A new topology on $B^{\ast }$-algebras arising from the Arens products

Abstract:

A locally convex topology $\mu$ is defined on a Banach algebra A. This topology arises naturally from considerations of the Arens products on the second conjugate space ${A^{ \ast \ast }}$ of A. The main result states that if A is a ${B^\ast }$-algebra on which the mapping $(a,b) \to ab$ is $\mu$-continuous for $\left \| a \right \| \leqq 1$, then the completion of A with respect to the uniformity generated by $\mu$ is linearly isomorphic to ${A^{ \ast \ast }}$. An example is included which shows that this continuity condition does not hold in general as announced by P. C. Shields.