On unconditionally convergent series in topological rings
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\subse...
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| Format: | Article |
| Language: | English |
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Vasyl Stefanyk Precarpathian National University
2022-06-01
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| Series: | Karpatsʹkì Matematičnì Publìkacìï |
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| Online Access: | https://journals.pnu.edu.ua/index.php/cmp/article/view/5655 |
| _version_ | 1797205813591277568 |
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| author | T.O. Banakh A.V. Ravsky |
| author_facet | T.O. Banakh A.V. Ravsky |
| author_sort | T.O. Banakh |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-24T08:57:05Z |
| format | Article |
| id | doaj.art-fdac066bb3bd481e9881d0a4420d4a6c |
| institution | Directory Open Access Journal |
| issn | 2075-9827 2313-0210 |
| language | English |
| last_indexed | 2024-04-24T08:57:05Z |
| publishDate | 2022-06-01 |
| publisher | Vasyl Stefanyk Precarpathian National University |
| record_format | Article |
| series | Karpatsʹkì Matematičnì Publìkacìï |
| spelling | doaj.art-fdac066bb3bd481e9881d0a4420d4a6c2024-04-16T07:10:59ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102022-06-0114126628810.15330/cmp.14.1.266-2884893On unconditionally convergent series in topological ringsT.O. Banakh0A.V. Ravsky1https://orcid.org/0000-0003-2542-6959Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine; Institute of Mathematics, Jan Kochanowski University in Kielce, 7 Uniwersytecka str., 25406, Kielce, PolandInstitute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, UkraineWe 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.https://journals.pnu.edu.ua/index.php/cmp/article/view/5655topological ringunconditional convergencelocally compact topological ringlocally compact abelian topological group |
| spellingShingle | T.O. Banakh A.V. Ravsky On unconditionally convergent series in topological rings Karpatsʹkì Matematičnì Publìkacìï topological ring unconditional convergence locally compact topological ring locally compact abelian topological group |
| title | On unconditionally convergent series in topological rings |
| title_full | On unconditionally convergent series in topological rings |
| title_fullStr | On unconditionally convergent series in topological rings |
| title_full_unstemmed | On unconditionally convergent series in topological rings |
| title_short | On unconditionally convergent series in topological rings |
| title_sort | on unconditionally convergent series in topological rings |
| topic | topological ring unconditional convergence locally compact topological ring locally compact abelian topological group |
| url | https://journals.pnu.edu.ua/index.php/cmp/article/view/5655 |
| work_keys_str_mv | AT tobanakh onunconditionallyconvergentseriesintopologicalrings AT avravsky onunconditionallyconvergentseriesintopologicalrings |