Radicals of some semigroup algebras

TL;DR

This paper investigates the Jacobson radical of semigroup-based algebras, especially focusing on the Stone–Čech compactification of certain cancellative, abelian semigroups, including the algebra (eta \u2115).

Contribution

It provides new insights into the structure of radicals in semigroup algebras, particularly for (eta \u2115), expanding understanding of their algebraic properties.

Findings

Determined the Jacobson radical of (eta ).

Analyzed the radical structure for semigroups  with specific properties.

Extended previous results on semigroup algebra radicals.

Abstract

In this paper we seek to determine the Jacobson radical of certain algebras based on semigroups, and in particular on the semigroups $(\beta S, \Box)$, where $S$ is a cancellative, countable, abelian semigroup and $\beta S$ is its Stone--\v{C}ech semigroup compactification. In particular, we wish to determine the radical of $\ell^{ 1}(\beta \N)$.