On tractability and ideal problem in non-associative operator algebras
type: Item
instance of: scholarly article
title: On tractability and ideal problem in non-associative operator algebras
zbMATH Open document ID: 1203.46032
author: Matej Brešar
author: V. S. Shulman
author: Yurii V. Turovskii
published in: Integral Equations and Operator Theory
publication date: 2010-08-23T00:00:00Z
review text: The paper centers around the following problem by \textit{W.\,Wojtyński} [Bull.\ Acad.\ Polon.\ Sci.\ 24, 797--801 (1976; Zbl 0356.47015)]: Does every closed Lie algebra of compact quasinilpotent operators on a Banach space contain a non-trivial closed Lie ideal? The authors give several partial answers and, approaching the problem from different directions, they find certain tools and introduce some concepts that might be of independent interest. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other results, it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtyński's problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 5774015
zbMATH Keywords: Lie algebra
zbMATH Keywords: Jordan algebra
zbMATH Keywords: ideal
zbMATH Keywords: elementary operator
zbMATH Keywords: compact operator
zbMATH Keywords: invariant subspace
zbMATH Keywords: triangularizability
zbMATH Keywords: tractability
reviewed by: Sergei S. Platonov
MaRDI profile type: Publication
full work available at URL: s00020-010-1781-z
OpenAlex ID: W2034908186
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DOI: 10.1007/S00020-010-1781-Z
description: scientific article; zbMATH DE number 5774015