Arens regularity of certain weighted semigroup algebras and countability

TL;DR

This paper investigates the conditions under which weighted semigroup algebras are Arens regular, revealing that for many semigroups, such regularity implies the semigroup is countable, thus linking algebraic and set-theoretic properties.

Contribution

It demonstrates that for a broad class of semigroups, Arens regularity of the weighted algebra implies the semigroup's countability, extending previous results.

Findings

Arens regularity implies countability for many semigroups

No uncountable group admits an Arens regular weighted algebra

Results connect algebraic regularity with set-theoretic size

Abstract

It is known that every countable semigroup admits a weight w for which the semigroup algebra l_1(S,w) is Arens regular and no uncountable group admits such a weight; see [4]. In this paper, among other things, we show that for a large class of semigroups, the Arens regularity of the weighted semigroup algebra l_1(S,w) implies the countability of S.