arXiv subjects

Gábor Lukács

Publications and source records attributed to Gábor Lukács.

17 recordsLinked to original sources

arXiv2021

Long colimits of topological groups IV: Spaces with socks

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support, or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors develop techniques for showing that products of certain spaces with CSHP, such as the Closed Long Ray and the Long Line, have CSHP again.

math.GN↗

arXiv2021

Long colimits of topological groups III: Homeomorphisms of products and coproducts

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in a number of ways, including as a colimit of homeomorphism groups with a given compact support, or as a subgroup of the homeomorphism group of its Stone-Čech compactification. A space is said to have the Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. The authors provide necessary and sufficient conditions for finite products of ordinals equipped with the order topology to have CSHP. In addition, necessary conditions are presented for finite products and coproducts of spaces to have CSHP.

math.GN↗

arXiv2020

Long colimits of topological groups II: Free groups and vector spaces

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological $\mathbb{R}$-vector space $V(X)$ on $X$. We show that $V(X)$ is a quotient of the free abelian topological group on $[-1,1]\times X$, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of $V(X)$ satisfy the so-called $\textit{algebraic colimit property}$ defined in the authors' previous work.

math.GN↗

arXiv2019

Long colimits of topological groups I: Continuous maps and homeomorphisms

The union of a directed family of topological groups can be equipped with two noteworthy topologies: the finest topology making each injection continuous, and the finest group topology making each injection continuous. This begs the question of whether the two topologies coincide. If the family is countable, the answer is well known in many cases. We study this question in the context of so-called long families, which are as far as possible from countable ones. As a first step, we present answers to the question for families of group-valued continuous maps and homeomorphism groups, and provide additional examples.

math.GN↗

arXiv2013

Compact-like abelian groups without non-trivial quasi-convex null sequences

In this paper, we study precompact abelian groups G that contain no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 0. We characterize groups with this property in the following classes of groups: (a) bounded precompact abelian groups; (b) minimal abelian groups; (c) totally minimal abelian groups; (d) ω-bounded abelian groups. We also provide examples of minimal abelian groups with this property, and show that there exists a minimal pseudocompact abelian group with the same property; furthermore, under Martin's Axiom, the group may be chosen to be countably compact minimal abelian.

math.GN↗

arXiv2012

On the quasi-component of pseudocompact abelian groups

In this paper, we describe the relationship between the quasi-component q(G) of a (perfectly) minimal pseudocompact abelian group G and the quasi-component q(\widetilde G) of its completion. Specifically, we characterize the pairs (C,A) of compact connected abelian groups C and subgroups A such that A \cong q(G) and C \cong q(\widetilde G). As a consequence, we show that for every positive integer n or n=ω, there exist plenty of abelian pseudocompact perfectly minimal n-dimensional groups G such that the quasi-component of G is not dense in the quasi-component of the completion of G.

math.GN↗

arXiv2010

On zero-dimensionality and the connected component of locally pseudocompact groups

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is dense in the component of the completion of G, and G/G_0 is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that G/G_0 may fail to be zero-dimensional even for totally minimal pseudocompact groups.

math.GN↗

arXiv2008

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that for every locally compact abelian group G, the following statements are equivalent: (i) G contains no sequence {x_n} such that {0} \cup {\pm x_n : n \in N} is infinite and quasi-convex in G, and x_n --> 0; (ii) one of the subgroups {g \in G : 2g=0 and {g \in G : 3g=0} is open in G; (iii) G contains an open compact subgroup of the form Z_2^κor Z_3^κfor some cardinal κ.

math.GN↗

arXiv2008

Quasi-convex sequences in the circle and the 3-adic integers

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J_3 of 3-adic integers. These sequences are determined by an increasing sequences of integers. For an increasing sequence \underline{a}=\{a_n\} of integers, put g_n=a_{n+1}-a_n. We prove that: (a) the set \{0\}\cup\{\pm 3^{-(a_n+1)} : n\in N\} is quasi-convex in T if and only if a_0>0 and g_n>1 for every n\in N; (b) the set \{0\}\cup\{\pm 3^{a_n} : n\in N\} is quasi-convex in the group J_3 of 3-adic integers if and only if g_n>1 for every n\in N. Moreover, we solve an open problem of Dikranjan and de Leo by providing a complete characterization of the sequences \underline{a} such that \{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences \underline{a} such that the set \{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in R.

