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On unconditionally convergent series in topological rings
- Source :
- Karpatsʹkì Matematičnì Publìkacìï, Vol 14, Iss 1, Pp 266-288 (2022)
- Publication Year :
- 2022
- Publisher :
- Vasyl Stefanyk Precarpathian National University, 2022.
-
Abstract
- We define a topological ring $R$ to be Hirsch, if for any unconditionally convergent series $\sum_{n\in\omega} x_i$ in $R$ and any neighborhood $U$ of the additive identity $0$ of $R$ there exists a neighborhood $V\subseteq R$ of $0$ such that $\sum_{n\in F} a_n x_n\in U$ for any finite set $F\subset\omega$ and any sequence $(a_n)_{n\in F}\in V^F$. We recognize Hirsch rings in certain known classes of topological rings. For this purpose we introduce and develop the technique of seminorms on actogroups. We prove, in particular, that a topological ring $R$ is Hirsch provided $R$ is locally compact or $R$ has a base at the zero consisting of open ideals or $R$ is a closed subring of the Banach ring $C(K)$, where $K$ is a compact Hausdorff space. This implies that the Banach ring $\ell_\infty$ and its subrings $c_0$ and $c$ are Hirsch. Applying a recent result of Banakh and Kadets, we prove that for a real number $p\ge 1$ the commutative Banach ring $\ell_p$ is Hirsch if and only if $p\le 2$. Also for any $p\in (1,\infty)$, the (noncommutative) Banach ring $L(\ell_p)$ of continuous endomorphisms of the Banach ring $\ell_p$ is not Hirsch.
Details
- Language :
- English, Ukrainian
- ISSN :
- 20759827 and 23130210
- Volume :
- 14
- Issue :
- 1
- Database :
- Directory of Open Access Journals
- Journal :
- Karpatsʹkì Matematičnì Publìkacìï
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.fdac066bb3bd481e9881d0a4420d4a6c
- Document Type :
- article
- Full Text :
- https://doi.org/10.15330/cmp.14.1.266-288