Rigidity results for group von Neumann algebras with diffuse center
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909348736794624 |
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| author | Chifan, Ionuţ Quero, Adriana Fernández Tan, Hui |
| author_facet | Chifan, Ionuţ Quero, Adriana Fernández Tan, Hui |
| contents | We introduce the first examples of groups $G$ with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, $\mathcal{L}(G)$. Specifically, assume that $G=A\times W$, where $A$ is an infinite abelian group and $W$ is an ICC wreath-like product group [CIOS22a; AMCOS23] with property (T) and trivial abelianization. Then whenever $H$ is an \emph{arbitrary} group such that $\mathcal{L}(G)$ is $\ast$-isomorphic to $\mathcal L(H)$ it must be the case that $H= B \times H_0$ where $B$ is infinite abelian and $H_0$ is isomorphic to $W$. Moreover, we completely describe the $\ast$-isomorphism between $\mathcal L(G)$ and $\mathcal L(H)$. This yields new applications to the classification of group C$^*$-algebras, including examples of non-amenable groups which are recoverable from their reduced C$^*$-algebras but not from their von Neumann algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity results for group von Neumann algebras with diffuse center Chifan, Ionuţ Quero, Adriana Fernández Tan, Hui Operator Algebras We introduce the first examples of groups $G$ with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, $\mathcal{L}(G)$. Specifically, assume that $G=A\times W$, where $A$ is an infinite abelian group and $W$ is an ICC wreath-like product group [CIOS22a; AMCOS23] with property (T) and trivial abelianization. Then whenever $H$ is an \emph{arbitrary} group such that $\mathcal{L}(G)$ is $\ast$-isomorphic to $\mathcal L(H)$ it must be the case that $H= B \times H_0$ where $B$ is infinite abelian and $H_0$ is isomorphic to $W$. Moreover, we completely describe the $\ast$-isomorphism between $\mathcal L(G)$ and $\mathcal L(H)$. This yields new applications to the classification of group C$^*$-algebras, including examples of non-amenable groups which are recoverable from their reduced C$^*$-algebras but not from their von Neumann algebras. |
| title | Rigidity results for group von Neumann algebras with diffuse center |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2403.01280 |