math.GN↗

arXiv2006

Notes on duality theories of abelian groups

In this notebook, I present duality theory (or theories) of abelian groups with some categorical and categorical topological flavour. I consider writing this notebook as a longer-term project, and its current content and presentation is "under development." In other words, all questions, comments, suggestions, and criticism is more than usually welcome. In Chapter 1, which is the only more-or-less ready part, the evaluation homomorphism A --> A^^ (of a group A into its bidual A^^) is studied, with particular attention to quasi-convexity and special groups (compact, discrete, LCA) and subgroups (open, compact). The treatment of LCA groups is incomplete, but the ultimate goal is to make it as self-contained as possible.

math.GN↗

arXiv2006

Hereditarily non-topologizable groups

A group G is non-topologizable if the only Hausdorff group topology that G admits is the discrete one. Is there an infinite group G such that H/N is non-topologizable for every subgroup H <= G and every normal subgroup N <| H? We show that a solution of this essentially group theoretic question provides a solution to the problem of c-compactness.

math.GR↗

arXiv2006

Bounded and unitary elements in pro-C^*-algebras

A pro-C^*-algebra is a (projective) limit of C^*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C^*-algebras can be seen as non-commutative k-spaces. An element of a pro-C^*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C^*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C^*-algebra. In this paper, we investigate pro-C^*-algebras from a categorical point of view. We study the functor (-)_b that assigns to a pro-C^*-algebra the C^*-algebra of its bounded elements, which is the dual of the Stone-Čech-compactification. We show that (-)_b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand-duality for commutative unital pro-C^*-algebras is also presented.

math.CT↗

arXiv2005

Almost maximally almost-periodic group topologies determined by T-sequences

A sequence $\{a_n\}$ in a group $G$ is a {\em $T$-sequence} if there is a Hausdorff group topology $τ$ on $G$ such that $a_n\stackrelτ\longrightarrow 0$. In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a $T$-sequence, and investigate special sequences in the Prüfer groups $\mathbb{Z}(p^\infty)$. We show that for $p\neq 2$, there is a Hausdorff group topology $τ$ on $\mathbb{Z}(p^\infty)$ that is determined by a $T$-sequence, which is close to being maximally almost-periodic--in other words, the von Neumann radical $\mathbf{n}(\mathbb{Z}(p^\infty),τ)$ is a non-trivial finite subgroup. In particular, $\mathbf{n}(\mathbf{n}(\mathbb{Z}(p^\infty),τ)) \subsetneq \mathbf{n}(\mathbb{Z}(p^\infty),τ)$. We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a $T$-sequence with non-trivial finite von Neumann radical.

math.GN↗

arXiv2005

Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group K and its subgroup H, one defines the g-closure g(H) of H in K as the subgroup consisting of $χ\in K$ such that $χ(a_n)\longrightarrow 0$ in T=R/Z for every sequence {a_n} in $\hat K$ (the Pontryagin dual of K) that converges to 0 in the topology that H induces on $\hat K$. We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the $G_δ$-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.

math.GN↗

arXiv2005

Lifted closure operators

In this paper, we study the properties of closure operators obtained as initial lifts along a reflector, and compactness with respect to them in particular. Applications in the areas of topology, topological groups and topological *-algebras are presented.

math.CT↗

arXiv2004

Hereditarily h-complete groups

A topological group G is h-complete if every continuous homomorphic image of G is (Raikov-)complete; we say that G is hereditarily h-complete if every closed subgroup of G is h-complete. In this paper, we establish open-map properties of hereditarily h-complete groups with respect to large classes of groups, and prove a theorem on the (total) minimality of subdirectly represented groups. Numerous applications are presented, among them: 1. Every hereditarily h-complete group with quasi-invariant basis is the projective limit of its metrizable quotients; 2. If every countable discrete hereditarily h-complete group is finite, then every locally compact hereditarily h-complete group that has small invariant neighborhoods is compact. In the sequel, several open problems are formulated.

math.GN↗

arXiv2004

On sequentially h-complete groups

A topological group $G$ is {\em sequentially $h$-complete} if all the continuous homomorphic images of $G$ are sequentially complete. In this paper we give necessary and sufficient conditions on a complete group for being compact, using the language of sequential $h$-completeness. In the process of obtaining such conditions, we establish a structure theorem for $ω$-precompact sequentially $h$-complete groups. As a consequence we obtain a reduction theorem for the problem of $c$-compactness.

math.GR↗