GENERALIZED σ-DERIVATION ON BANACH ALGEBRAS

Abstract. Let A be a Banach algebra and M be a Banach A-bimodule. We say that a linear mapping δ: A →M is a generalized σ-derivation whenever there exists a σ-derivation d: A →M such that δ(ab) = δ(a)σ(b) + σ(a)d(b), for all a, b ∈ A. Giving some facts concerning generalized σ-derivations, we prove that if A is unital and if δ: A → A is a generalized σ-derivation and there exists an element a ∈ A such that d(a) is invertible, then δ is continuous if and only if d is continuous. We also show that ifM is unital, has no zero divisor and δ: A → M is a generalized σ-derivation such that d(1) 6 = 0, then ker(δ) is a bi-ideal of A and ker(δ) = ker(σ) = ker(d), where 1 denotes the unit element of A. 1