Analysis of the Plancherel weight and factoriality of the group von Neumann algebras of non-unimodular almost unimodular groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916955390214144 |
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| author | Miyamoto, Yuki |
| author_facet | Miyamoto, Yuki |
| contents | Let $G$ be a locally compact group, $L(G)$ be its group von Neumann algebra equipped with the Plancherel weight $φ_G$. In this paper, we consider the following two questions. (1) When is the restriction of $φ_G$ to the subalgebra generated by a closed subgroup $H$ semifinite? If so, is it equal (up to a constant) to $φ_H$? (2) When is $L(G)$ a factor? We give a complete answer to (1), and when $G$ is second countable, $G_1 := \text{ker}Δ_G$ is open in $G$ (called almost unimodular) and admits a sufficiently large non-unimodular part, we provide an answer to (2). When $L(G)$ is a factor, we also provide the formula of the S-invariant of $L(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21253 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of the Plancherel weight and factoriality of the group von Neumann algebras of non-unimodular almost unimodular groups Miyamoto, Yuki Operator Algebras Let $G$ be a locally compact group, $L(G)$ be its group von Neumann algebra equipped with the Plancherel weight $φ_G$. In this paper, we consider the following two questions. (1) When is the restriction of $φ_G$ to the subalgebra generated by a closed subgroup $H$ semifinite? If so, is it equal (up to a constant) to $φ_H$? (2) When is $L(G)$ a factor? We give a complete answer to (1), and when $G$ is second countable, $G_1 := \text{ker}Δ_G$ is open in $G$ (called almost unimodular) and admits a sufficiently large non-unimodular part, we provide an answer to (2). When $L(G)$ is a factor, we also provide the formula of the S-invariant of $L(G)$. |
| title | Analysis of the Plancherel weight and factoriality of the group von Neumann algebras of non-unimodular almost unimodular groups |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2505.21253 |