arXiv subjects

María V. Ferrer

Publications and source records attributed to María V. Ferrer.

11 recordsLinked to original sources

arXiv2026

Duality and Baire like properties in topological abelian groups

This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. In particular, extending the notion of a $g$-barrelled group, we introduce and study the classes of $κ$-barrelled and $ω$-barrelled groups. Our first main result establishes that every $ω$-barrelled group is $\aleph_0$-barrelled (i.e., if every weakly convergent sequence in the dual group is equicontinuous, then every metrizable weakly compact subset of the dual group is also equicontinuous). From this, we deduce that the free topological abelian group $A(K)$ over a metrizable compact space $K$ is determined by the subgroup generated by any of its dense subsets. Furthermore, we prove that if $G$ is a $σ$-compact, $ω$-barrelled group, then its dual group $\widehat{G}_κ$ equipped with the compact-open topology is sequentially complete (complete if $G$ is hemicompact). We also solve an open question posed by Trigos-Arrieta by constructing an explicit example of an $ω$-barrelled metrizable group that fails to be $g$-barrelled. Finally, we investigate the dualization of groups lacking infinite compact subsets, proving that for a totally bounded abelian group $G$, every compact subset of $G$ is finite if and only if its dual group $\widehat{G}$, endowed with the finite-open topology, is unordered Baire-like.

math.FA↗

arXiv2026

Weak split extensions of topological Abelian groups

In the category of topological Abelian groups, we consider the usual notion of an extension $E=(B \to X \to A)$ of $B$ by $A$, together with the notion of a weakly split extension, i.e., an extension for which the projection $X \to A$ admits a continuous section $A \to X$. Given a weakly split extension $E$, the topological Abelian group $X$ is homeomorphic to $B \times A$, although in general it is not algebraically isomorphic to $B \times A$. For two topological Abelian groups $A$ and $B$, we study the Abelian group $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ of weakly split extensions of $B$ by $A$, modulo extension isomorphisms. We show that $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ can be described as the group of all continuous sum structures defined on the product space $B \times A$ (up to topological isomorphism), with $B$ as a topological subgroup and $A$ as a topological quotient. We also provide an alternative description of $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ as a quotient $Z_c(A,B)/B_c(A,B)$, where $Z_c(A,B)$ consists of cocycles given by continuous maps $A \times A \to B$, and $B_c(A,B)$ denotes the corresponding coboundaries. Furthermore, we compare $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ with the group of standard extensions $E_A(A,B)$, where $A$ and $B$ denote the underlying Abelian groups, and relate these constructions by means of a six-term exact sequence. Although the Bohr topology of discrete Abelian groups has been investigated by many workers, there still remain many parts that are not well understood. Here, as an application of the methods developed in the paper, new examples of nontrivial $ws$-extensions for discrete Abelian groups equipped with the Bohr topology are provided and some related open questions are also proposed.

math.AT↗

arXiv2025

Bohr compactification and Chu duality of non-abelian locally compact groups

The \emph{Bohr compactification} of an arbitrary topological group $G$ is defined as the group compactification $(bG,b)$ with the following universal property: for every continuous homomorphism $h$ from $G$ into a compact group $K$ there is a continuous homomorphism $h^{b}$ from $bG$ into $K$ extending $h$ in the sense that $h=h^b \circ b$. The Bohr compactification $(bG,b)$ is the unique (up to equivalence) largest compactification of $G$. Although, for locally compact Abelian groups, the Bohr compactification is a big monster, for non-Abelian groups the situation is much more interesting and it can be said that all options are possible. Here we are interested in locally compact groups whose Bohr compactification is \emph{small}. Among other results, we characterize when the Bohr the Bohr compactification of a locally compact group is topologically isomorphic to its Chu or unitary quasi-dual. Our results fixe some incorrect statements appeared in the literature.

math.GR↗

arXiv2024

Tensor products of topological abelian groups and Pontryagin duality

Let $G$ be the group of all $\ZZ$-valued homomorphisms of the Baer-Specker group $\ZZ^\NN$. The group $G$ is algebraically isomorphic to $\ZZ^{(\NN)}$, the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on $\ZZ^\NN$, becomes a non reflexive prodiscrete group. It was an open question to find its dual group $\hat{G}$. Here, we answer this question by proving that $\hat{G}$ is topologically isomorphic to $\ZZ^\NN\otimes_\mathcal{Q}\TT$, the (locally quasi-convex) tensor product of $\ZZ^\NN$ and $\TT$. Furthermore, we investigate the reflexivity properties of the groups of $C_p(X,\ZZ)$, the group of all $\ZZ$-valued continuous functions on $X$ equipped with the pointwise convergence topology, and $A_p(X)$, the free abelian group on a $0$-dimensional space $X$ equipped with the topology $t_p(C(X,\ZZ))$ of pointwise convergence topology on $C(X,\ZZ)$. In particular, we prove that $\hat{A_p(X)}\simeq C_p(X,\ZZ)\otimes_\mathcal{Q}\TT$ and we establish the existence of $0$-dimensional spaces $X$ such that $C_p(X,\ZZ)$ is Pontryagin reflexive.

