Adiabatic groupoid, crossed product by $\R_+^*$ and Pseudodifferential calculus

TL;DR

This paper explores the relationship between adiabatic groupoids, crossed products by positive reals, and pseudodifferential calculus, establishing Morita equivalences and expressing operators via integrals on groupoids.

Contribution

It constructs an explicit Morita equivalence linking pseudodifferential operators on a Lie groupoid with those on its adiabatic groupoid, advancing the understanding of their algebraic structures.

Findings

Established Morita equivalence between pseudodifferential operator sequences

Expressed pseudodifferential operators as integrals over smoothing operators

Connected the algebraic structures of $G$ and its adiabatic groupoid

Abstract

We consider the crossed product $G_{ga}$ by $\R_+^*$ of the adiabatic groupoid associated with any Lie groupoid $G$. We construct an explicit Morita equivalence between the exact sequence of order 0 pseudodifferential operators on $G$ and (a restriction of) the natural exact sequence associated with $G_{ga}$. As an important intermediate step, we express a pseudodifferential operator on $G$ as an integral associated to a smoothing operator on the adiabatic groupoid $G_{ad}$ of $G$.