On $T$-sequences and characterized subgroups

Abstract

Let $X$ be a compact metrizable abelian group and $\mathbf{u}=\{u_n\}$ be a sequence in its dual $X^{\wedge }$. Set $s_{\mathbf{u}}(X)=\{x:(u_n,x)\to 1\}$ and ${\mathbb{T}}_0^H=\{(z_n)\in {\mathbb{T}}^{\infty }:z_n\to 1\}$. Let $G$ be a subgroup of $X$. We prove that $G=s_{\mathbf{u}}(X)$ for some $\mathbf{u}$ iff it can be represented as some dually closed subgroup $G_{\mathbf{u}}$ of ${\mathrm{Cl}}_{X}G\times {\mathbb{T}}_0^H$. In particular, $s_{\mathbf{u}}(X)$ is polishable. Let $\mathbf{u}=\{u_n\}$ be a $T$-sequence. Denote by $(\hat{X},\mathbf{u})$ the group $X^{\wedge }$ equipped with the finest group topology in which $u_n\to 0$. It is proved that $(\hat{X},\mathbf{u}{)}^{\wedge }=G_{\mathbf{u}}$ and $\mathbf{n}(\hat{X},\mathbf{u})=s_{\mathbf{u}}(X{)}^{\perp }$. We also prove that the group generated by a Kronecker set can not be characterized.

References in corpus (1)

• Groups of quasi-invariance and the Pontryagin duality

Cited by in corpus (1)

• Which infinite abelian groups admit an almost maximally almost-periodic group topology?