Vanishing of dimensions and nonexistence of spectral triples on compact Vilenkin groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915271078313984 |
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| author | Biswas, Surajit Saurabh, Bipul |
| author_facet | Biswas, Surajit Saurabh, Bipul |
| contents | We compute the spectral dimension, the dimension of a symmetric random walk, and the Gelfand-Kirillov dimension for compact Vilenkin groups. As a result, we show that these dimensions are zero for any compact, totally disconnected, metrizable topological group. We provide an explicit description of the $K$-groups for compact Vilenkin groups. We express the generators of the $K_0$-groups in terms of the corresponding matrix coefficients for two specific examples: the group of $p$-adic integers and the $p$-adic Heisenberg group. Finally, we prove the nonexistence of a natural class of spectral triples on the group of $p$-adic integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vanishing of dimensions and nonexistence of spectral triples on compact Vilenkin groups Biswas, Surajit Saurabh, Bipul Operator Algebras Probability Quantum Algebra 58B32, 58B34, 46L80 We compute the spectral dimension, the dimension of a symmetric random walk, and the Gelfand-Kirillov dimension for compact Vilenkin groups. As a result, we show that these dimensions are zero for any compact, totally disconnected, metrizable topological group. We provide an explicit description of the $K$-groups for compact Vilenkin groups. We express the generators of the $K_0$-groups in terms of the corresponding matrix coefficients for two specific examples: the group of $p$-adic integers and the $p$-adic Heisenberg group. Finally, we prove the nonexistence of a natural class of spectral triples on the group of $p$-adic integers. |
| title | Vanishing of dimensions and nonexistence of spectral triples on compact Vilenkin groups |
| topic | Operator Algebras Probability Quantum Algebra 58B32, 58B34, 46L80 |
| url | https://arxiv.org/abs/2505.01439 |