Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G,\tau )}^{\wedge }$ is a dense -closed subgroup of the compact group ${\left(G_d\right)}^{\wedge }$, where $G_d$ is the group G with the discrete...

Let $X$ be an Abelian topological group. A subgroup $H$ of $X$ is characterized if there is a sequence $𝐮=\left\{u_n\right\}$ 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 $c_0\left(X\right)$ be the group of all $X$-valued null sequences and ${𝔲}_0$ be the uniform topology on $c_0\left(X\right)$. If $X$ is compact we prove that $c_0\left(X\right)$ is a characterized subgroup of $X^{\mathbb{N}}$ if and only if $X\cong {𝕋}^n\times F$, where $n\ge 0$ and $F$ is a finite Abelian group. For every compact Abelian...

We study some embeddings of suitably topologized spaces of vector-valued smooth functions on topological groups, where smoothness is defined via differentiability along continuous one-parameter subgroups. As an application, we investigate the canonical correspondences between the universal enveloping algebra, the invariant local operators, and the convolution algebra of distributions supported at the unit element of any finite-dimensional Lie group, when one passes from finite-dimensional Lie groups...