Approximate Identity and Arens Regularity of Some Banach Algebras

Abstract

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