Approximate amenability of certain semigroup algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra l(1)(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then l(1)(S) is approximately amenable if and only if G is amenable. Moreover l(1)(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M-a(S)-measurable subset B of S and s is an element of S the set sB is M-a(S)-measurable, it is proved that if the measure algebra M-a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.