Derivations mapping into the radical
type: Item
instance of: scholarly article
title: Derivations mapping into the radical
zbMATH Open document ID: 0714.46038
DOI: 10.1007/BF01246745
author: Martin Mathieu
author: G. J. Murphy
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: 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)].
reviewed by: Martin Mathieu
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46L57
zbMATH DE Number: 4175757
zbMATH Keywords: Posner's theorems
zbMATH Keywords: Kleinecke-Shirokov theorem
zbMATH Keywords: Singer-Wermer theorem
zbMATH Keywords: local and global versions
zbMATH Keywords: bounded derivations on non-commutative Banach algebras
zbMATH Keywords: bounded centralising derivation
zbMATH Keywords: radical
MaRDI profile type: Publication
cites work: Q3215824
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full work available at URL: https://doi.org/10.1007/bf01246745
OpenAlex ID: W2010997366
description: scientific article; zbMATH DE number 4175757