Banach Algebras on Semigroups and on Their Compactifications

Let $S$ be a (discrete) semigroup, and let $\ell^{\,1}(S)$ be the Banach algebra which is the semigroup algebra of $S$. This studies the structure of this Banach algebra and of its second dual. It determines exactly when $\ell^{\,1}(S)$ is amenable as a Banach algebra, and discusses its amenability constant, showing that there are `forbidden values' for this constant.