Sums of commutators in non-commutative Banach function spaces.
type: Item
instance of: scholarly article
title: Sums of commutators in non-commutative Banach function spaces.
zbMATH Open document ID: 1062.46053
DOI: 10.1016/S0022-1236(03)00235-0
author: Thierry Fack
published in: Journal of Functional Analysis
publication date: 2004-03-14T00:00:00Z
review text: In [J. Reine Angew. Math. 504, 115--125 (1998; Zbl 0919.47022)], \textit{N. J. Kalton} gave a complete spectral characterization of \([I,B(H)]\) for a large class of two-sided ideals \(I\) of compact operators. Here, \([I,B(H)]\) denotes the linear span of commutators \([A,B]=AB-BA\) with \(A\in I\) and \(B\in B(H)\). The importance of the theorem stems from its applications to noncommutative geometry. Fack extends the result, replacing \(B(H)\) with a type \(II_\infty\) factor \(M\) and \(I\) with the set \(I_X(M,\tau)\) of elements of \(M\) belonging to a noncommutative Banach function space \(X(M,\tau)\) associated with a rearrangement invariant Köthe function space \(X\) on \([0,+\infty[\). The main theorem gives conditions under which \(T\) belongs to \(I_X(M,\tau)\) in terms of Brown's spectral measure of \(T\). The result implies that the Dixmier trace of a quasinilpotent element in \(L^{1,\infty}(M,\tau)\) is always zero. This very well written paper contains a wealth of material on noncommutative Banach function spaces.
Mathematics Subject Classification ID: 46L52
Mathematics Subject Classification ID: 47L20
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 2055848
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: commutators
zbMATH Keywords: generalized s-numbers
zbMATH Keywords: noncommutative Banach function spaces
reviewed by: Stanisław Goldstein
MaRDI profile type: Publication
cites work: Q3738112
cites work: The action functional in non-commutative geometry
cites work: Q4837918
cites work: Q5565478
cites work: Non-commutative Banach function spaces
cites work: Q5336643
cites work: Commutator structure of operator ideals
cites work: Q4216765
cites work: Q3955927
cites work: Sommes de commutateurs dans les algèbres de von Neumann finies continues
cites work: Generalized s-numbers of \(\tau\)-measurable operators
cites work: Q3852455
cites work: Trace-class operators and commutators
cites work: Spectral characterization of sums of commutators I
cites work: Non-commutative Lorentz spaces associated with a semi-finite von Neumann algebra and applications
cites work: Q3289300
cites work: Q4136031
cites work: On commutators in ideals of compact operators
cites work: Q3857335
cites work: Q3857325
description: scientific article; zbMATH DE number 2055848