LAUSR

Abstract Let A be unital prime Banach algebra over ℝ or ℂ with centre and G 1, G 2 be open subsets of be a continuous linear generalized skew derivation, and be a continuous linear map. We prove that must be commutative if one of the following conditions holds: For each a ∈ G 1 , b ∈ G 2, there exists an integer m ∈ Z >1 depending on a and b such that either . For each a ∈ G 1 , b ∈ G 2, 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].