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Type: Journal article
Title: Noncommutative residues and a characterisation of the noncommutative integral
Author: Lord, S.
Sukochev, F.
Citation: Proceedings of the American Mathematical Society, 2011; 139(1):243-257
Publisher: Amer Mathematical Soc
Issue Date: 2011
ISSN: 0002-9939
Statement of
Steven Lord and Fedor A. Sukochev
Abstract: We continue the study of the relationship between Dixmier traces and noncommutative residues initiated by A. Connes. The utility of the residue approach to Dixmier traces is shown by a characterisation of the noncommutative integral in Connes' noncommutative geometry (for a wide class of Dixmier traces) as a generalised limit of vector states associated to the eigenvectors of a compact operator (or an unbounded operator with compact resolvent). Using the characterisation, a criteria involving the eigenvectors of a compact operator and the projections of a von Neumann subalgebra of bounded operators is given so that the noncommutative integral associated to the compact operator is normal, i.e. satisfies a monotone convergence theorem, for the von Neumann subalgebra. Flat tori, noncommutative tori, and a link with the QUE property of manifolds are given as examples.
Rights: © 2010 American Mathematical Society. The copyright for this article reverts to public domain after 28 years from publication.
DOI: 10.1090/S0002-9939-2010-10472-0
Grant ID: ARC
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