McDuff and Prime von Neumann algebras arising from Thompson-Like Groups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916421027495936 |
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| author | de Santiago, Rolando DeBonis, Patrick Khan, Krishnendu |
| author_facet | de Santiago, Rolando DeBonis, Patrick Khan, Krishnendu |
| contents | In this paper we show that the cloning system construction of Skipper and Zaremsky [SZ21], under sufficient conditions, gives rise to Thompson-Like groups which are stable; in particular, these are McDuff groups in the sense of Deprez and Vaes [DV18]. This answers a question of Bashwinger and Zaremsky posed in [BZ23] in the affirmative. In the opposite direction, we show that the group von Neumann algebra for the Higman-Thompson groups $T_d$ and $V_d$ are both prime II$_1$ factors. This follows from a new deformation/rigidity argument for a certain class of groups which admit a proper cocycle into a quasi-regular representation that is not necessarily weakly $\ell^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_08345 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | McDuff and Prime von Neumann algebras arising from Thompson-Like Groups de Santiago, Rolando DeBonis, Patrick Khan, Krishnendu Operator Algebras Dynamical Systems Group Theory 46L36 In this paper we show that the cloning system construction of Skipper and Zaremsky [SZ21], under sufficient conditions, gives rise to Thompson-Like groups which are stable; in particular, these are McDuff groups in the sense of Deprez and Vaes [DV18]. This answers a question of Bashwinger and Zaremsky posed in [BZ23] in the affirmative. In the opposite direction, we show that the group von Neumann algebra for the Higman-Thompson groups $T_d$ and $V_d$ are both prime II$_1$ factors. This follows from a new deformation/rigidity argument for a certain class of groups which admit a proper cocycle into a quasi-regular representation that is not necessarily weakly $\ell^2$. |
| title | McDuff and Prime von Neumann algebras arising from Thompson-Like Groups |
| topic | Operator Algebras Dynamical Systems Group Theory 46L36 |
| url | https://arxiv.org/abs/2312.08345 |