Generalized Derivations and Generalized Amenability of Banach Algebras

Amenability is a cohomological property of Banach algebras which was introduced by Johnson in [14]. Let A be a Banach algebra, and suppose that X is a Banach A−bimodule such that the following statements hold ∥a · x∥ ≤ ∥a∥∥x∥ and ∥x · a∥ ≤ ∥a∥∥x∥ for each a ∈ A and x ∈ X. We can define the right and left actions of A on dual space X∗ of X via ⟨x, λ · a⟩ = ⟨a · x, λ⟩ ⟨x, a · λ⟩ = ⟨x · a, λ⟩, for each a ∈ A, x ∈ X and λ ∈ X∗. Suppose that X is a Banach A-bimodule. A derivation D : A → X is a linear map which satisfies D(ab) = a · D(b) + D(a) · b for each a, b ∈ A and it is called Jordan derivation in case D(x2) = D(x) ·x+x ·D(x) for each x ∈ A. It is clear that every derivation is a Jordan derivation. A derivation δ is said to be inner if there exists a x ∈ X such that δ(a) = δx(a) = a ·x−x ·a for each a ∈ A. We denote the linear space of bounded derivations from A intoX by Z1(A, X) and the linear subspace of inner derivations byN1(A, X). We consider the quotient space H1(A, X) = Z1(A, X)/N1(A, X), it is called the first Hochschild cohomology group of A with coefficients in X. The Banach algebra A is said to be amenable if H1(A, X∗) = {0} for each Banach A-bimodules X. The Banach algebra A is called weakly amenable if, H1(A,A∗) = {0} (for more details