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

Abstract

From the viewpoint of $\ast$-homomorphism on $C^{\ast }$-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 $C^{\ast }$-algebra bundle $E_{hom}$ of $M$ constructed from quasi-Riesz transforms on $\mathbb{G}$, we show that there exists a surjective $\ast$-homomorphism $${\mathrm{sym}}_M:{\Pi }_M\to C_b(E_{hom})$$ such that $$\ker ({\mathrm{sym}}_M)=\mathcal{K}(L_2(M,\nu ))\subset {\Pi }_M$$ where the domain ${\Pi }_M\subset \mathcal{B}(L_2(M,\nu ))$ is a $C^{\ast }$-algebra and $C_b(E_{hom})$ is the $C^{\ast }$-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}.