\sigma-weak amenability of Banach algebras

Let \mathcal{A} be a Banach algebra, \sigma be continuous homomorphism on \mathcal{A} with \overline{\sigma(\mathcal{A})}=\mathcal{A}. The bounded linear map D : \mathcal{A}\to\mathcal{A}^* is \sigma-derivation, ifD(ab) = D(a) \sigma(b) + \sigma(a)  D(b)\quad (a, b\in \mathcal{A}).We say that A is \sigma-weakly amenable, when for each bounded derivation D : \mathcal{A}\to\mathcal{A}^*, there exists a^*\in \mathcal{A}^* such that D(a) = \sigma(a) a^*-a^*\sigma(a). For a commutative Banach algebra \mathcal{A}, we show \mathcal{A} is \sigma-weakly amenable if and only if every \sigma-derivation from \mathcal{A} into a \sigma-symmetric Banach \mathcal{A}-bimodule X is zero. Also, we show that a commutative Banach algebra \mathcal{A} is \sigma-weakly amenable if and only if A^\# is \sigma^\#-weakly amenable, where \sigma^\#(a + \alpha) = \sigma(a) +\alpha.