Person:
Martín Peinador, Elena

First Name

Elena

Last Name

Martín Peinador

URI

https://hdl.handle.net/20.500.14352/83483

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Matemáticas

Area

Geometría y Topología

Identifiers

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Now showing 1 - 10 of 49

On strongly reflexive topological groups
(Applied General Topology, 2001) Chasco, M.J.; Martín Peinador, Elena
Let Gˆ denote the Pontryagin dual of an abelian topological group G. Then G is reflexive if it is topologically isomorphic to Gˆˆ, strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of Gˆ is reflexive. It is well known that locally compact abelian (LCA) groups are strongly reflexive. W. Banaszczyk [Colloq. Math. 59 (1990), no. 1, 53–57], extending an earlier result of R. Brown, P. J. Higgins and S. A. Morris [Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32], showed that all countable products and sums of LCA groups are strongly reflexive. L. Aussenhofer [Dissertationes Math. (Rozprawy Mat.) 384 (1999), 113 pp.] showed that all Čech-complete nuclear groups are strongly reflexive. It is an open question whether the strongly reflexive groups are exactly the Čech-complete nuclear groups and their duals. A Hausdorff topological group G is almost metrizable if and only if it has a compact subgroup K such that G/K is metrizable [W. Roelcke and S. Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill, New York, 1981]. In this paper it is shown that the annihilator of a closed subgroup of an almost metrizable group G is topologically isomorphic to the dual of the corresponding Hausdorff quotient, and an analogous statement holds for the character group of G. It then follows that an almost metrizable group is strongly reflexive only if its Hausdorff quotients and those of its dual are reflexive.
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.
Krein’s Theorem in the Context of Topological Abelian Groups
(Axioms, 2022) Borsich, Tayomara; Domínguez Pérez, Xabier E; Martín Peinador, Elena
A topological abelian group G is said to have the quasi-convex compactness property (briefly, qcp) if the quasi-convex hull of every compact subset of G is again compact. In this paper we prove that there exist locally quasi-convex metrizable complete groups G which endowed with the weak topology associated to their character groups G∧, do not have the qcp. Thus, Krein’s Theorem, a well known result in the framework of locally convex spaces, cannot be fully extended to locally quasi-convex groups. Some features of the qcp are also studied.
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.
Sistemas L de rayos y sumabilidad
(Contribuciones matemáticas : Libro-homenaje al profesor Francisco Botella Radúan, 1986) Plans, A.; Martín Peinador, Elena; Outerelo Dominguez, Enrique
The work under consideration fits into the following general circle of problems: Given a Banach space B which possesses a distinguished basis (ei) i∈N and a bounded linear operator A:B→C, to what extent does the sequence (Aei) constitute some sort of basis (ai) on C, where ai=Aei? It turns out to be more suitable to work with systems of rays (ri)i∈N (that is, one-dimensional subspaces) such that ai∈ri. Compact operators A are excluded for what turn out to be obvious reasons, and the operators A are required to be injective. This leads at various points to a consideration of cases: the range R(A) of A is a closed subspace; and R(A) is dense in C but not equal to C. The paper is devoted principally to the case B=C=l2 with the distinguished basis (ei) being a complete orthonormal set (c.o.s.). There are also results applying to lp, p≠2. A sequence (ai) is said to be doubly bounded (d.b.) provided that 00 for some particular c.o.s. (e′i) contained in a previously fixed linear subspace dense in l 2. Henceforth A represents a noncompact injective operator with R(A)≠R(A) ¯ ¯ ¯ . It is proved that there exists a c.o.s. (e′i) such that the L-system (Ae′i) is d.b., complete in R(A) ¯ ¯ ¯ and heterogonal in blocks. Furthermore, it is shown that for a noncompact A, the following are equivalent: (i) N(A)=0. (ii) There exists a c.o.s. (ei) such that (Aei) is a strong M base which is d.b. in R(A) ¯ ¯ ¯ . Definition: A sequence (ai) is minimal if ai is not in the closed linear subspace spanned by the aj, j≠i. An M-base is a complete minimal sequence (ai) such that ⋂ ∞ i=1 [a i ,a i+1 ,⋯]=0 where [a i ,a i+1 ⋯] represents the closed linear subspace spanned by ai, a i+1,⋯. A further theorem states that given a sequence of rays (ri) the following are equivalent: (I) (ri) is an SLR; (II) for (ai∈ri∖{0}) one has ∑ ∞ i=1||ai||2<∞ if and only if (ai) is summable. A final section of the paper is devoted to lp, p≠2. It is shown here that if p>2 and (xn) is a d.b. sequence in lp then the following are equivalent: (1) (xn) is weakly p-summable; (2) ∑ ∞ 1 ξ n x n converges unconditionally if and only if ∑ ∞ 1 |ξ n | p′ <∞, where 1/p+1/p′ =1. For p<2, there are no d.b. weakly summable sequences in lp.
Completeness properties of locally quasi-convex groups
(Topology and its Applications, 2000) Bruguera Padró, M. Montserrat; Chasco, M.J.; Martín Peinador, Elena; Tarieladze, Vaja
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. (C) 2001 Elsevier Science B.V.
Special Issue: Algebra meets Topology Special Issue on Dikran Dikranjan's 60th Birthday Preface
(Topology and its Applications, 2012) Ardanza Trevijano, Sergio; Chasco, M.J.; Hofmann, Karl H.; Martín Peinador, Elena; Shakhmatov, Dmitri
A characterization of K-analyticity of groups of continuous homomorphisms
(Boletín de la Sociedad Matemática Mexicana, 2008) Kąkol, Jerzy; Martín Peinador, Elena; Moll, Santiago
For an abelian locally compact group X let X^p be the group of continuous homomorphisms from X into the unit circle T of the complex plane endowed with the pointwise convergence topology. It is proved that X is metrizable iff X^p is K-analytic iff X endowed with its Bohr topology σ(X,X^) has countable tightness. Using this result, we establish a large class of topological groups with countable tightness which are not sequential, so neither Fréchet-Urysohn
Completeness properties of group topologies for R
(Topology and its Applications, 2015) Martín Peinador, Elena; Stevens, T. Christine
We study the completeness properties of several different group topologies for the additive group of real numbers, and we also compute the corresponding dual groups. We first present two metrizable connected group topologies on R with topologically isomorphic dual groups, one of which is noncomplete and arcwise connected and the other one is compact (therefore complete), but not arcwise connected. Using a theorem about T -sequences and adapting a result about weakened analytic groups, we then describe a method for obtaining Hausdorff group topologies R that are strictly weaker than the usual topology and are complete. They are not Baire, and consequently not metrizable.
Bounded duality in topological abelian groups
(Functional Analysis and Continuous Optimization. IMFACO 2022, 2023) Martín Peinador, Elena; Chasco, M. J.; Amigó, J. M.; Cánovas, M. J.; López-Cerdá, M. A.; López-Pellicer, M.
We define the β-duality for topological Abelian groups by means of the notion of Hejcman of boundedness in uniform spaces. A real locally convex space considered as an Abelian topological group is β-reflexive iff it is reflexive in the ordinary sense for locally convex spaces. Thus, β-reflexivity is the natural extension to Abelian topological groups of the well-known notion of reflexivity. We prove: 1) A locally compact Abelian group is β-reflexive. 2) A β-reflexive metrizable group is reflexive in Pontryagin sense. 3) The β-bidual of a metrizable group is also a metrizable group.