Aspects of the Theory of Derivations

We survey some old and new results in the theory of derivations on Banach algebras. Although our overview is broad ranging, our principal interest is in recent results concerning conditions on a derivation implying that its range is contained in the radical of the algebra. In this paper we present a survey of some aspects of the theory of derivations on Banach algebras. No attempt is made at completeness; rather, our intention is to cover the basic theory and to discuss some recent results. We are principally interested in certain results generalising the Singer–Wermer theorem; more precisely, in certain generalisations involving conditions on a derivation that ensure that its range is contained in the radical of the algebra. We also discuss other results, and some problems, related to this generalised Singer–Wermer theory. A very important aspect of the theory of derivations centres on the question of innerness of all the derivations on a given Banach algebra. We give a brief overview of this theory and of the closely related results pertaining to the problem of whether all the derivations of an algebra can be made “inner” in some bigger algebra. We shall occasionally include (easier) proofs to give some indication of the methods used in this area. Also, we shall give details of some examples where it is easy to do so. We shall be concerned, for the most part, in this exposition only with bounded everywhere-defined derivations on Banach algebras, with only occasional references to unbounded derivations. There exists a well-developed theory of denselydefined derivations that has a wide variety of applications. The interested reader is referred to Sakai’s recent book [19] for details. This book also contains an extensive bibliography. 1991 Mathematics Subject Classification: 46H99, 46J99, 47B47. The paper is in final form and no version of it will be published elsewhere.