Banach space properties forcing a reflexive, amenable Banach algebra to be trivial (Q5953116)

It is an open question whether there exist infinite-dimensional reflexive amenable Banach algebras \({\mathcal A}\). It is known from the literature [see \textit{B. E. Johnson}, Proc. Am. Math. Soc. 119, 1259-1258 (1993; Zbl 0813.43002)] that this is impossible if all maximal left ideals are complemented in \({\mathcal A}\). In particular, no amenable Banach algebra can be a Hilbert space. In this paper, the author is able to replace the complemented subspace condition by conditions related to the approximation property. So his result applies also to \({\mathcal L}^p\)-spaces in the sense of Lindenstrauss.