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

Abstract:

Let $A$ be a ${B^{\ast }}$-algebra, ${A^{{\ast }{\ast }}}$ its second conjugate space and $\pi$ the canonical embedding of $A$ into ${A^{{\ast }{\ast }}}$. ${A^{{\ast }{\ast }}}$ is a ${B^{\ast }}$-algebra under the Arens product. Our main result states that $A$ is a dual algebra if and only if $\pi (A)$ is a two-sided ideal of ${A^{{\ast }{\ast }}}$. Gulick has shown that for a commutative $A,\;\pi (A)$ is an ideal if and only if the carrier space of $A$ is discrete. As this is equivalent to $A$ being a dual algebra, Gulick’s result thus carries over to the general ${B^{\ast }}$-algebra.