  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 find some relations between it and the topological
center of $A^{**}$. We also extend this new definition into 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.
