%0 Journal Article %T $(-1)$-Weak Amenability of Second Dual of Real Banach Algebras %J Sahand Communications in Mathematical Analysis %I University of Maragheh %Z 2322-5807 %A Alihoseini, Hamidreza %A Alimohammadi, Davood %D 2018 %\ 11/01/2018 %V 12 %N 1 %P 59-88 %! $(-1)$-Weak Amenability of Second Dual of Real Banach Algebras %K Banach algebra‎ %K ‎Banach module‎ %K ‎Complexification‎ %K ‎Derivation‎ %K ‎$(-1)$-Weak amenability %R 10.22130/scma.2018.88929.466 %X 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. %U https://scma.maragheh.ac.ir/article_34113_087079dab0bf46a0162249a173f41f59.pdf