An example of an infinite amenable group with the ISR property
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929301092302848 |
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| author | Jiang, Yongle Zhou, Xiaoyan |
| author_facet | Jiang, Yongle Zhou, Xiaoyan |
| contents | Let $G$ be $S_{\mathbb{N}}$, the finitary permutation (i.e. permutations with finite support) group on positive integers $\mathbb{N}$. We prove that $G$ has the invariant von Neumann subalgebras rigidity (ISR, for short) property as introduced in Amrutam-Jiang's work. More precisely, every $G$-invariant von Neumann subalgebra $P\subseteq L(G)$ is of the form $L(H)$ for some normal sugbroup $H\lhd G$ and in this case, $H=\{e\}, A_{\mathbb{N}}$ or $G$, where $A_{\mathbb{N}}$ denotes the finitary alternating group on $\mathbb{N}$, i.e. the subgroup of all even permutations in $S_{\mathbb{N}}$. This gives the first known example of an infinite amenable group with the ISR property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08061 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An example of an infinite amenable group with the ISR property Jiang, Yongle Zhou, Xiaoyan Operator Algebras Let $G$ be $S_{\mathbb{N}}$, the finitary permutation (i.e. permutations with finite support) group on positive integers $\mathbb{N}$. We prove that $G$ has the invariant von Neumann subalgebras rigidity (ISR, for short) property as introduced in Amrutam-Jiang's work. More precisely, every $G$-invariant von Neumann subalgebra $P\subseteq L(G)$ is of the form $L(H)$ for some normal sugbroup $H\lhd G$ and in this case, $H=\{e\}, A_{\mathbb{N}}$ or $G$, where $A_{\mathbb{N}}$ denotes the finitary alternating group on $\mathbb{N}$, i.e. the subgroup of all even permutations in $S_{\mathbb{N}}$. This gives the first known example of an infinite amenable group with the ISR property. |
| title | An example of an infinite amenable group with the ISR property |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2312.08061 |