arXiv subjects

Salvador Hernández

Publications and source records attributed to Salvador Hernández.

21 records · Page 2Linked to original sources

arXiv2012

The weights of closed subgroups of a locally compact group

Let $G$ be an infinite locally compact group and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le w(G)$ for the weight $w(G)$ of $G$. It is shown that there is a closed subgroup $N$ of $G$ with $w(N)=\aleph$. Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group $G$ and $\aleph$ a cardinal satisfying $\aleph_0\le\aleph\le \lw(G)$, where $\lw(G)$ is the local weight of $G$, there are either no infinite compact subgroups at all or there is a compact subgroup $N$ of $G$ with $w(N)=\aleph$. (3) For an infinite abelian group $G$ there exists a properly ascending family of locally quasiconvex group topologies on $G$, say, $(τ_\aleph)_{\aleph_0\le \aleph\le \card(G)}$, such that $(G,τ_\aleph)\hat{\phantom{m}}\cong\hat G$. Items (2) and (3) are shown in Section 5.

math.GR↗

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↗

arXiv2005

Some new results on the Chu duality of discrete groups

This paper deals mainly with the Chu duality of discrete groups. Among other results, we give sufficient conditions for an $FC$ group to satisfy Chu duality and characterize when the Chu quasi-dual and the Takahashi quasi-dual of a group $G$ coincide. As a consequence, it follows that when $G$ is a weak sum of a family of finite simple groups, if the exponent of the groups in the family is bounded then $G$ satisfies Chu duality; on the other hand, if the exponent of the groups goes to infinite then the Chu quasi-dual of $G$ coincide with its Takahashi quasi-dual. We also present examples of discrete groups whose Chu quasi-duals are not locally compact and examples of discrete Chu reflexive groups which contain non-trivial sequences converging in the Bohr topology of the groups. Our results systematize some previous work and answer some open questions in the subject.

math.GN↗