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Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday

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A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: closed (15 March 2022) | Viewed by 22711

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Special Issue Editors

Prof. Dr. Elena Martín-Peinador

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Guest Editor

Institute of Interdisciplinary Mathematics and Department of Algebra Geometry and Topology, Complutense University of Madrid, 28040 Madrid, Spain
Interests: topology; functional analysis; topological groups

Prof. Dr. Mikhail Tkachenko

E-Mail Website
Co-Guest Editor

Department of Mathematics, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, C.P. 09340, Mexico City, Mexico
Interests: topological algebra (semitopological, paratopological and topological groups); general topology

Prof. Dr. T. Christine Stevens

E-Mail Website
Co-Guest Editor

Department of Mathematics and Statistics, Saint Louis University, 220 N. Grand Blvd., Saint Louis, MO 63103, USA
Interests: topological groups

Prof. Dr. Xabier Domínguez

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Co-Guest Editor

Departamento de Matemáticas, Universidade da Coruña, E.T.S. I. de Caminos, Canales y Puertos, Campus de Elviña, 15071 A Coruña, Spain
Interests: topological groups; duality of topological abelian groups

Special Issue Information

Dear Colleagues,

The interaction between topology and functional analysis has been a wellspring of powerful mathematical ideas and developments since the early stages of both disciplines. One of the many important programs originating within this framework can be described as borrowing some of the tools and concepts of topological vector space theory to study the structure and duality properties of abelian topological groups. Such a viewpoint turns out to be particularly useful, for instance, when dealing with Pontryagin duality and reflexivity outside the class of locally compact groups.

We are preparing a Special Issue in Axioms under the title “Topology and Functional Analysis”, to showcase recent work on these and related topics and pay tribute to María Jesús Chasco's mathematical career on the occasion of her 65th birthday. She has been a key contributor to the field for thirty years and maintains a very fruitful collaboration network with other researchers from Spain and abroad.

Full research papers on these topics, as well as review papers, can be included in this issue. Submitted papers should not exceed 20 pages and should follow the usual guidelines for publication in Axioms.

Prof. Dr. Elena Martín-Peinador
Prof. Dr. Mikhail Tkachenko
Prof. Dr. T. Christine Stevens
Prof. Dr. Xabier Domínguez
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Topological groups
Locally convex spaces
Duality theory
Topological algebra
Descriptive topology
Geometric topology
Dynamical systems

Published Papers (15 papers)

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Editorial

Jump to: Research, Review

5 pages, 213 KiB

Open AccessEditorial

Advance in Topology and Functional Analysis in Honour of María Jesús Chasco’s 65th Birthday

by Xabier Domínguez, Elena Martín-Peinador, T. Christine Stevens and Mikhail Tkachenko

Axioms 2024, 13(4), 266; https://doi.org/10.3390/axioms13040266 - 18 Apr 2024

Viewed by 335

Abstract

We are honoured to present this Special Issue of Axioms with the title “Topology and Functional Analysis” to showcase recent work on this and related topics and to provide an opportunity for María Jesús Chasco’s friends and colleagues to pay tribute to her [...] Read more.

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

Research

Jump to: Editorial, Review

32 pages, 464 KiB

Open AccessArticle

Permutations, Signs, and Sum Ranges

by Sergei Chobanyan, Xabier Domínguez, Vaja Tarieladze and Ricardo Vidal

Axioms 2023, 12(8), 760; https://doi.org/10.3390/axioms12080760 - 01 Aug 2023

Viewed by 738

Abstract

The sum range

SR [x; X]

, for a sequence

x = {(x_{n})}_{n \in N}

of elements of a topological vector space X, is defined as the set of all elements

s \in X

for which there [...] Read more.

The sum range

SR [x; X]

, for a sequence

x = {(x_{n})}_{n \in N}

of elements of a topological vector space X, is defined as the set of all elements

s \in X

for which there exists a bijection (=permutation)

π : N \to N

, such that the sequence of partial sums

{(\sum_{k = 1}^{n} x_{π (k)})}_{n \in N}

converges to s. The sum range problem consists of describing the structure of the sum ranges for certain classes of sequences. We present a survey of the results related to the sum range problem in finite- and infinite-dimensional cases. First, we provide the basic terminology. Next, we devote attention to the one-dimensional case, i.e., to the Riemann–Dini theorem. Then, we deal with spaces where the sum ranges are closed affine for all sequences, and we include some counterexamples. Next, we present a complete exposition of all the known results for general spaces, where the sum ranges are closed affine for sequences satisfying some additional conditions. Finally, we formulate two open questions. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

15 pages, 335 KiB

Open AccessFeature PaperArticle

Krein’s Theorem in the Context of Topological Abelian Groups

by Tayomara Borsich, Xabier Domínguez and Elena Martín-Peinador

Axioms 2022, 11(5), 224; https://doi.org/10.3390/axioms11050224 - 12 May 2022

Viewed by 1777

Abstract

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 [...] Read more.

