$L^p$ coarse Baum-Connes conjecture via $C_{0}$ coarse geometry
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917399104585728 |
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| author | Wang, Hang Wang, Yanru Zhang, Jianguo Zhou, Dapeng |
| author_facet | Wang, Hang Wang, Yanru Zhang, Jianguo Zhou, Dapeng |
| contents | In this paper, we investigate the $L^{p}$ coarse Baum-Connes conjecture for $p\in [1,\infty)$ via $C_{0}$ coarse structure, which is a refinement of the bounded coarse structure on a metric space. We prove that the $C_{0}$ version of the $L^{p}$ coarse Baum-Connes conjecture holds for a finite-dimensional simplicial complex equipped with a uniform spherical metric. Using this result, we construct an obstruction group for the $L^{p}$ coarse Baum-Connes conjecture. As an application, we show that the obstruction group vanishes under the assumption of finite asymptotic dimension, thereby providing a new proof of the $L^{p}$ coarse Baum-Connes conjecture in this case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_05333 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $L^p$ coarse Baum-Connes conjecture via $C_{0}$ coarse geometry Wang, Hang Wang, Yanru Zhang, Jianguo Zhou, Dapeng Functional Analysis K-Theory and Homology Operator Algebras 46L80, 55U10, 58B34 In this paper, we investigate the $L^{p}$ coarse Baum-Connes conjecture for $p\in [1,\infty)$ via $C_{0}$ coarse structure, which is a refinement of the bounded coarse structure on a metric space. We prove that the $C_{0}$ version of the $L^{p}$ coarse Baum-Connes conjecture holds for a finite-dimensional simplicial complex equipped with a uniform spherical metric. Using this result, we construct an obstruction group for the $L^{p}$ coarse Baum-Connes conjecture. As an application, we show that the obstruction group vanishes under the assumption of finite asymptotic dimension, thereby providing a new proof of the $L^{p}$ coarse Baum-Connes conjecture in this case. |
| title | $L^p$ coarse Baum-Connes conjecture via $C_{0}$ coarse geometry |
| topic | Functional Analysis K-Theory and Homology Operator Algebras 46L80, 55U10, 58B34 |
| url | https://arxiv.org/abs/2311.05333 |