Radicals of some semigroup algebras

In this paper we seek to determine the Jacobson radical of certain algebras based on semigroups, and in particular on the semigroups (βS,□), where S is a cancellative, countable, abelian semigroup and βS is its Stone–Čech semigroup compactification. In particular, we wish to determine the radical of ℓ  1(βℕ)