Sigma-weak amenability of Banach algebras

Abstract

Let A be a Banach algebra,  be continuous homomorphism on A with (A) = A. The boundedlinear map D : A ! A is 􀀀derivation, ifD(ab) = D(a)
(b) + (a)
D(b) (a; b 2 A):We say that A is -weakly amenable, when for each bounded derivation D : A ! A, there existsa 2 A such that D(a) = (a)
a 􀀀 a
(a). For a commutative Banach algebra A, we showA is 􀀀weakly amenable if and only if every 􀀀derivation from A into a 􀀀symmetric BanachA􀀀bimodule X is zero. Also, we show that a commutative Banach algebra A is 􀀀weakly amenableif and only if A# is #􀀀weakly amenable, where #(a + ) = (a) + .