Applications of a result of Aron, Hervés, and Valdivia to the homology of Banach algebras. (Q2760135)

Let \((A_n)\) be a sequence of associative Banach algebras and \(l_\infty (A_n)\) the Banach algebra of all sequences \((f_n)\), with \(f_n\in A_n\) for all \(n\in \mathbb{N}\), and \(\sup_n| f_n |_{A_n}\) finite with coordinatewise multiplication and norm \(\| \cdot\|\) defined for each \((f_n)\in l_\infty (A_n)\) by \(\| (f_n)\|=\sup_n\| f_n\|_{A_n}\). Conditions for which there exists a surjective homomorphism from \(l_\infty (A_n)\) onto \(A^{**}\) (the second dual of \(A\)) are found. By this result and a result of \textit{R. M. Aron}, \textit{C. Hervés} and \textit{M. Valdivia} [J. Funct. Anal. 52, 189--204 (1983; Zbl 0517.46019)] it is shown that there is a sequence \((A_n)\) of finite dimensional (hence amenable) Lipschitz algebras \(A_n\) such that \(l_\infty (A_n)\) is not even weakly amenable. Moreover, it is shown that there exists a sequence \((A_n)\) of finite dimensional (hence amenable) \(C^*\)-algebras \(A_n\) such that \(l_\infty (A_n)\) is not amenable.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00063].