Noncommutative Residues and a Characterisation of the Noncommutative Integral

TL;DR

This paper explores the connection between noncommutative residues and Dixmier traces, providing a new characterization of the noncommutative integral as a generalized limit of eigenvector states, with implications for normality criteria.

Contribution

It introduces a novel characterization of the noncommutative integral using eigenvector-based limits and establishes criteria for its normality within von Neumann subalgebras.

Findings

Characterization of the noncommutative integral as a generalized limit of eigenvector states.

Criteria for the normality of the noncommutative integral in von Neumann subalgebras.

Extension of residue approach to a broader class of Dixmier traces.

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), 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.