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Derivations mapping into the radical. II. (English) Zbl 0733.46023
In 1955 I. M. Singer and J. Wermer formulated the conjecture that every derivation on a commutative Banach algebra maps into the radical, and proved the result in the bounded case [Math. Ann. 129, 260–264 (1955; Zbl 0067.35101)]. It took more than thirty years until M. P. Thomas verified this conjecture in [Ann. Math. (2) 128, 435–460 (1988; Zbl 0681.47016)]. Since then a number of non-commutative versions of the Singer-Wermer theorem were obtained. In [Arch. Math. 57, No. 5, 469–474 (1991; Zbl 0714.46038)] the first-named author jointly with G. J. Murphy proved the result that every bounded centralising derivation on an arbitrary Banach algebra maps into the radical. In the present paper, the boundedness assumption is removed using a classical result of Posner on centralising derivations of prime rings and Thomas’ theorem.

MSC:
46H05 General theory of topological algebras
46J45 Radical Banach algebras
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