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

TL;DR

This paper establishes that for a Lie groupoid, the corona algebras of the full and reduced $C^*$-algebras, derived from order 0 pseudodifferential calculus, are identical to the principal symbol algebra.

Contribution

It proves the equality of corona algebras and principal symbol algebras for order 0 pseudodifferential operators on Lie groupoids, clarifying their structure.

Findings

Corona algebras of full and reduced $C^*$-algebras are equal.

Both corona algebras are isomorphic to the principal symbol algebra.

Results apply to the analysis of pseudodifferential operators on Lie groupoids.

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.