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

2024/10/29 by David Farrell, Fedor Sukochev, Farrell, David +5 · 1 citation
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Numerical methods for differential equations #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2410.22125

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 \mathbbG be a stratified Lie group, and let M be a filtered manifold with a \mathbbG-atlas and a smooth positive density ν. For the C^*-algebra bundle E_hom of M constructed from quasi-Riesz transforms on \mathbbG, we show that there exists a surjective *-homomorphism \rm sym_M:Π_M→ C_b(E_hom) such that \rm ker(\rm sym_M)=K(L₂(M,ν))⊂ Π_M where the domain Π_M\subsetB(L₂(M,ν)) 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 \citeDAO3,DAO4.

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