Internal sequential commutation and single generation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929319495860224 |
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| author | Gao, David Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory Tan, Hui |
| author_facet | Gao, David Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory Tan, Hui |
| contents | We extract a precise internal description of the sequential commutation equivalence relation introduced in [KEP23] for tracial von Neumann algebras. As an application we prove that if a tracial von Neumann algebra $N$ is generated by unitaries $\{u_i\}_{i\in \mathbb{N}}$ such that $u_i\sim u_j$ (i.e, there exists a finite set of Haar unitaries $\{w_i\}_{i=1}^{n}$ in $N^\mathcal{U}$ such that $[u_i, w_1]= [w_k, w_{k+1}]=[w_n,u_j]=0$ for all $1\leq k< n$) then $N$ is singly generated. This generalizes and recovers several known single generation phenomena for II$_1$ factors in the literature with a unified proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12380 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Internal sequential commutation and single generation Gao, David Elayavalli, Srivatsav Kunnawalkam Patchell, Gregory Tan, Hui Operator Algebras Functional Analysis Group Theory We extract a precise internal description of the sequential commutation equivalence relation introduced in [KEP23] for tracial von Neumann algebras. As an application we prove that if a tracial von Neumann algebra $N$ is generated by unitaries $\{u_i\}_{i\in \mathbb{N}}$ such that $u_i\sim u_j$ (i.e, there exists a finite set of Haar unitaries $\{w_i\}_{i=1}^{n}$ in $N^\mathcal{U}$ such that $[u_i, w_1]= [w_k, w_{k+1}]=[w_n,u_j]=0$ for all $1\leq k< n$) then $N$ is singly generated. This generalizes and recovers several known single generation phenomena for II$_1$ factors in the literature with a unified proof. |
| title | Internal sequential commutation and single generation |
| topic | Operator Algebras Functional Analysis Group Theory |
| url | https://arxiv.org/abs/2404.12380 |