On skew derivations and generalized skew derivations in Banach algebras

Abstract

Let ? be a unital prime Banach algebra over ℝ or ℂ with centre and G_1,G₂ be open subsets of ?, F : ? → ? be a continuous linear generalized skew derivation, and D : ? → ? be a continuous linear map. We prove that ? must be commutative if one of the following conditions holds:
(a) For each a ∈ G₁; b ∈ G₂, there exists an integer m ∈ Z^>1 depending on a and b
such that either .
(b) For each a ∈  G₁; b ∈  G₂, there exists an integer m ∈ Z ^>1 depending on a and b
such that either .
These results generalize a number of theorems of this type. In particular, as an application, we give an affirmative answer to some questions posed in [21].

Mathematics Subject Classication (2010): 16W25, 46J10.