  Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We define
$\tilde{Z}_1(A^{**})$ as a weak topological center of $A^{**}$ with respect to
first Arens product and we will find some relations between this concept and
the topological center of $A^{**}$. We also extend this new definition into the
module actions and find relationship between weak topological center of module
actions and reflexivity or Arens regularity of some Banach algebras, and we
investigate some applications of this new definition in the weak amenability of
some Banach algebras.
