Document Type : Research Paper


Department of Mathematics, Faculty of Science, Arak University, 38156-8-8349, Arak, Iran.


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.


Main Subjects

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