(-1)-Weak Amenability of Second Dual of Real Banach Algebras

Let (A,broken vertical bar broken vertical bar . broken vertical bar broken vertical bar) be a real Banach algebra, a complex algebra A(C) be a complexification of A and broken vertical bar broken vertical bar broken vertical bar . broken vertical bar broken vertical bar broken vertical bar be an algebra norm on AC satisfying a simple condition together with the norm broken vertical bar broken vertical bar . broken vertical bar broken vertical bar on A. In this paper we first show that A* is a real Banach A**-module if and only if (A(C))* is a complex Banach (A(C))**-module. Next we prove that A** is (-1)-Weakly amenable if and only if (A(C))** is (-1)-Weakly amenable. Finally, we give some examples of real Banach algebras which their second duals of some them are and of others are not (-1)-Weakly amenable.