\(\phi\)-amenability and character amenability of some classes of Banach algebras (Q2839883)

Let \(A\) be a Banach algebra, and let \(\phi\) be a character (i.e., non-zero multiplicative linear functional) on \(A\). Then \(A\) is said to be \(\phi\)-amenable if, for every Banach \(A\)-bimodule \(X\) such that \(a\cdot x=\phi(a)x\) for all \(a\in A\) and \(x\in X\), each continuous derivation from \(A\) into the dual \(X^*\) is inner. The authors call \(A\) character amenable if \(A\) is \(\phi\)-amenable for each character \(\phi\) and \(A\) has a bounded right approximate identity. The authors study the above introduced notions for the semigroup algebra \(\ell^1(S)\) corresponding to a semilattice \(S\). Further, they characterize the character amenability of \(\ell^1(S)\) in the case where \(S\) is a uniformly locally finite inverse semigroup.