Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals
type: Item
instance of: scholarly article
title: Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals
zbMATH Open document ID: 1170.47057
published in: Journal of Functional Analysis
publication date: 2009-03-03T00:00:00Z
review text: The article is devoted to Lie and Banach Lie algebras \({\mathcal L}\) of bounded linear operators on an infinite-dimensional complex Banach space \(X\). Their subclass with Lie subalgebras \({\mathcal L}_0\) having a nontrivial closed invariant subspace in \(X\) of finite codimension or dimension is studied. A Lie subalgebra \({\mathcal L}_0\) of a Lie algebra \(\mathcal L\) is called nondegenerate in \(\mathcal L\), if \({\mathcal L}_0\) has finite codimension in \(\mathcal L\) and \({\mathcal L} = {\mathcal L}_0 + [{\mathcal L}_0, {\mathcal L}]\). If a chain of embedded Lie subalgebras \({\mathcal L}_0 = {\mathcal L}^0 \subseteq {\mathcal L}^1 \subseteq \dots \subseteq {\mathcal L}^p = {\mathcal L}\) is such that \({\mathcal L}^j\) is nondegenerate in \({\mathcal L}^{j+1}\) for each \(j\), then \({\mathcal L}_0\) is called related to \(\mathcal L\). One the main results of the paper is the following Theorem 8.1. Let \({\mathcal L}_0 \subseteq L \subseteq {\mathcal L}\) be Lie subalgebras of an algebra \({\mathcal B}(X)\) of all bounded operators on an infinite-dimensional complex Banach space X and let \(\dim ({\mathcal L}_0) = \infty \). Suppose that \({\mathcal L}_0\) has a closed invariant subspace \(X_0\). {\parindent=6mm \begin{itemize}\item[(i)] Let \({\mathcal L}_0\) be related to \(L\) and let \(L\) be a Lie ideal of \({\mathcal L}\). If \(\text{codim} (X_0)<\infty \) or \(\dim(X_0)<\infty \), then \(\mathcal L\) is reducible. \item[(ii)] Let \({\mathcal L}_0\) be a Lie ideal of \(L\), let also \(\text{codim}({\mathcal L}_0)<\infty \) and let \(L\) be related to \(\mathcal L\). If \(\text{codim}(X_0)<\infty \) and \({\mathcal L}_0\) has no operator-nilpotent Lie ideals of finite codimension, then \(\mathcal L\) is reducible. \end{itemize}} Moreover, invariant subspaces of operator Lie algebras are investigated. These results are applied to studies of Lie ideals in Banach Lie algebras, particularly, to the problem whether there exist nontrivial closed Lie …
Mathematics Subject Classification ID: 47L10
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 5522892
zbMATH Keywords: Lie algebras of bounded operators
zbMATH Keywords: invariant subspaces
author: E. Kissin
author: V. S. Shulman
author: Yurii V. Turovskii
reviewed by: Sergey V. Ludkovsky
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.10.012
OpenAlex ID: W2093743829
cites work: Quasi-Ideals of Lie Algebras II
cites work: On the cohomology of soluble Lie algebras
cites work: Lie algebras of bounded operators
cites work: Q5614712
cites work: Lie ideals: from pure algebra to C*-algebras
cites work: Q5521075
cites work: On Some Reflexive Algebras of Operators and the Operator Lie Algebras of Their Derivations
cites work: On normed Lie algebras with sufficiently many subalgebras of codimension I
cites work: Q3762468
cites work: On the structure of algebraic algebras
cites work: On the closed Lie ideals of certain \(C^*\)-algebras
cites work: Associative and Lie subalgebras of finite codimension
cites work: Q5686535
cites work: Simultaneous triangularization
cites work: A Solution to the Invariant Subspace Problem on the Space l 1
cites work: Joint spectral radius, operator semigroups, and a problem of W. Wojtyński
cites work: Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action
cites work: Lie algebras all of whose maximal subalgebras have codimension one
cites work: Volterra semigroups have invariant subspaces
cites work: Q4152980
DOI: 10.1016/J.JFA.2008.10.012
description: scientific article; zbMATH DE number 5522892