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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.17794 |
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| _version_ | 1866911397193973760 |
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| author | McDonald, Edward |
| author_facet | McDonald, Edward |
| contents | The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold $M$ whose tangent bundle $TM$ is equipped with a nested family $H$ of sub-bundles $H_0\leq H_1 \leq \cdots \leq TM$ which defines a filtration of the Lie algebra of vector fields on $M.$ Differential operators on Carnot manifolds have their order measured in terms of the filtration defined by $H,$ and the algebra of differential operators can be extended to an algebra of pseudodifferential operators. Recently, Dave-Haller and Couchet-Yuncken proposed definitions of a residue functional on the algebra of pseudodifferential operators adapted to a Carnot manifold. We prove that Connes' trace theorem holds in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17794 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Connes trace theorem for Carnot manifolds McDonald, Edward Functional Analysis 58J42 The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. A Carnot manifold is a manifold $M$ whose tangent bundle $TM$ is equipped with a nested family $H$ of sub-bundles $H_0\leq H_1 \leq \cdots \leq TM$ which defines a filtration of the Lie algebra of vector fields on $M.$ Differential operators on Carnot manifolds have their order measured in terms of the filtration defined by $H,$ and the algebra of differential operators can be extended to an algebra of pseudodifferential operators. Recently, Dave-Haller and Couchet-Yuncken proposed definitions of a residue functional on the algebra of pseudodifferential operators adapted to a Carnot manifold. We prove that Connes' trace theorem holds in this setting. |
| title | Connes trace theorem for Carnot manifolds |
| topic | Functional Analysis 58J42 |
| url | https://arxiv.org/abs/2601.17794 |