TY - JOUR
AU - Banakh, T.O.
AU - Ravsky, A.V.
PY - 2022/06/30
Y2 - 2024/05/22
TI - On unconditionally convergent series in topological rings
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 14
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.14.1.266-288
UR - https://scijournals.pnu.edu.ua/index.php/cmp/article/view/5655
SP - 266-288
AB - <p>We define a topological ring $R$ to be <em>Hirsch</em>, 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.</p>
ER -