A survey on reflexivity of abelian topological groups

Chasco, M.J.; Dikranjan, Dikran; Martín-Peinador, Elena

doi:10.1016/j.topol.2012.04.012

Cited by 14 publications

(14 citation statements)

References 77 publications

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“…Since |X| = |Z N | = c, the topological groups Z X and Z c are topologically isomorphic. Thus, we obtain the following Recall that a topological abelian group G is called strongly reflexive if all closed subgroups and all quotient groups of G are reflexive; see, for example, [6]. Our last corollary provides a counter-example to [14,Corollary 4.8].…”

Section: Proposition 23 and Corollary 32 Imply The Followingmentioning

confidence: 86%

A countable free closed non-reflexive subgroup of ℤ^{𝔠}

Ferrer

Hernández

Shakhmatov

2017

Proc. Amer. Math. Soc.

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We prove that the group G = Hom(Z N , Z) of all homomorphisms from the Baer-Specker group Z N to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual of G is discrete. As G is non-discrete, it is not reflexive. Since G can be viewed as a closed subgroup of the Tychonoff product Z c of continuum many copies of the integers Z, this provides an example of a group described in the title, thereby answering Problem 11 from [J. Galindo, L. Recorder-Núñez, M. Tkachenko, Reflexivity of prodiscrete topological groups, J. Math. Anal. Appl. 384 (2011), 320-330]. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.

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Section: Proposition 23 and Corollary 32 Imply The Followingmentioning

confidence: 86%

A countable free closed non-reflexive subgroup of ℤ^{𝔠}

Ferrer

Hernández

Shakhmatov

2017

Proc. Amer. Math. Soc.

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“…The pseudocompletion X ▽ can be conceived as an envelope (in the sense of [6]) of the locally convex space X in the class PC of all pseudocomplete locally convex spaces with respect to the same class PC: (11) X ▽ = Env PC PC X = Env PC X (this fact follows from Theorem 3.1 and the remark at page 50 of the work [6]). This operation is similar to the usual completion in that it does not change the topology of the space X (i.e.…”

Section: Stereotype Spacesmentioning

confidence: 99%

Kernels and cokernels in the category of augmented involutive stereotype algebras

Akbarov

2019

J. Algebra Appl.

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We prove several properties of kernels and cokernels in the category of augmented involutive stereotype algebras: (1) this category has kernels and cokernels, (2) the cokernel is preserved under the passage to the group stereotype algebras, and (3) the notion of cokernel allows to prove that the continuous envelope [Formula: see text] of the group algebra of a compact buildup of an abelian locally compact group is an involutive Hopf algebra in the category of stereotype spaces [Formula: see text]. The last result plays an important role in the generalization of the Pontryagin duality for arbitrary Moore groups.

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“…The exactness is equivalent to the property of all continuous (with respect to the current topology) characters of a closed subgroup to be extendable to continuous characters of the full group. In the terminology of [2], in order the sequences to be exact, the closed image of each term of the sequence in the next term needs to be a dually embedded subgroup of the next term. 4.6.…”

Section: 5mentioning

confidence: 99%

Generalised Kawada–Satake method for Mackey functors in class field theory

Fesenko

Востоков

Yoon

2018

European Journal of Mathematics

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We propose and study a generalised Kawada-Satake method for Mackey functors in the class field theory of positive characteristic. The root of this method is in the use of explicit pairings, such as the Artin-Schreier-Witt pairing, for groups describing abelian extensions. We separate and simplify the algebraic component of the method and discuss a relation between the existence theorem in class field theory and topological reflexivity with respect to the explicit pairing. We apply this method to derive higher local class field theory of positive characteristic, using advanced properties of topological Milnor K-groups of such fields. CONTENTS

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A survey on reflexivity of abelian topological groups

Cited by 14 publications

References 77 publications

A countable free closed non-reflexive subgroup of ℤ^{𝔠}

A countable free closed non-reflexive subgroup of ℤ^{𝔠}

Kernels and cokernels in the category of augmented involutive stereotype algebras

Generalised Kawada–Satake method for Mackey functors in class field theory

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