On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences
Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 1, pp. 73-99
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Let $X$ be an Abelian topological group.
A subgroup $H$ of $X$ is characterized
if there is a sequence
$\mathbf{u} = \{u_n\}$ in the dual
group of $X$ such that
$H= \{x\in X: \; (u_n,x)\to 1\}$.
We reduce the study of characterized
subgroups of $X$ to the study of
characterized subgroups of compact
metrizable Abelian groups.
Let $c_0(X)$ be the group of all
$X$-valued null sequences and
$\mathfrak{u}_0$ be the uniform
topology on $c_0(X)$. If $X$ is compact
we prove that $c_0(X)$ is a characterized
subgroup of $X^\mathbb{N}$ if and only
if $X\cong \mathbb T^n\times F$, where
$n\geq 0$ and $F$ is a finite Abelian
group. For every compact Abelian group
$X$, the group $c_0(X)$ is a
$\mathfrak{g}$-closed subgroup of
$X^\mathbb N$. Some general properties
of $(c_0(X),\mathfrak{u}_0)$ and its
dual group are given. In particular,
we describe compact subsets of
$(c_0(X),\mathfrak{u}_0)$.
Let $X$ be an Abelian topological group.
A subgroup $H$ of $X$ is characterized
if there is a sequence
$\mathbf{u} = \{u_n\}$ in the dual
group of $X$ such that
$H= \{x\in X: \; (u_n,x)\to 1\}$.
We reduce the study of characterized
subgroups of $X$ to the study of
characterized subgroups of compact
metrizable Abelian groups.
Let $c_0(X)$ be the group of all
$X$-valued null sequences and
$\mathfrak{u}_0$ be the uniform
topology on $c_0(X)$. If $X$ is compact
we prove that $c_0(X)$ is a characterized
subgroup of $X^\mathbb{N}$ if and only
if $X\cong \mathbb T^n\times F$, where
$n\geq 0$ and $F$ is a finite Abelian
group. For every compact Abelian group
$X$, the group $c_0(X)$ is a
$\mathfrak{g}$-closed subgroup of
$X^\mathbb N$. Some general properties
of $(c_0(X),\mathfrak{u}_0)$ and its
dual group are given. In particular,
we describe compact subsets of
$(c_0(X),\mathfrak{u}_0)$.
Classification :
22A10, 43A40, 54H11
Keywords: group of null sequences; $T$-sequence; characterized subgroup; $T$-characterized subgroup; $\mathfrak{g}$-closed subgroup
Keywords: group of null sequences; $T$-sequence; characterized subgroup; $T$-characterized subgroup; $\mathfrak{g}$-closed subgroup
@article{CMUC_2014_55_1_a6,
author = {Gabriyelyan, S. S.},
title = {On characterized subgroups of {Abelian} topological groups $X$ and the group of all $X$-valued null sequences},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {73--99},
year = {2014},
volume = {55},
number = {1},
mrnumber = {3160827},
zbl = {06383786},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2014_55_1_a6/}
}
TY - JOUR AU - Gabriyelyan, S. S. TI - On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences JO - Commentationes Mathematicae Universitatis Carolinae PY - 2014 SP - 73 EP - 99 VL - 55 IS - 1 UR - http://geodesic.mathdoc.fr/item/CMUC_2014_55_1_a6/ LA - en ID - CMUC_2014_55_1_a6 ER -
%0 Journal Article %A Gabriyelyan, S. S. %T On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences %J Commentationes Mathematicae Universitatis Carolinae %D 2014 %P 73-99 %V 55 %N 1 %U http://geodesic.mathdoc.fr/item/CMUC_2014_55_1_a6/ %G en %F CMUC_2014_55_1_a6
Gabriyelyan, S. S. On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences. Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 1, pp. 73-99. http://geodesic.mathdoc.fr/item/CMUC_2014_55_1_a6/