A Banach algebra A is called
amenable if all bounded derivations into dual Banach A-modules are inner. Let S be
a semigroup and let l1(S) be the corresponding discrete convolution algebra. This
paper is on the theme: “On the hypothesis that l1(S) is amenable, what
conclusions can be drawn about the (algebraic) structure of S?” We give a
complete characterization of commutative semigroups carrying amenable
semigroup algebras. If S is commutative, then l1(S) is amenable if and only if
S is a finite semilattice of groups, that is, there is a finite semilattice Y
and disjoint commutative groups Gα(α ∈ Y ) such that S =⋃α∈YGα and
GαGβ⊆ Gαβ(α,β ∈ Y ).
