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A groupoid approach to the Wodzicki residue

Nathan Couchet, Robert Yuncken

Abstract

Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis and Guillemin. In this article we give a definition of the Wodzicki residue, using the langage of r-fibered distributions in the context of filtered manifolds. We show that this groupoidal residue behaves like a trace on the algebra of pseudodifferential operators on filtered manifolds and coincides with the usual residue Wodzicki in the case where the manifold is trivially filtered. Moreover, in the context of Heisenberg calculus, we show that the groupoidal residue coincides with Ponge's definition for contact and codimension 1 foliation Heisenberg manifolds.

A groupoid approach to the Wodzicki residue

Abstract

Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis and Guillemin. In this article we give a definition of the Wodzicki residue, using the langage of r-fibered distributions in the context of filtered manifolds. We show that this groupoidal residue behaves like a trace on the algebra of pseudodifferential operators on filtered manifolds and coincides with the usual residue Wodzicki in the case where the manifold is trivially filtered. Moreover, in the context of Heisenberg calculus, we show that the groupoidal residue coincides with Ponge's definition for contact and codimension 1 foliation Heisenberg manifolds.
Paper Structure (5 sections, 11 theorems, 75 equations)

This paper contains 5 sections, 11 theorems, 75 equations.

Table of Contents

  1. Introduction.
  2. Basic properties of the Wodzicki residue on a filtered manifold.
  3. Pseudo-homogeneous functions and kernels
  4. The Wodziciki residue coincides with the groupoidal residue
  5. The noncommutative residue on a filtered manifold

Key Result

Lemma 1.3

Let $M$ be a filtered manifold and $\mathbbm{k} \in \boldsymbol{\Psi}_{\text{vEY}}^{- d_H}(\mathbb{T}_HM)$. For every $x \in M$, the function defined by: is a group homomorphism from $(\mathbb{R}_+^*,\times)$ to $(\mathbb{C},+)$. More precisely, there exists a constant $r_x \in \mathbb{C}$ such that for all $s>0$:

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Definition 1.4
  • proof : Proof of Lemma \ref{['759']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 18 more