Lw∗wc and Rw∗wc and weak amenability of banach algebras

We introduce some new concepts as lef t − weak∗ − weak convergence property [Lw∗wc−property] and right−weak∗− weak convergence property [Rw∗wc−property] 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. Let D : A → A ∗ be a surjective derivation. If D ۰۰ is a derivation, then A is Arens regular.