On unconditionally convergent series in topological rings

Authors

  • T.O. Banakh Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine; Institute of Mathematics, Jan Kochanowski University in Kielce, 7 Uniwersytecka str., 25406, Kielce, Poland
  • A.V. Ravsky Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine https://orcid.org/0000-0003-2542-6959
https://doi.org/10.15330/cmp.14.1.266-288

Keywords:

topological ring, unconditional convergence, locally compact topological ring, locally compact Abelian topological group
Published online: 2022-06-30

Abstract

We define a topological ring R to be Hirsch, if for any unconditionally convergent series nωxi in R and any neighborhood U of the additive identity 0 of R there exists a neighborhood VR of 0 such that nFanxnU for any finite set Fω and any sequence (an)nFVF. 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 and its subrings c0 and c are Hirsch. Applying a recent result of Banakh and Kadets, we prove that for a real number p1 the commutative Banach ring p is Hirsch if and only if p2. Also for any p(1,), the (noncommutative) Banach ring L(p) of continuous endomorphisms of the Banach ring p is not Hirsch.

How to Cite
(1)
Banakh, T.; Ravsky, A. On Unconditionally Convergent Series in Topological Rings. Carpathian Math. Publ. 2022, 14, 266-288.