  In this note, we study the Arens regularity of projective tensor product
$A\hat{\otimes}B$ whenever $A$ and $B$ are Arens regular. We establish some new
conditions for showing that the Banach algebras $A$ and $B$ are Arens regular
if and only if $A\hat{\otimes}B$ is Arens regular. We also introduce some new
concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and
right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach
algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have
$Rw^*wc-$property and $Lw^*wc-$property. Then if $A^{**}$ is weakly amenable,
it follows that $A$ is weakly amenable. We also offer some results concerning
the relation between these properties with some special derivation
$D:A\rightarrow A^*$. We obtain some conclusions in the Arens regularity of
Banach algebras.
