Banach Algebras on Semigroups and their Compactifications – revised version after submission to the Memoirs American Mathematical Society 8 September 2006

Let S be a semigroup, and let ` (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 ` (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 ` (S) is the Banach algebraM(βS) of measures on the Stone-Čech compactification βS of S, whereM(βS) and βS are taken with the first Arens product 2. 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(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 ofM(βS) is just ` (S), and so ` (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 ` (βS) and M(βS). We shall also discuss analogous results for related spaces such as WAP (S) and LUC(G). 2000 Mathematics Subject Classification. Primary 43A10, 43A20; secondary 46J10.