  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 $\ell^{ 1}(\beta \N)$.
