Arens multiplication and a characterization of $w^{\ast }$-algebras

Abstract:

Let $\mathfrak {A}$ be a Banach algebra which is the dual of a normed linear space $\mathfrak {X}$. Suppose the multiplication in $\mathfrak {A}$ is a continuous function of each factor separately in the weak* topology. We show that the natural projection of ${\mathfrak {A}^{ \ast \ast }} = {\mathfrak {X}^{ \ast \ast \ast }}$ onto $\mathfrak {A} = {\mathfrak {X}^ \ast }$ is a homomorphism with respect to either Arens’ multiplication. From this we derive a simple proof of a variant form of Sakai’s characterization of ${W^ \ast }$-algebras.