$L^p$-spectral triples and $p$-quantum compact metric spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909985132249088 |
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| author | Delfín, Alonso Farsi, Carla Packer, Judith |
| author_facet | Delfín, Alonso Farsi, Carla Packer, Judith |
| contents | For $p \in [1, \infty)$, we generalize the concept of classical spectral triples by extending the framework from Hilbert spaces to $L^p$-spaces, and from C*-algebras to $L^p$-operator algebras. In addition, we define an $L^p$-spectral triple to be metric when the state space of the algebra has a $p$-quantum compact metric space structure. Specifically, we construct $L^p$-spectral triples for reduced $L^p$-group algebras of countable discrete groups with proper length functions and also for $L^p$ UHF-algebras of infinite tensor product type, the latter inspired by E. Christensen and C. Ivan's construction of a Dirac operator on AF C*-algebras. We prove that $L^p$-spectral triples associated with $L^p$-group algebras (provided that the length function is of bounded doubling) and those associated with $L^p$ UHF-algebras are always metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13735 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^p$-spectral triples and $p$-quantum compact metric spaces Delfín, Alonso Farsi, Carla Packer, Judith Functional Analysis Operator Algebras Primary 46H15, 46H35, 46L89, 58B34, Secondary 43A15 For $p \in [1, \infty)$, we generalize the concept of classical spectral triples by extending the framework from Hilbert spaces to $L^p$-spaces, and from C*-algebras to $L^p$-operator algebras. In addition, we define an $L^p$-spectral triple to be metric when the state space of the algebra has a $p$-quantum compact metric space structure. Specifically, we construct $L^p$-spectral triples for reduced $L^p$-group algebras of countable discrete groups with proper length functions and also for $L^p$ UHF-algebras of infinite tensor product type, the latter inspired by E. Christensen and C. Ivan's construction of a Dirac operator on AF C*-algebras. We prove that $L^p$-spectral triples associated with $L^p$-group algebras (provided that the length function is of bounded doubling) and those associated with $L^p$ UHF-algebras are always metric. |
| title | $L^p$-spectral triples and $p$-quantum compact metric spaces |
| topic | Functional Analysis Operator Algebras Primary 46H15, 46H35, 46L89, 58B34, Secondary 43A15 |
| url | https://arxiv.org/abs/2411.13735 |