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 $S$. The authors study the structure of this Banach algebra and of its second dual. The authors determine exactly when $ell^{,1}(S)$ is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are 'forbidden values' for this constant.