Noncommutative Residues and a Characterisation of the Noncommutative Integral

Details Noncommutative Residues and a Characterisation of the Noncommutative Integral Noncommutative Residues and a Characterisation of the Noncommutative Integral

2009-05-02

arxiv.org/abs/0905.0187v1

Proc. Amer. Math. Soc., (2010)

Mathematics

Operator Algebras

15 pages

Scientific paper

10.1090/S0002-9939-2010-10472-0

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), i.e. as a generalised quantum limit. 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.

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