Filtered Calculus and crossed products by R-actions

TL;DR

This paper establishes an isomorphism linking the kernel of the tangent groupoid's C*-algebra for filtered manifolds to a crossed product involving pseudodifferential operators, extending classical calculus results.

Contribution

It introduces a novel isomorphism for filtered manifolds' tangent groupoid C*-algebra, utilizing a structure result for graded nilpotent Lie groups.

Findings

Isomorphism between kernel of tangent groupoid C*-algebra and crossed product.

Decomposition of principal symbol algebra for filtered calculus.

Extension of Epstein and Melrose's decomposition to contact manifolds.

Abstract

We show an isomorphism between the kernel of the C*-algebra of the tangent groupoid of a filtered manifold and the crossed product of the order 0 pseudodifferential operators in the associated filtered calculus by a natural R-action. This isomorphism is constructed in the same way as in the classical pseudodifferential calculus by Debord and Skandalis. The proof however relies on a structure result for the C*-algebra of graded nilpotent Lie groups which did not appear in the commutative case. A consequence of this structure result is a decomposition of the principal symbol algebra, generalizing the decomposition of Epstein and Melrose in the case of contact manifolds.