Noncommutative geometry on the Berkovich projective line
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917392095903744 |
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| author | Khalkhali, Masoud Tageddine, Damien |
| author_facet | Khalkhali, Masoud Tageddine, Damien |
| contents | We construct several $C^*$-algebras and spectral triples associated to the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}({\mathbb{C}_p})$. In the commutative setting, we construct a spectral triple as a direct limit over finite $\mathbb{R}$-trees. More general $C^*$-algebras generated by partial isometries are also presented. We use their representations to associate a Perron-Frobenius operator and a family of projection valued measures. Finally, we show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of the crossed product algebra with a Schottky subgroup of $\mathrm{PGL}_2(\mathbb{C}_p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02593 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noncommutative geometry on the Berkovich projective line Khalkhali, Masoud Tageddine, Damien Functional Analysis Number Theory Operator Algebras Quantum Algebra 58B34 We construct several $C^*$-algebras and spectral triples associated to the Berkovich projective line $\mathbb{P}^1_{\mathrm{Berk}}({\mathbb{C}_p})$. In the commutative setting, we construct a spectral triple as a direct limit over finite $\mathbb{R}$-trees. More general $C^*$-algebras generated by partial isometries are also presented. We use their representations to associate a Perron-Frobenius operator and a family of projection valued measures. Finally, we show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of the crossed product algebra with a Schottky subgroup of $\mathrm{PGL}_2(\mathbb{C}_p)$. |
| title | Noncommutative geometry on the Berkovich projective line |
| topic | Functional Analysis Number Theory Operator Algebras Quantum Algebra 58B34 |
| url | https://arxiv.org/abs/2411.02593 |