Person:
Tarieladze, Vaja

First Name

Vaja

Last Name

Tarieladze

URI

https://hdl.handle.net/20.500.14352/82396

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Matemáticas

Area

Geometría y Topología

Identifiers

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Now showing 1 - 8 of 8

Eberlein–Šmulyan theorem for Abelian topological groups
(Journal of the London Mathematical Society. Second Series, 2004) Bruguera Padró, M. Montserrat; Martín Peinador, Elena; Tarieladze, Vaja
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.
A property of Dunford-Pettis type in topological groups
(Proceedings of the American Mathematical Society, 2003) Martín Peinador, Elena; Tarieladze, Vaja
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.
Group valued null sequences and metrizable non-Mackey groups
(Forum Mathematicum, 2014) Dikranjan, Dikran; Martín Peinador, Elena; Tarieladze, Vaja
For a topological abelian group X we topologize the group c0(X) of all X-valued null sequences in a way such that when X= the topology of c0() coincides with the usual Banach space topology of the classical Banach space c0. If X is a non-trivial compact connected metrizable group, we prove that c0(X) is a non-compact Polish locally quasi-convex group with countable dual group c0(X). Surprisingly, for a compact metrizable X, countability of c0(X) leads to connectedness of X. Our principal application of the above results is to the class of locally quasi-convex Mackey groups (LQC-Mackey groups). A topological group (G,) from a class of topological abelian groups will be called a Mackey group in or a -Mackey group if it has the following property: if is a group topology in G such that (G,) and (G,) has the same character group as (G,), then . Based upon the results obtained for c0(X), we provide a large family of metrizable precompact (hence, locally quasi-convex) connected groups which are not LQC-Mackey. Namely, we show that for a connected compact metrizable group X0, the group c0(X), endowed with the topology induced from the product topology on X, is a metrizable precompact connected group which is not a Mackey group in LQC. Since metrizable locally convex spaces always carry the Mackey topology – a well-known fact from Functional Analysis –, our results prove that a Mackey theory for abelian groups is not a simple traslation of items known to hold for locally convex spaces. This paper is a contribution to the Mackey theory for groups, where properties of a topological nature like compactness or connectedness have an important role.
Countable powers of compact Abelian groups in the uniform topology and cardinality of their dual groups
(2014) Dikranjan, Dikran; Martín Peinador, Elena; Tarieladze, Vaja
We equip the product of countably many copies of a compact Abelian group X with the uniform topology, and study some properties of the topological group G thus obtained. In particular, we determine the cardinality of the dual group of G, when X is the circle group: it is precisely 2^c.
On Mackey topology for groups
(Studia Mathematica, 1999) Martín Peinador, Elena; Chasco, M.J.; Tarieladze, Vaja
The present paper is a contribution to fill in a gap existing between the theory of topological vector spaces and that of topological abelian groups. Topological vector spaces have been extensively studied as part of Functional Analysis. It is natural to expect that some important and elegant theorems about topological vector spaces may have analogous versions for abelian topological groups. The main obstruction to get such versions is probably the lack of the notion of convexity in the framework of groups. However, the introduction of quasi-convex sets and locally quasi-convex groups by Vilenkin [26] and the work of Banaszczyk [1] have paved the way to obtain theorems of this nature. We study here the group topologies compatible with a given duality. We have obtained, among others, the following result: for a complete metrizable topological abelian group, there always exists a finest locally quasi-convex topology with the same set of continuous characters as the original topology. We also give a description of this topology as an G-topology and we prove that, for the additive group of a complete metrizable topological vector space, it coincides with the ordinary Mackey topology.
Lindelöf spaces C(X) over topological groups
(Forum Mathematicum, 2008) Kąkol, Jerzy; López Pellicer, Manuel; Martín Peinador, Elena; Tarieladze, Vaja
Theorem 1 proves (among the others) that for a locally compact topological group X the following assertions are equivalent: (i) X is metrizable and sigma-compact. (ii) C-p(X) is analytic. (iii) C-p(X) is K-analytic. (iv) C-p(X) is Lindelof. (v) C-c(X) is a separable metrizable and complete locally convex space. (vi) C,(X) is compactly dominated by irrationals. This result supplements earlier results of Corson, Christensen and Calbrix and provides several applications, for example, it easily applies to show that: (1) For a compact topological group X the Eberlein, Talagrand, Gul'ko and Corson compactness are equivalent and any compact group of this type is metrizable. (2) For a locally compact topological group X the space C-p(X) is Lindelof iff C-c(X) is weakly Lindelof. The proofs heavily depend on the following result of independent interest: A locally compact topological group X is metrizable iff every compact subgroup of X has countable tightness (Theorem 2). More applications of Theorem 1 and Theorem 2 are provided.
Conway’s Question: The Chase for Completeness
(Applied Categorical Structures, 2007) Dikranjan, Dikran; Martín Peinador, Elena; Tarieladze, Vaja
We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–Čech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Raĭkov completion . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.
On Ultrabarrelled Spaces, their Group Analogs and Baire Spaces
(Descriptive Topology and Functional Analysis II, 2019) Dominguez, X; Martín Peinador, Elena; Tarieladze, Vaja
Let E and F be topological vector spaces and let G and Y be topological abelian groups. We say that E is sequentially barrelled with respect to F if every sequence (un)n∈N of continuous linear maps from E to F which converges pointwise to zero is equicontinuous. We say that G is barrelled with respect to F if every set H of continuous homomorphisms from G to F, for which the set H(x) is bounded in F for every x∈E, is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every H⊆CHom(G,Y) which is compact in the product topology of YG is equicontinuous. We prove that - a barrelled normed space may not be sequentially barrelled with respect to a complete metrizable locally bounded topological vector space, - a topological group which is a Baire space is barrelled with respect to any topological vector space, - a topological group which is a Namioka space is g-barrelled with respect to any metrizable topological group, - a protodiscrete topological abelian group which is a Baire space may not be g-barrelled (with respect to R/Z).