G^{\land}

, 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. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

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38 pages, 574 KiB

Open AccessFeature PaperArticle

Aspects of Differential Calculus Related to Infinite-Dimensional Vector Bundles and Poisson Vector Spaces

by Helge Glöckner

Axioms 2022, 11(5), 221; https://doi.org/10.3390/axioms11050221 - 09 May 2022

Cited by 1 | Viewed by 1689

Abstract

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter

k_{R}

-spaces and locally convex spaces E such that

E \times E

is a

k_{R}

-space. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

22 pages, 415 KiB

Open AccessArticle

On the Group of Absolutely Summable Sequences

by Lydia Außenhofer

Axioms 2022, 11(5), 218; https://doi.org/10.3390/axioms11050218 - 07 May 2022

Viewed by 1326

Abstract

For an abelian topological group G, the sequence group

ℓ^{1} (G)

of all absolutely summable sequences in G is studied. It is shown that

ℓ^{1} (G)

is a Pontryagin reflexive group in case G is a [...] Read more.

For an abelian topological group G, the sequence group

ℓ^{1} (G)

of all absolutely summable sequences in G is studied. It is shown that

ℓ^{1} (G)

is a Pontryagin reflexive group in case G is a reflexive metrizable group or an LCA group. Further,

ℓ^{1} (G)

has the Schur property if and only if G has it and

ℓ^{1} (G)

is a Schwartz group if and only if G is linearly topologized. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

31 pages, 506 KiB

Open AccessArticle

A Distinguished Subgroup of Compact Abelian Groups

by Dikran Dikranjan, Wayne Lewis, Peter Loth and Adolf Mader

Axioms 2022, 11(5), 200; https://doi.org/10.3390/axioms11050200 - 24 Apr 2022

Cited by 1 | Viewed by 1713

Abstract

Here “group” means additive abelian group. A compact group G contains

δ

–subgroups, that is, compact totally disconnected subgroups

Δ

such that

G / Δ

is a torus. The canonical subgroup

Δ (G)

of G that is the sum of all [...] Read more.

Here “group” means additive abelian group. A compact group G contains

δ

–subgroups, that is, compact totally disconnected subgroups

Δ

such that

G / Δ

is a torus. The canonical subgroup

Δ (G)

of G that is the sum of all

δ

–subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of

Δ (G)

when considering only finite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G, we prove the following:

Δ (G)

contains

tor (G)

, is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G, and

G / Δ (G)

is torsion-free and divisible;

Δ (G)

is a functorial subgroup of G, it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G. The subgroup

Δ (G)

appeared before in the literature as

td (G)

motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor, preserves proper exactness of short sequences of compact groups. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

8 pages, 261 KiB

Open AccessFeature PaperArticle

Bounded Sets in Topological Spaces

by Cristina Bors, María V. Ferrer and Salvador Hernández

Axioms 2022, 11(2), 71; https://doi.org/10.3390/axioms11020071 - 10 Feb 2022

Viewed by 1927

Abstract

Let G be a monoid that acts on a topological space X by homeomorphisms such that there is a point

x_{0} \in X

with

G U = X

for each neighbourhood U of

x_{0}

. A subset A of X is [...] Read more.

Let G be a monoid that acts on a topological space X by homeomorphisms such that there is a point

x_{0} \in X

with

G U = X

for each neighbourhood U of

x_{0}

. A subset A of X is said to be G-bounded if for each neighbourhood U of

x_{0}

there is a finite subset F of G with

A \subseteq F U

. We prove that for a metrizable and separable G-space X, the bounded subsets of X are completely determined by the bounded subsets of any dense subspace. We also obtain sufficient conditions for a G-space X to be locally G-bounded, which apply to topological groups. Thereby, we extend some previous results accomplished for locally convex spaces and topological groups. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

7 pages, 237 KiB

Open AccessArticle

Series with Commuting Terms in Topologized Semigroups

by Alberto Castejón, Eusebio Corbacho and Vaja Tarieladze

Axioms 2021, 10(4), 237; https://doi.org/10.3390/axioms10040237 - 24 Sep 2021

Cited by 1 | Viewed by 1149

Abstract

We show that the following general version of the Riemann–Dirichlet theorem is true: if every rearrangement of a series with pairwise commuting terms in a Hausdorff topologized semigroup converges, then its sum range is a singleton. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

11 pages, 253 KiB

Open AccessArticle

An Expository Lecture of María Jesús Chasco on Some Applications of Fubini’s Theorem

by Alberto Castejón, María Jesús Chasco, Eusebio Corbacho and Virgilio Rodríguez de Miguel

Axioms 2021, 10(3), 225; https://doi.org/10.3390/axioms10030225 - 14 Sep 2021

Viewed by 1370

Abstract

The usefulness of Fubini’s theorem as a measurement instrument is clearly understood from its multiple applications in Analysis, Convex Geometry, Statistics or Number Theory. This article is an expository paper based on a master class given by the second author at the University of Vigo and is devoted to presenting some Applications of Fubini’s theorem. In the first part, we present Brunn–Minkowski’s and Isoperimetric inequalities. The second part is devoted to the estimations of volumes of sections of balls in

R^{n}

. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

6 pages, 243 KiB

Open AccessArticle

Normed Spaces Which Are Not Mackey Groups

by Saak Gabriyelyan

Axioms 2021, 10(3), 217; https://doi.org/10.3390/axioms10030217 - 08 Sep 2021

Cited by 2 | Viewed by 1181

Abstract

It is well known that every normed (even quasibarrelled) space is a Mackey space. However, in the more general realm of locally quasi-convex abelian groups an analogous result does not hold. We give the first examples of normed spaces which are not Mackey [...] Read more.

