Traces of compact operators and the noncommutative residue

TL;DR

This paper extends the noncommutative residue to a broader class of pseudo-differential operators, generalizes Connes' trace theorem, and explores the implications for non-measurable operators in noncommutative geometry.

Contribution

It broadens the noncommutative residue concept and generalizes Connes' trace theorem, revealing non-uniqueness of traces for certain pseudo-differential operators.

Findings

Extended noncommutative residue to operators of order -d

Generalized Connes' trace theorem to new operator classes

Identified non-measurable pseudo-differential operators in noncommutative geometry

Abstract

We extend the noncommutative residue of M. Wodzicki on compactly supported classical pseudo-differential operators of order $-d$ and generalise A. Connes' trace theorem, which states that the residue can be calculated using a singular trace on compact operators. Contrary to the role of the noncommutative residue for the classical pseudo-differential operators, a corollary is that the pseudo-differential operators of order $-d$ do not have a `unique' trace; pseudo-differential operators can be non-measurable in Connes' sense. Other corollaries are given clarifying the role of Dixmier traces in noncommutative geometry \`{a} la Connes, including the definitive statement of Connes' original theorem.