On unconditionally convergent series in topological rings
| Autor: | T.O. Banakh, A.V. Ravsky |
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| Jazyk: | English<br />Ukrainian |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Karpatsʹkì Matematičnì Publìkacìï, Vol 14, Iss 1, Pp 266-288 (2022) |
| Druh dokumentu: | article |
| ISSN: | 2075-9827 2313-0210 |
| DOI: | 10.15330/cmp.14.1.266-288 |
| Popis: | 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. |
| Databáze: | Directory of Open Access Journals |
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