On density of transitive operator algebras
type: Item
instance of: scholarly article
title: On density of transitive operator algebras
zbMATH Open document ID: 1195.46061
author: Jaeseong Heo
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-02-19T00:00:00Z
review text: Let \(G\) be a discrete group and let \({\mathcal H}\) be the Hilbert space \(l^{2}(G)\) with the usual inner product. For each \(g\in G\), let \(\lambda_{g}\) denote the left translation of functions in \({\mathcal H}\) by \(g^{-1}\). Let \(L(G)\) be the von Neumann algebra generated by \(\{ \lambda_{g}:g\in G\} \). Similarly, let \(R(G)\) be the corresponding von Neumann algebra generated by right translations. Then \(R(G)=L(G)'\). Let \(\mathbf{F}_{\infty}\) be the free group with countable infinite set of generators \(\{ a_{i}:i=1,2,\dots\}\). Consider two sums \(\lambda_{a_{1}}+\lambda_{a_{2}}\) and \(\lambda_{a_{3}}+\lambda_{a_{4}}\) of unitary generators in the free von Neumann algebra \(L(\mathbf{F}_{\infty})\). Then the main result of the paper says that the set \( \{ \lambda_{a_{1}}+\lambda_{a_{2}}, \lambda_{a_{3}}+\lambda_{a_{4}} \} \) together with the commutant \(R(\mathbf{F}_{\infty})\) generate \({\mathcal B}({\mathcal H})\) as a von Neumann algebra. The author also discusses relations between the transitive algebra problem and the invariant subspace problem relative to some von Neumann algebras.
reviewed by: Vladimir S. Pilidi
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47C15
zbMATH DE Number: 5673280
zbMATH Keywords: transitive set
zbMATH Keywords: transitive operator algebra
zbMATH Keywords: invariant subspace
zbMATH Keywords: free group factor
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.10.060
OpenAlex ID: W2044368213
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DOI: 10.1016/J.JMAA.2009.10.060
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description: scientific article; zbMATH DE number 5673280