Banach Algebras on Semigroups and on Their Compactifications

Let S be a (discrete) semigroup, and let ℓ 1(S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ℓ 1(S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are 'forbidden values' for this constant. The second dual of ℓ 1(S) is the Banach algebra M(βS) of measures on the Stone-Čech compactification βS of S, where M(βS) and βS are taken with the first Arens product □. We shall show that S is finite whenever M(βS) is amenable, and we shall discuss when M(βS) is weakly amenable. We shall show that the second dual of L 1(G), for G a locally compact group, is weakly amenable if and only if G is finite. We shall also discuss left-invariant means on S as elements of the space M(βS), and determine their supports. We shall show that, for each weakly cancellative and nearly right cancellative semigroup S, the topological centre of M(βS) is just ℓ 1 (S), and so ℓ 1 (S) is strongly Arens irregular; indeed, we shall considerably strengthen this result by showing that, for such semigroups S, there are two-element subsets of βS S that are determining for the topological centre; for more general semigroups S, there are finite subsets of βS S with this property. We have partial results on the radical of the algebras ℓ 1(βS) and M(βS). We shall also discuss analogous results for related spaces such as WAP(S) and LUC(G). © 2009 American Mathematical Society.