Examples of compact quantum groups with $\operatorname{\mathsf{L}^{\!\infty}}(\mathbb{G})$ a factor
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866909303691018240 |
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| author | Krajczok, Jacek Sołtan, Piotr M. |
| author_facet | Krajczok, Jacek Sołtan, Piotr M. |
| contents | For each $λ\in\left]0,1\right]$ we exhibit an uncountable family of compact quantum groups $\mathbb{G}$ such that the von Neumann algebra $\mathsf{L}^{\!\infty}(\mathbb{G})$ is the injective factor of type $\mathrm{III}_λ$ with separable predual. We also show that uncountably many injective factors of type $\mathrm{III}_0$ arise as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for some compact quantum group $\mathbb{G}$. To distinguish between our examples we introduce invariants related to the scaling group modeled on the Connes invariant $T$ for von Neumann algebras and investigate the connection between so obtained invariants of $\mathbb{G}$ and the Connes invariants $T(\mathsf{L}^{\!\infty}(\mathbb{G}))$, $S(\mathsf{L}^{\!\infty}(\mathbb{G}))$. In the final section we show that factors of type $\mathrm{I}$ cannot be obtained as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for a non-trivial compact quantum group $\mathbb{G}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_10976 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Examples of compact quantum groups with $\operatorname{\mathsf{L}^{\!\infty}}(\mathbb{G})$ a factor Krajczok, Jacek Sołtan, Piotr M. Operator Algebras Functional Analysis 46L67, 46L36 For each $λ\in\left]0,1\right]$ we exhibit an uncountable family of compact quantum groups $\mathbb{G}$ such that the von Neumann algebra $\mathsf{L}^{\!\infty}(\mathbb{G})$ is the injective factor of type $\mathrm{III}_λ$ with separable predual. We also show that uncountably many injective factors of type $\mathrm{III}_0$ arise as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for some compact quantum group $\mathbb{G}$. To distinguish between our examples we introduce invariants related to the scaling group modeled on the Connes invariant $T$ for von Neumann algebras and investigate the connection between so obtained invariants of $\mathbb{G}$ and the Connes invariants $T(\mathsf{L}^{\!\infty}(\mathbb{G}))$, $S(\mathsf{L}^{\!\infty}(\mathbb{G}))$. In the final section we show that factors of type $\mathrm{I}$ cannot be obtained as $\mathsf{L}^{\!\infty}(\mathbb{G})$ for a non-trivial compact quantum group $\mathbb{G}$. |
| title | Examples of compact quantum groups with $\operatorname{\mathsf{L}^{\!\infty}}(\mathbb{G})$ a factor |
| topic | Operator Algebras Functional Analysis 46L67, 46L36 |
| url | https://arxiv.org/abs/2203.10976 |