On unconditionally convergent series in topological rings
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Keywords:
topological ring, unconditional convergence, locally compact topological ring, locally compact Abelian topological groupAbstract
We define a topological ring R to be Hirsch, if for any unconditionally convergent series ∑n∈ωxi in R and any neighborhood U of the additive identity 0 of R there exists a neighborhood V⊆R of 0 such that ∑n∈Fanxn∈U for any finite set F⊂ω and any sequence (an)n∈F∈VF. 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 ℓ∞ and its subrings c0 and c are Hirsch. Applying a recent result of Banakh and Kadets, we prove that for a real number p≥1 the commutative Banach ring ℓp is Hirsch if and only if p≤2. Also for any p∈(1,∞), the (noncommutative) Banach ring L(ℓp) of continuous endomorphisms of the Banach ring ℓp is not Hirsch.