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

Let (A,∥⋅∥) be a real Banach algebra, a complex algebra AC be a complexification of A and ∥|⋅∥| be an algebra norm on AC satisfying a simple condition together with the norm ∥⋅∥ on A. In this paper we first show that A∗ is a real Banach A∗∗-module if and only if (AC)∗ is a complex Banach (AC)∗∗-module. Next we prove that A∗∗ is (−1)-weakly amenable if and only if (AC)∗∗ 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.