The dual and bidual of certain 𝐴*-algebras

Abstract

It is well known that every B ∗ {B^\ast } -algebra is Arens’ regular and that its bidual is a B ∗ {B^\ast } -algebra. Wong has asked whether a dual A ∗ {A^\ast } -algebra of the first kind is Arens’ regular. It is shown that this is true in the topologically simple case; in the course of the proof it is shown that in this case the bidual is, modulo its radical, an A ∗ {A^\ast } -algebra of the first kind.