Noncommutative residues and a characterisation of the noncommutative integral

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dc.contributor.authorLord, S.
dc.contributor.authorSukochev, F.
dc.date.issued2011
dc.description.abstractWe 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.
dc.description.statementofresponsibilitySteven Lord and Fedor A. Sukochev
dc.identifier.citationProceedings of the American Mathematical Society, 2011; 139(1):243-257
dc.identifier.doi10.1090/S0002-9939-2010-10472-0
dc.identifier.issn0002-9939
dc.identifier.issn1088-6826
dc.identifier.orcidLord, S. [0000-0002-6142-5358]
dc.identifier.urihttp://hdl.handle.net/2440/63726
dc.language.isoen
dc.publisherAmer Mathematical Soc
dc.relation.grantARC
dc.rights© 2010 American Mathematical Society. The copyright for this article reverts to public domain after 28 years from publication.
dc.source.urihttps://doi.org/10.1090/s0002-9939-2010-10472-0
dc.titleNoncommutative residues and a characterisation of the noncommutative integral
dc.typeJournal article
pubs.publication-statusPublished

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