Algebra involutions on the bidual of a Banach algebra (Q1062251)

It is shown that a necessary condition for the bidual of a Banach algebra A (possessing a bounded two-sided approximate identity) to admit an algebra involution with respect to the first (second) Arens product is that the dual of A is essential as a right (left) A-module. This implies that the bidual of the group algebra of any infinite, non-discrete locally compact Hausdorff group does not admit any algebra involution with respect to either Arens product. Consequently, the main result of the paper ''P-commutativity of the Banach algebra \(L_ 1^{**}(G)''\) by \textit{R. S. Doran} and \textit{W. Tiller} [ibid. 43, 85-86 (1983; Zbl 0526.46049)] only applies to the trivial case of finite abelian groups.