Approximate Identity and Arens Regularity of Some Banach Algebras

Abstract

Let $A$ be a Banach algebra with the second dual $A^{\ast \ast }$. If $A$ has a bounded approximate identity $(=BAI)$, then $A^{\ast \ast }$ is unital if and only if $A^{\ast \ast }$ has a $wea{k}^{\ast }boundedapproximate$$identity(=W^{\ast }BAI)$. If $A$ is Arens regular and $A$ \noindent has a BAI, then $A^{\ast }$ factors on both sides. In this paper we introduce new concepts $L{W}^{\ast }W$ and $R{W}^{\ast }W$- property and we show that under certain conditions if $A$ has $L{W}^{\ast }W$ and $R{W}^{\ast }W$- property, then $A$ is Arens regular and also if $A$ is Arens regular, then $A$ has $L{W}^{\ast }W$ and $R{W}^{\ast }W$- property. We also offer some applications of these new concepts for the special algebras $l^1(G),L^1(G),M(G)$, and $A(G)$.

15 page