On the range of a derivation

Abstract

Let be a Banach algebra and a derivation. In this paper, it is proved, under certain conditions, that , where is the Jacobson radical of . Moreover, we prove that if is unital and is a continuous derivation, then , where denotes the set of all primitive ideals such that is commutative, denotes the set of all maximal (modular) ideals such that is commutative, and is the set of all non-zero multiplicative linear functionals from into . In addition, we present several results about the range of a derivation on algebras having the property ( ).