On unconditionally convergent series in topological rings

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 $p\in (2,\infty )$.

23 pages