Primitive ideals and derivations on non-commutative Banach algebras
type: Item
instance of: scholarly article
title: Primitive ideals and derivations on non-commutative Banach algebras
zbMATH Open document ID: 0739.47014
author: Marc P. Thomas
published in: Pacific Journal of Mathematics
publication date: 1993-01-01T00:00:00Z
review text: The Singer-Wermer conjecture states that if \(D\) is a (possibly unbounded) derivation on a commutative Banach algebra then the range of \(D\) is contained in the (Jacobson) radical of the algebra. This conjecture is now known to be true. However, it is still not currently known whether or not the Singer-Wermer conjecture on derivations extends to non- commutative Banach algebras in the following sense: if \(D\) is a (possibly unbounded) derivation then is \(D(P)\subseteq P\) for all primitive ideals \(P\) of the algebra? This has become known as the non-commutative version of the Singer-Wermer conjecture. We first correct an automatic continuity result in the literature concerning which (and how many) primitive idelas can fail to be invariant. Using this result together with some representation theory we prove a theorem about derivations whose second iteration annihilates some element (specifically, \(D^ 2a=0\) implies that \(Da\) is quasinilpotent). This theorem does not require commutativity of the algebra and it is easily seen to imply the Singer-Wermer conjecture. The proof itself is done by contradiction in which the remaining case leads to a new derivation on a commutative subalgebra, and this case can be contradicted by the arguments used in the proof of the Singer-Wermer conjecture.
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46H40
Mathematics Subject Classification ID: 46H15
zbMATH DE Number: 4216255
zbMATH Keywords: Jacobson radical
zbMATH Keywords: derivation on a commutative Banach algebra
zbMATH Keywords: non- commutative Banach algebras
zbMATH Keywords: non-commutative version of the Singer-Wermer conjecture
zbMATH Keywords: automatic continuity
zbMATH Keywords: primitive idelas
zbMATH Keywords: derivations whose second iteration annihilates some element
MaRDI profile type: Publication
full work available at URL: pjm.1993.159.139
OpenAlex ID: W2093248102
DOI: 10.2140/PJM.1993.159.139
description: scientific article; zbMATH DE number 4216255