A $C^{*}$-Algebraic Approach To Principal Symbol Calculus On Filtered Manifolds

TL;DR

This paper develops a $C^{*}$-algebraic framework for principal symbol calculus on filtered manifolds, extending previous results without assuming compactness or specific lattice structures.

Contribution

It introduces a $C^{*}$-algebraic principal symbol map for filtered manifolds modeled on stratified Lie groups, generalizing prior approaches.

Findings

Established a surjective $*$-homomorphism for principal symbols

Connected the kernel to compact operators on $L_2$ spaces

Removed assumptions on lattice structures and compactness

Abstract

From the viewpoint of $*$-homomorphism on $C^{*}$-algebras, we establish the principal symbol mapping for filtered manifolds which are locally isomorphic to stratified Lie groups. Let $\mathbb{G}$ be a stratified Lie group, and let $M$ be a filtered manifold with a $\mathbb{G}$-atlas and a smooth positive density $\nu$. For the $C^{*}$-algebra bundle $E_{hom}$ of $M$ constructed from quasi-Riesz transforms on $\mathbb{G}$, we show that there exists a surjective $*$-homomorphism $${\rm sym}_{M}:\Pi_{M}\to C_{b}(E_{hom})$$ such that $${\rm ker}({\rm sym}_{M})=\mathcal{K}(L_{2}(M,\nu))\subset \Pi_{M}$$ where the domain $\Pi_{M}\subset\mathcal{B}(L_{2}(M,\nu))$ is a $C^{*}$-algebra and $C_{b}(E_{hom})$ is the $C^{*}$-algebra of bounded continuous sections of $E_{hom}$. Especially, we do not make any assumptions on the lattice of the osculating group of $M$ or the assumption of compactness on manifolds in \cite{DAO3,DAO4}.