Irregular abelian semigroups with weakly amenable semigroup algebra

In this paper we give counterexamples for the open problem, posed by Blackmore (Semigroup Forum 55:359–377, 1987) of whether weak amenability of a semigroup algebra ℓ ¹(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but ℓ ¹(S) is weakly amenable. Examples are then given.