| Title:
|
On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences (English) |
| Author:
|
Gabriyelyan, S. S. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
73-99 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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
$(c_0(X),\mathfrak{u}_0)$. (English) |
| Keyword:
|
group of null sequences |
| Keyword:
|
$T$-sequence |
| Keyword:
|
characterized subgroup |
| Keyword:
|
$T$-characterized subgroup |
| Keyword:
|
$\mathfrak{g}$-closed subgroup |
| MSC:
|
22A10 |
| MSC:
|
43A40 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 06383786 |
| idMR:
|
MR3160827 |
| . |
| Date available:
|
2014-01-17T09:37:11Z |
| Last updated:
|
2016-04-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143569 |
| . |
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