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Noncommutative residues and a characterisation of the noncommutative integral

Authors: Steven Lord and Fedor A. Sukochev
Journal: Proc. Amer. Math. Soc. 139 (2011), 243-257
MSC (2010): Primary 46L51, 47B10, 58B34; Secondary 58J42, 58C35
Published electronically: July 9, 2010
MathSciNet review: 2729087
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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.

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Additional Information

Steven Lord
Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia

Fedor A. Sukochev
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia

Keywords: Dixmier trace, zeta functions, noncommutative integral, noncommutative geometry, normal, noncommutative residue
Received by editor(s): May 27, 2009
Received by editor(s) in revised form: May 28, 2009, and March 1, 2010
Published electronically: July 9, 2010
Additional Notes: This research was supported by the Australian Research Council
Communicated by: Varghese Mathai
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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