  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 $A(G)$.
