On characterized subgroups of Abelian topological groups $X$ and the group of all $X$-valued null sequences

Gabriyelyan, S. S.

Commentationes Mathematicae Universitatis Carolinae, Tome 55 (2014) no. 1, pp. 73-99.

Résumé

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 $(c_0(X),\mathfrak{u}_0)$.

Détail

Classification : 22A10, 43A40, 54H11
Keywords: group of null sequences; $T$-sequence; characterized subgroup; $T$-characterized subgroup; $\mathfrak{g}$-closed subgroup