Derivations mapping into the socle, III

Nadia Boudi¹ ; Peter Šemrl²

¹ Département de Mathématiques Université Moulay Ismail Faculté des Sciences, BP 11201 Zitoune, Meknes, Morocco
² Faculty of Mathematics and Physics University of Ljubljana Jadranska 19, SI-1000 Ljubljana, Slovenia

Studia Mathematica, Tome 197 (2010) no. 2, pp. 141-155

Résumé

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.

Détail

DOI : 10.4064/sm197-2-2

Keywords: banach algebra continuous derivation each element range has finite spectrum series papers has proved derivation inner derivation implemented element socle modulo radical precise formulation statement found introduction paper twofold extend result where necessarily continuous complete description maps semisimple