Derivations mapping into the radical (Q750832)

The ideas investigated in this paper centre around two classical results of Banach algebra theory, the Kleinecke-Shirokov theorem and the Singer- Wermer theorem. New light is shed on these results by proving local and global versions in the setting of bounded derivations on non-commutative Banach algebras. For example, every bounded centralising derivation maps into the radical which may be viewed both as a global version of the Kleinecke-Shirokov theorem as well as a non-commutative version of the Singer-Wermer theorem. Apart from the reduction to the primitive case (which uses a theorem due to Sinclair) the proofs are purely algebraic wherefore the results may be true in the unbounded situation too. Related results were obtained independently by \textit{M. Brešar} and \textit{J. Vukman} in [Proc. Am. Math. Soc. 110, 7-16 (1990; Zbl 0703.16020)].