Weak forms of amenability for Banach algebras (Q2891980)

In [Stud. Math. 177, No. 1, 81--96 (2006; Zbl 1117.46030)], \textit{H. G. Dales}, \textit{R. J. Loy} and \textit{Y. Zhang} proved that the Banach sequence algebras \(\ell^p(\omega)\), \(1\leq p<\infty\), \(\omega\in[1,+\infty)^I\), are not approximately amenable. In the present paper, the authors prove that, for a given set \(I\), a family \(\{\mathfrak{U_i}\}_{i\in I}\) of Banach algebras with unit, and \(\omega=(a_i)\in[1,+\infty)^I\), if \(1\leq p<\infty\), then \(\ell^p((\mathfrak{U_i),\omega)}\) is amenable (respectively, approximately amenable) if and only if the set \(I\) is finite, and, for each \(i\in I\), \(\mathfrak{U_i}\) is amenable (respectively, approximately amenable). Also, they investigate applications to compact groups. For example, they show that, if \(G\) is an infinite compact group, then the convolution Banach algebra \(L^2(G)\) is not approximately amenable.