Weak amenability of discrete semigroup algebras (Q1365573)

A Banach algebra is weakly amenable if all bounded derivations from the algebra into its dual are inner. This paper studies the weak amenability of weighted semigroup algebras \(\ell^1 (S, w )\). The first main result is that if \(S\) is completely regular (that is, a union of groups), then \(\ell^1 (S) \) is weakly amenable. If \(S\) is commutative then the weak amenability of \(\ell^1 (S, w)\) for one weight \(w\) implies the weak amenability of \(\ell^1 (S)\). If a commutative semigroup \(S\) has a homomorphic image in which the set of singular elements (\(s\) is singular if there is no \(t\) for which both \(sts=s\) and \(tst = t \)) is non-empty but finite, then \(\ell^1 (S,w)\) is not weakly amenable for all weights \(w\). In the final section non-commutative semigroups are considered. Take \(S\) to be any semigroup, possibly with zero \(0.\) Consider the three conditions: (C1) when \(u,v,w,z \in S\) and \(uv=wz \neq 0\), then there is \(a \in S \) with \(ua =w\) and \(v = az ;\) (C2) when \(u,v,w,z \neq 0\) in \(S\) and \(vu =0\) and \(uv = wz\), then \( zw =0 ;\) (C3) when \(u,v,w,z \in S\), \(u,v \neq 0\), and \(uv=vu =0\), then there are \(b,c \in S\) with \(u=bc\) and \(cv = vb =0\). If \(S\) satisfies all three conditions, then \(\ell^1 (S)\) is weakly amenable. This class of semigroups includes all Rees matrix semigroups.