math.FA↗

arXiv2021

Homomorphic encoders of profinite abelian groups I

Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp (c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset $\{g_n : n\in\N\}$ that topologically generates the group and whose elements $g_n$ all have finite support. As a consequence, sufficient conditions are obtained that allow us to encode, by means of a topological group isomorphism, order controllable profinite abelian groups. Some applications of these results to group codes will appear subsequently \cite{FH:2021}.

math.GR↗

arXiv2021

Homomorphic encoders of profinite abelian groups II

Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp (c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable group codes. It is proved that if $G$ is an order controllable, shift invariant, group code over a finite abelian group $H$, then $G$ possesses a finite canonical generating set. Furthermore, our construction also yields that $G$ is algebraically conjugate to a full group shift.

math.GN↗

arXiv2021

The weak compactification of locally compact groups

We further investigate the weak topology generated by the irreducible unitary representations of a group $G$. A deep result due to Ernest \cite{Ernest1971} and Hughes \cite{Hughes1973} asserts that every weakly compact subset of a locally compact (LC) group $G$ is compact in the LC-topology, generalizing thereby a previous result of Glicksberg \cite{glicks1962} for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the original topology of LC groups. Among others, we deal with the preservation of countably compactness, pseudocompactness and functional boundedness.

math.GN↗

arXiv2018

On the structure of abelian profinite groups

A subgroup $G$ of a product $\prod\limits_{i\in\mathbb{N}}G_i$ is \emph{rectangular} if there are subgroups $H_i$ of $G_i$ such that $G=\prod\limits_{i\in\mathbb{N}}H_i$. We say that $G$ is \emph{weakly rectangular} if there are finite subsets $F_i\subseteq \mathbb{N}$ and subgroups $H_i$ of $\bigoplus\limits_{j\in F_i} G_j$ that satisfy $G=\prod\limits_{i\in\mathbb{N}}H_i$. %We say that $G$ is a \emph{subdirect product} of the family $\{G_i\}_{i\in I}$ if $G$ is weakly rectangular and %$G\cap\bigoplus\limits_{i\in I} G_i=\bigoplus\limits_{i\in\mathbb{N}}H_i$. In this paper we discuss when a closed subgroup of a product is weakly rectangular. Some possible applications to the theory of group codes are also highlighted.

math.GR↗

arXiv2018

Interpolation sets in spaces of continuous metric-valued functions

Let $X$ and $M$ be a topological space and metric space, respectively. If $C(X,M)$ denotes the set of all continuous functions from X to M, we say that a subset $Y$ of $X$ is an \emph{$M$-interpolation set} if given any function $g\in M^Y$ with relatively compact range in $M$, there exists a map $f\in C(X,M)$ such that $f_{|Y}=g$. In this paper, motivated by a result of Bourgain in \cite{Bourgain1977}, we introduce a property, stronger than the mere \emph{non equicontinuity} of a family of continuous functions, that isolates a crucial fact for the existence of interpolation sets in fairly general settings. As a consequence, we establish the existence of $I_0$ sets in every nonprecompact subset of a abelian locally $k_ω$-groups. This implies that abelian locally $k_ω$-groups strongly respects compactness.

math.GN↗

arXiv2014

Representation of group isomorphisms. The compact case

Let $G$ be a discrete group and let $\mathcal A$ and $\mathcal B$ be two subgroups of $G$-valued continuous functions defined on two $0$-dimensional compact spaces $X$ and $Y$. A group isomorphism $H$ defined between $\mathcal A$ and $\mathcal B$ is called \textit{separating} when for each pair of maps $f,g\in \mathcal A$ satisfying that $f^{-1}(e_G)\cup g^{-1}(e_G)=X$, it holds that $Hf^{-1}(e_G)\cup Hg^{-1}(e_G)=Y$. We prove that under some mild conditions every separating isomorphism $H:\mathcal A\longrightarrow \mathcal B$ can be represented by means of a continuous function $h: Y\longrightarrow X$ as a weighted composition operator. As a consequence we establish the equivalence of two subgroups of continuous functions if there is a biseparating isomorphism defined between them.

math.GN↗

arXiv2010

Dual topologies on non-abelian groups

The notion of locally quasi-convex abelian group, introduce by Vilenkin, is extended to maximally almost-periodic non-necessarily abelian groups. For that purpose, we look at certain bornologies that can be defined on the set $\hbox{rep}(G)$ of all finite dimensional continuous representations on a topological group $G$ in order to associate well behaved group topologies (dual topologies) to them. As a consequence, the lattice of all Hausdorff totally bounded group topologies on a group $G$ is shown to be isomorphic to the lattice of certain special subsets of $\hbox{rep}(G_d)$. Moreover, generalizing some ideas of Namioka, we relate the structural properties of the dual topological groups to topological properties of the bounded subsets belonging to the associate bornology. In like manner, certain type of bornologies that can be defined on a group $G$ allow one to define canonically associate uniformities on the dual object $\hat G$. As an application, we prove that if for every dense subgroup $H$ of a compact group $G$ we have that $\hat H$ and $\hat G$ are uniformly isomorphic, then $G$ is metrizable. Thereby, we extend to non-abelian groups some results previously considered for abelian topological groups.

math.GN↗