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Exact Sequence of $0$-Order Pseudodifferential Operators on a Lie Groupoid

Mahsa Naraghi

TL;DR

This work extends the exact sequence framework for order-0 pseudodifferential operators from smooth compact manifolds to the setting of Lie groupoids, addressing both the full and reduced groupoid C*-algebras. The authors develop the order-0 calculus on a groupoid, prove that negative-order operators embed into $C^*(G)$, and show that order-0 operators act as multipliers, with a principal symbol living on $C_0(S^*A_G)$. They then extend these constructions to multipliers and establish symbol-approximation results that culminate in isomorphisms between the corona algebras and the principal symbol algebra, for both $C^*(G)$ and $C^*_r(G)$. The main contribution is a unified exact-sequence picture $0 o C^*(G) o Psi^*(G) o C_0(S^*A_G) o 0$ (and its reduced analogue), which broadens index-theory tools to singular spaces modeled by Lie groupoids, such as foliations. This provides a robust, algebraic-analytic framework for analyzing elliptic operators on groupoids and their boundary/collar behavior via corona algebras.

Abstract

Associated to a Lie groupoid, there are two $C^*$-algebras: the full and the reduced one. The associated order $0$ pseudodifferential calculus gives rise to multiplier algebras of both. We prove that both associated corona algebras are equal to the natural (commutative) principal symbol algebras.

Exact Sequence of $0$-Order Pseudodifferential Operators on a Lie Groupoid

TL;DR

This work extends the exact sequence framework for order-0 pseudodifferential operators from smooth compact manifolds to the setting of Lie groupoids, addressing both the full and reduced groupoid C*-algebras. The authors develop the order-0 calculus on a groupoid, prove that negative-order operators embed into , and show that order-0 operators act as multipliers, with a principal symbol living on . They then extend these constructions to multipliers and establish symbol-approximation results that culminate in isomorphisms between the corona algebras and the principal symbol algebra, for both and . The main contribution is a unified exact-sequence picture (and its reduced analogue), which broadens index-theory tools to singular spaces modeled by Lie groupoids, such as foliations. This provides a robust, algebraic-analytic framework for analyzing elliptic operators on groupoids and their boundary/collar behavior via corona algebras.

Abstract

Associated to a Lie groupoid, there are two -algebras: the full and the reduced one. The associated order pseudodifferential calculus gives rise to multiplier algebras of both. We prove that both associated corona algebras are equal to the natural (commutative) principal symbol algebras.
Paper Structure (13 sections, 8 theorems, 23 equations)

This paper contains 13 sections, 8 theorems, 23 equations.

Table of Contents

  1. Preliminaries
  2. Lie groupoids
  3. C*-algebras of a Lie Groupoid.
  4. The convolution algebra.
  5. The full C*-algebra
  6. The reduced $C^*$-algebra.
  7. Pseudodifferential operators on Lie groupoids
  8. Exact Sequences of Pseudodifferential Operators of Order 0
  9. Pseudodifferential operators of negative order
  10. Pseudodifferential operators of order $0$ are multipliers
  11. Extensions to multipliers
  12. Approximation of the principal symbol
  13. The exact sequences

Key Result

Proposition 2.1

For $m<0$, we have $\mathcal{P}^{m}_c(G)\subset C^*(G)$. More precisely, there is an algebra morphism $j:\mathcal{P}^{m}_c(G)\to C^*(G)$ whose restriction to $C_c^\infty(G)$ is the inclusion of $C_c(G)$ in the $C^*$-algebra of $G$.

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 11 more