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Connes trace theorem for Carnot manifolds

Edward McDonald

TL;DR

This work extends Connes' trace theorem to Carnot manifolds by unifying recent tangent-groupoid–based residues (Dave–Haller and Couchet–Yuncken) and showing that any normalised trace on the weak trace class yields $\varphi(T)=\frac{1}{d_H}\mathrm{Res}(T)$ for $T\in \Psi^{-{d_H}}_H(M)$. The authors develop the van Erp–Yuncken calculus, including the $H$-tangent groupoid, zoom action, local traces, and principal cosymbols, to define and compare the DH and CY residues. They prove that these residues coincide and correspond to the generalized Wodzicki residue in the Carnot setting, with a meromorphic-trace framework via holomorphic families. The results connect spectral Weyl-type asymptotics to residue functionals, extending Connes' trace theorem beyond classical elliptic theory and providing a robust tool for noncommutative geometry on filtered manifolds. This has potential implications for index theory, noncommutative geometry, and spectral geometry on sub-Riemannian and hypoelliptic settings.

Abstract

The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold $M$ whose tangent bundle $TM$ is equipped with a nested family $H$ of sub-bundles $H_0\leq H_1 \leq \cdots \leq TM$ which defines a filtration of the Lie algebra of vector fields on $M.$ Differential operators on Carnot manifolds have their order measured in terms of the filtration defined by $H,$ and the algebra of differential operators can be extended to an algebra of pseudodifferential operators. Recently, Dave-Haller and Couchet-Yuncken proposed definitions of a residue functional on the algebra of pseudodifferential operators adapted to a Carnot manifold. We prove that Connes' trace theorem holds in this setting.

Connes trace theorem for Carnot manifolds

TL;DR

This work extends Connes' trace theorem to Carnot manifolds by unifying recent tangent-groupoid–based residues (Dave–Haller and Couchet–Yuncken) and showing that any normalised trace on the weak trace class yields for . The authors develop the van Erp–Yuncken calculus, including the -tangent groupoid, zoom action, local traces, and principal cosymbols, to define and compare the DH and CY residues. They prove that these residues coincide and correspond to the generalized Wodzicki residue in the Carnot setting, with a meromorphic-trace framework via holomorphic families. The results connect spectral Weyl-type asymptotics to residue functionals, extending Connes' trace theorem beyond classical elliptic theory and providing a robust tool for noncommutative geometry on filtered manifolds. This has potential implications for index theory, noncommutative geometry, and spectral geometry on sub-Riemannian and hypoelliptic settings.

Abstract

The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold whose tangent bundle is equipped with a nested family of sub-bundles which defines a filtration of the Lie algebra of vector fields on Differential operators on Carnot manifolds have their order measured in terms of the filtration defined by and the algebra of differential operators can be extended to an algebra of pseudodifferential operators. Recently, Dave-Haller and Couchet-Yuncken proposed definitions of a residue functional on the algebra of pseudodifferential operators adapted to a Carnot manifold. We prove that Connes' trace theorem holds in this setting.
Paper Structure (18 sections, 20 theorems, 192 equations)

This paper contains 18 sections, 20 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Background material
  3. Carnot manifolds
  4. Traces on $\mathcal{L}_{1,\infty}$
  5. The Wodzicki residue and Connes' trace theorem
  6. Main results
  7. Relation to Weyl laws and an open problem
  8. Preliminaries on the van Erp-Yuncken calculus
  9. Densities and fibred distributions
  10. The $H$-tangent groupoid
  11. The zoom action and the zoom vector field
  12. The van Erp-Yuncken calculus
  13. The local trace
  14. The principal symbol mapping and $H$-elliptic operators
  15. Holomorphic families of pseudodifferential operators
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $M$ be a compact $d$-dimensional manifold and let $T\in \Psi^{-d}(M)$ be a classical pseudodifferential operator of order $-d.$ Let $\nu$ be a density on $M.$ Then $T$ can be identified with an element of $\mathcal{L}_{1,\infty}(L_2(M,\nu))$ and for any normalised trace $\varphi$ on $\mathcal{L}

Theorems & Definitions (48)

  • Theorem 2.1: Connes' trace theorem
  • Theorem 2.2
  • Theorem 2.3
  • Conjecture 1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • ...and 38 more