeness and k-completeness, introduced analogously by means of the ω-closure and the k-closure. We prove that the groups A( X) and F( X) are ω-complete ( k-complete) iff X is ω-closed ( k-closed) in the Dieudonné completion μX of X.

Invariance of compactness for the Bohr topology☆☆Research partially supported by Spanish DGES, grant number PB96-1075, and Fundació Caixa Castelló, grant number P1B98-24.

We define the g-extension of a topological Abelian group G as the set of all characters on G ̂ such that the restriction to every equicontinuous subset of G ̂ is continuous with respect to the pointwise convergence topology. A g-group is a topological Abelian group (G,τ) such that its g-extension coincides with its completion. The Bohr topology of a topological group (G,τ) is the topology that the group inherits as a subset of its Bohr compactification. A topological group (G,τ) respects a property P if the subsets A of G that satisfy the property P are exactly the same for the Bohr topology and for the original topology of the group [Trigos-Arrieta, J. Pure Appl. Algebra 70 (1991) 199]. All groups here are assumed to be Abelian. We prove that every complete g-group when endowed with its Bohr topology is a μ -space. As a consequence, we obtain that for a complete g-group the properties of respecting functionally boundedness, pseudocompactness, countable compactness and compactness are all equivalent and a characterization of this property is also provided. Finally, we extend a theorem of Rosenthal about the existence of sequences equivalent to the ℓ 1 -basis. We prove that for a Čech-complete g-group the property of respecting compactness is equivalent to the existence of conveniently placed sequences equivalent to the ℓ 1 -basis.

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.