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

Let $ (A,| cdot |) $ be a real Banach algebra, a complex algebra $ A_mathbb{C} $ be a complexification of $ A $ and $ | | cdot | | $ be an algebra norm on  $ A_mathbb{C}  $  satisfying a simple condition together with the norm $ | cdot | $ on $ A$.  In this paper we first show that $ A^* $ is a real Banach $ A^{**}$-module if and only if $ (A_mathbb{C})^* $ is a complex Banach $ (A_mathbb{C})^{**}$-module. Next  we prove that $ A^{**} $ is $ (-1)$-weakly  amenable if and only if $ (A_mathbb{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.