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220 KiB

Open AccessEditorial

An Overview of Topological Groups: Yesterday, Today, Tomorrow

by Sidney A. Morris

Axioms 2016, 5(2), 11; https://doi.org/10.3390/axioms5020011 - 5 May 2016

Viewed by 4651

Abstract

It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of [...] Read more.

It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups.[...] Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

Research

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296 KiB

Open AccessArticle

Non-Abelian Pseudocompact Groups

by W. W. Comfort and Dieter Remus

Axioms 2016, 5(1), 2; https://doi.org/10.3390/axioms5010002 - 12 Jan 2016

Cited by 3 | Viewed by 4406

Abstract

Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2^|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 2^{2^|K|} -many strictly finer pseudocompact topological group refinements. [...] Read more.

Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2^|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 2^{2^|K|} -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudocompact subgroup and a strictly finer pseudocompact topological group refinement. (Theorems (A), (B) and (C) become false if the non-metrizable hypothesis is omitted.) With a detailed view toward the relevant literature, the present authors ask: What happens to (A), (B), (C) and to similar known facts about pseudocompact abelian groups if the abelian hypothesis is omitted? Are the resulting statements true, false, true under certain natural additional hypotheses, etc.? Several new results responding in part to these questions are given, and several specific additional questions are posed. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

300 KiB

Open AccessArticle

Free Boolean Topological Groups

by Ol’ga Sipacheva

Axioms 2015, 4(4), 492-517; https://doi.org/10.3390/axioms4040492 - 3 Nov 2015

Cited by 14 | Viewed by 5287

Abstract

Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is placed on the application [...] Read more.

Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

352 KiB

Open AccessArticle

Characterized Subgroups of Topological Abelian Groups

by Dikran Dikranjan, Anna Giordano Bruno and Daniele Impieri

Axioms 2015, 4(4), 459-491; https://doi.org/10.3390/axioms4040459 - 16 Oct 2015

Cited by 9 | Viewed by 4787

Abstract

A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (v_n) of characters of X if H = {x ∈ X : v_n(x) → 0 in T}. [...] Read more.

A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (v_n) of characters of X if H = {x ∈ X : v_n(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of auto-characterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proven to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

336 KiB

Open AccessArticle

Locally Quasi-Convex Compatible Topologies on a Topological Group

by Lydia Außenhofer, Dikran Dikranjan and Elena Martín-Peinador

Axioms 2015, 4(4), 436-458; https://doi.org/10.3390/axioms4040436 - 13 Oct 2015

Cited by 9 | Viewed by 5010

Abstract

For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. [...] Read more.

For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9). Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

192 KiB

Open AccessArticle

Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds

by Morris W. Hirsch

Axioms 2015, 4(3), 313-320; https://doi.org/10.3390/axioms4030313 - 27 Jul 2015

Cited by 2 | Viewed by 4541

Abstract

I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

263 KiB

Open AccessArticle

Pro-Lie Groups: A Survey with Open Problems

by Karl H. Hofmann and Sidney A. Morris

Axioms 2015, 4(3), 294-312; https://doi.org/10.3390/axioms4030294 - 24 Jul 2015

Cited by 16 | Viewed by 5991

Abstract

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete [...] Read more.

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

235 KiB

Open AccessArticle

Lindelöf Σ-Spaces and R-Factorizable Paratopological Groups

by Mikhail Tkachenko

Axioms 2015, 4(3), 254-267; https://doi.org/10.3390/axioms4030254 - 10 Jul 2015

Cited by 3 | Viewed by 4459

Abstract

We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition, G is regular, then it is totally w-narrow and satisfies cel_w [...] Read more.

We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition, G is regular, then it is totally w-narrow and satisfies cel_w(G) ≤ w, and the Hewitt–Nachbin completion of G is again an R-factorizable paratopological group. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

283 KiB

Open AccessArticle

On T-Characterized Subgroups of Compact Abelian Groups

by Saak Gabriyelyan

Axioms 2015, 4(2), 194-212; https://doi.org/10.3390/axioms4020194 - 19 Jun 2015

Cited by 4 | Viewed by 4748

Abstract

A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group [...] Read more.

A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group \(X\) is \(T\)-characterized if there is a \(T\)-sequence \(\mathbf{u} =\{ u_n \}\) in the dual group of \(X\), such that \(H=\{ x\in X: \; (u_n, x)\to 1 \}\). We show that a closed subgroup \(H\) of \(X\) is \(T\)-characterized if and only if \(H\) is a \(G_\delta\)-subgroup of \(X\) and the annihilator of \(H\) admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group \(X\) are \(T\)-characterized if and only if \(X\) is metrizable and connected. We prove that every compact Abelian group \(X\) of infinite exponent has a \(T\)-characterized subgroup, which is not an \(F_{\sigma}\)-subgroup of \(X\), that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427–2442). Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

Review

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267 KiB

Open AccessReview

Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review

by Viorel Nitica and Andrei Török

Axioms 2015, 4(1), 84-101; https://doi.org/10.3390/axioms4010084 - 4 Feb 2015

Cited by 2 | Viewed by 5076

Abstract

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and [...] Read more.

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

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