Sigma-weak amenability of Banach algebras

Document Type : Research Paper


Department of Mathematics, Persian Gulf University, Bushehr, 75168, Iran


Let A be a Banach algebra,  be continuous homomorphism on A with (A) = A. The bounded
linear map D : A ! A is 􀀀derivation, if
D(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 exists
a 2 A such that D(a) = (a)  a 􀀀 a  (a). For a commutative Banach algebra A, we show
A is 􀀀weakly amenable if and only if every 􀀀derivation from A into a 􀀀symmetric Banach
A􀀀bimodule X is zero. Also, we show that a commutative Banach algebra A is 􀀀weakly amenable
if and only if A# is #􀀀weakly amenable, where #(a + ) = (a) + .