What Can Be Expected from the Image of σ-Derivations on Banach Algebras?

The main purpose of this paper is to obtain some results on the image of -derivations on Banach algebras. One of the main results of this paper is to prove that if is a commutative Banach algebra and is a continuous -derivation such that is a continuous homomorphism, and , then , where denotes the Jacobson radical of . Moreover, we obtain Sinclair's theorem for -derivations without assuming continuity. Indeed, under certain conditions, we prove that if is a -derivation on a Banach algebra , then for every primitive ideal of . Some other related results are also discussed. [ABSTRACT FROM AUTHOR]