On unconditionally convergent series in topological rings

TL;DR

This paper introduces the concept of Hirsch topological rings, characterizes them in various classes, and explores which classical Banach rings like al_, c_0, c, and al_p are Hirsch, revealing new structural insights.

Contribution

It defines Hirsch rings, develops seminorm techniques, and characterizes Hirsch property in several known classes of topological rings, including Banach rings.

Findings

al_, c_0, c are Hirsch rings.

al_p is Hirsch for p in [1,2].

al_p nd al_q are not Hirsch for p nd q istinct in [1,].

Abstract

We define a topological ring $R$ to be \emph{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. Also we prove that for every $p\in[1,2]$ the Banach ring $\ell_p$ is Hirsch. On the other hand, for any distinct numbers $p,q\in[1,\infty]$ the commutative Banach ring $\ell_p\oplus i\ell_q$ is not Hirsch. 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. We do not know whether the Banach rings $\ell_p$ are Hirsch for $p\in(2,\infty)$.