A $C^{*}$-Algebraic Approach To Principal Symbol Calculus On Filtered Manifolds
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909371712143360 |
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| author | Farrell, David Sukochev, Fedor Yang, Fulin Zanin, Dmitriy |
| author_facet | Farrell, David Sukochev, Fedor Yang, Fulin Zanin, Dmitriy |
| contents | 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 $ν$. 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 $${\rm sym}_{M}:Π_{M}\to C_{b}(E_{hom})$$ such that $${\rm ker}({\rm sym}_{M})=\mathcal{K}(L_{2}(M,ν))\subset Π_{M}$$ where the domain $Π_{M}\subset\mathcal{B}(L_{2}(M,ν))$ 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 \cite{DAO3,DAO4}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A $C^{*}$-Algebraic Approach To Principal Symbol Calculus On Filtered Manifolds Farrell, David Sukochev, Fedor Yang, Fulin Zanin, Dmitriy Operator Algebras Differential Geometry Functional Analysis 58C50, 46L87, 58A50, 47L80 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 $ν$. 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 $${\rm sym}_{M}:Π_{M}\to C_{b}(E_{hom})$$ such that $${\rm ker}({\rm sym}_{M})=\mathcal{K}(L_{2}(M,ν))\subset Π_{M}$$ where the domain $Π_{M}\subset\mathcal{B}(L_{2}(M,ν))$ 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 \cite{DAO3,DAO4}. |
| title | A $C^{*}$-Algebraic Approach To Principal Symbol Calculus On Filtered Manifolds |
| topic | Operator Algebras Differential Geometry Functional Analysis 58C50, 46L87, 58A50, 47L80 |
| url | https://arxiv.org/abs/2410.22125 |