Traces of compact operators and the noncommutative residue

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 à la Connes, including the definitive statement of Connes' original theorem.

arXiv commentVersion change: added information on Nigel Kalton. *Nigel Kalton (1946-2010). The author passed away during production of this paper

References in corpus (10)

• String Theory and Noncommutative Geometry
• Noncommutative Geometry and Matrix Theory: Compactification on Tori
• Gravity coupled with matter and foundation of non-commutative geometry
• The Spectral Action Principle
• D-branes and the Noncommutative Torus
• Moyal Planes are Spectral Triples
• Dimensions and singular traces for spectral triples, with applications to fractals
• Dixmier Traces as Singular Symmetric Functionals and Applications to Measurable Operators
• Measures from Dixmier Traces and Zeta Functions
• Lidskii-type formulae for Dixmier traces

Cited by in corpus (14)

• Quantum differentiability on quantum tori
• Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds
• Connes integration formula for the noncommutative plane
• Cwikel-Solomyak estimates on tori and Euclidean spaces
• Weyl's laws and Connes' integration formulas for matrix-valued $L\log L$-Orlicz potentials
• An Application of Singular Traces to Crystals and Percolation
• Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry
• Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature
• Connes Integration Formula without singular traces
• A $C^{\ast }$-algebraic approach to the principal symbol II
• Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori
• A brief survey of Nigel Kalton's work on interpolation and related topics
• Banach limits and traces on ${\mathcal{L}}_{1,\infty }$
• Fubini theorem in noncommutative geometry