Amenability of certain Banach algebras with application to measure algebras on foundation semigroups (Q1420717)

A Lau algebra is a Banach algebra \(A\) such that \(A\) is the predual of a von Neumann algebra \(M\) and the identity 1 of \(M\) is a multiplicative linear functional on \(A\). The class of Lau algebras was introduced by the reviewer (called \(F\)-algebras) in [Fundam. Math. 118, 161--175 (1983; Zbl 0545.46051)]. Examples of Lau algebras include the group algebra and the Fourier algebra of a locally compact group as well as the measure algebra \(M_a(S)\) of a topological semigroup \(S\). In this paper, the author generalizes a result of \textit{G. A. Willis} [J. Funct. Anal. 92, No. 1, 202--263 (1990; Zbl 0726.43001)] to separable Lau algebras \(A\) and shows that if \(A\) has a bounded approximate identity 1 and the collection \({\mathcal J}\) of \(J_b=\) norm closure of \(\{a- ab: a\in A\}\), \(b\) a positive norm one element in \(A\), has a unique maximal ideal, then \(A\) is left amenable. Conversely, if \(A\) is left amenable, then \(J\) has a unique maximal ideal. In this case, the unique maximal ideal is \(I_0(A)= \{a\in A:\langle a,1\rangle= 0\}\). He also shows that if \(S\) is a foundation semigroup with identity and the measure algebra \(M_a(S)\) is separable and left amenable, then there exists a discrete probability measure \(\mu\) on \(S\) such that \(I_0(M_a(S))= J_\mu\).