When is a ϕ-derivation continuous and where can its image be found?

Abstract

This paper has two main objectives. One of them is to obtain some conditions under which the image of a [Formula: see text]-derivation on a Banach algebra [Formula: see text] is contained in the Jacobson radical of [Formula: see text] and other purpose of this study is to present some results about the automatic continuity of [Formula: see text]-derivations in Banach algebras.