Derivations mapping into the socle, III
Tom 197 / 2010
Studia Mathematica 197 (2010), 141-155 MSC: Primary 47B47. DOI: 10.4064/sm197-2-2
Let $A$ be a Banach algebra, and let $d: A \to A$ be a continuous derivation such that each element in the range of $d$ has a finite spectrum. In a series of papers it has been proved that such a derivation is an inner derivation implemented by an element from the socle modulo the radical of $A$ (a precise formulation of this statement can be found in the Introduction). The aim of this paper is twofold: we extend this result to the case where $d$ is not necessarily continuous, and we give a complete description of such maps in the semisimple case.