Amenability of discrete convolution algebras, the commutative case
Niels Grønbæk · Pacific Journal of Mathematics · 1990
A Banach algebra 21 is called amenable if all bounded derivations into dual Banach 2l-modules are inner.Let S be a semigroup and let l ι (S) be the corresponding discrete convolution algebra.This paper is on the theme: "On the hypothesis that l ι (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 l ι (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 G Q (a e Y) such that S = \J aEY G a and G a G β C G aβ (a,βeY).1. Preliminaries.We shall need some elementary semigroup theory.We prefer to keep our exposition self-contained, so although most of what follows can be found in standard texts on the subject,