The Arens product and duality in $B^{\ast }$-algebras. II

Abstract:

Let $A$ be a commutative ${B^ \ast }$-algebra, $\Phi$ its carrier space and ${A^ \ast }$ the conjugate space of $A$. Let $A’$ be the closed subspace of ${A^ \ast }$ spanned by $\Phi$. We show that $A$ is a dual algebra if and only if $A’ = {A^ \ast }$ and for each $x \in A$, the mapping ${T_x}:f \to f \ast x$ is a weakly completely continuous operator on ${A^ \ast }$. This improves an early result by B. J. Tomiuk and the author. A similar result holds for general ${B^ \ast }$-algebras.