Noncommutative residues and a characterisation of the noncommutative integral

Lord, Steven; Sukochev, Fedor

doi:10.1090/S0002-9939-2010-10472-0

Noncommutative residues and a characterisation of the noncommutative integral
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by Steven Lord and Fedor A. Sukochev PDF

Proc. Amer. Math. Soc. 139 (2011), 243-257 Request permission

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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