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

8 pages, 246 KiB

Open AccessArticle

On Self-Aggregations of Min-Subgroups

by Carlos Bejines, Sergio Ardanza-Trevijano and Jorge Elorza

Axioms 2021, 10(3), 201; https://doi.org/10.3390/axioms10030201 - 24 Aug 2021

Cited by 2 | Viewed by 1405

Abstract

Preservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-subgroups has only been studied for binary aggregation functions. However, results concerning preservation of the min-subgroup structure under binary aggregations do not generalize to aggregation functions with arbitrary input size since they are not associative. In this article, we prove that arbitrary self-aggregation functions preserve the min-subgroup structure. Moreover, we show that whenever the aggregation function is strictly increasing on its diagonal, a min-subgroup and its self-aggregation have the same level sets. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

12 pages, 332 KiB

Open AccessArticle

Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

by Mikhail G. Tkachenko

Axioms 2021, 10(3), 167; https://doi.org/10.3390/axioms10030167 - 28 Jul 2021

Viewed by 1255

Abstract

This study is on the factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let

D = \prod_{i \in I} D_{i}

be a product of paratopological [...] Read more.

D = \prod_{i \in I} D_{i}

be a product of paratopological groups, S be a dense subgroup of D, and

χ

a continuous character of S. Then one can find a finite set

E \subset I

and continuous characters

χ_{i}

D_{i}

, for

i \in E

, such that

χ = (\prod_{i \in E} χ_{i} \circ p_{i}) S

, where

p_{i} : D \to D_{i}

is the projection. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

7 pages, 295 KiB

Open AccessArticle

Distinguished Property in Tensor Products and Weak* Dual Spaces

by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López

Axioms 2021, 10(3), 151; https://doi.org/10.3390/axioms10030151 - 08 Jul 2021

Cited by 3 | Viewed by 1337

Abstract

A local convex space E is said to be distinguished if its strong dual

E_{β}^{'}

has the topology

β (E^{'}, {(E_{β}^{'})}^{'})

, i.e., if

E_{β}^{'}

is barrelled. The distinguished property [...] Read more.

A local convex space E is said to be distinguished if its strong dual

E_{β}^{'}

has the topology

β (E^{'}, {(E_{β}^{'})}^{'})

, i.e., if

E_{β}^{'}

is barrelled. The distinguished property of the local convex space

C_{p} (X)

of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and

C_{p}

-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space

C_{p} (X)

is distinguished if and only if any function

f \in R^{X}

belongs to the pointwise closure of a pointwise bounded set in

C (X)

. The extensively studied distinguished properties in the injective tensor products

C_{p} (X) \otimes_{ε} E

and in

C_{p} (X, E)

contrasts with the few distinguished properties of injective tensor products related to the dual space

L_{p} (X)

C_{p} (X)

endowed with the weak* topology, as well as to the weak* dual of

C_{p} (X, E)

. To partially fill this gap, some distinguished properties in the injective tensor product space

L_{p} (X) \otimes_{ε} E

are presented and a characterization of the distinguished property of the weak* dual of

C_{p} (X, E)

for wide classes of spaces X and E is provided. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

3 pages, 197 KiB

Open AccessArticle

On Factoring Groups into Thin Subsets

by Igor Protasov

Axioms 2021, 10(2), 89; https://doi.org/10.3390/axioms10020089 - 14 May 2021

Viewed by 1159

Abstract

A subset X of a group G is called thin if, for every finite subset F of G, there exists a finite subset H of G such that

F x \cap F y = \emptyset

, [...] Read more.

A subset X of a group G is called thin if, for every finite subset F of G, there exists a finite subset H of G such that

F x \cap F y = \emptyset

x F \cap y F = \emptyset

for all distinct

x, y \in X \ H

. We prove that every countable topologizable group G can be factorized

G = A B

into thin subsets

A, B

. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

Review

Jump to: Editorial, Research

13 pages, 286 KiB

Open AccessReview

Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century

by Karl H. Hofmann and Sidney A. Morris

Axioms 2021, 10(3), 190; https://doi.org/10.3390/axioms10030190 - 17 Aug 2021

Cited by 3 | Viewed by 2384

Abstract

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to

R^{I} \times C

for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra

R [G]

of a finite group G to its representation algebra

R (G, R)

, via the natural duality of the topological vector space

R^{I}

to the vector space

R^{(I)}

, for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

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