Upgraded free independence phenomena for random unitaries
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912509074604032 |
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| author | Jekel, David Elayavalli, Srivatsav Kunnawalkam |
| author_facet | Jekel, David Elayavalli, Srivatsav Kunnawalkam |
| contents | We study upgraded free independence phenomena for unitary elements $u_1$, $u_2$, \dots representing the large-$n$ limit of Haar random unitaries, showing that free independence extends to several larger algebras containing $u_j$ in the ultraproduct of matrices $\prod_{n \to \mathcal{U}} M_n(\mathbb{C})$. Using a uniform asymptotic freeness argument and volumetric analysis, we prove free independence of the Pinsker algebras $\mathcal{P}_j$ containing $u_j$. The Pinsker algebra $\mathcal{P}_j$ is the maximal subalgebra containing $u_j$ with vanishing $1$-bounded entropy defined by Hayes; $\mathcal{P}_j$ in particular contains the relative commutant $\{u_j\}' \cap \prod_{n \to \mathcal{U}} M_n(\mathbb{C})$, more generally any unitary that can be connected to $u_j$ by a sequence of commuting pairs of Haar unitaries, and any unitary $v$ such that $v\mathcal{P}_j v^* \cap \mathcal{P}_j$ is diffuse. Through an embedding argument, we go back and deduce analogous free independence results for $\mathcal{M}^{\mathcal{U}}$ when $\mathcal{M}$ is a free product of Connes embeddable tracial von Neumann algebras $\mathcal{M}_i$, which thus yields (in the Connes-embeddable case) a generalization and a new proof of Houdayer--Ioana's results on free independence of approximate commutants. It also yields a new proof of the general absorption results for Connes-embeddable free products obtained by the first author, Hayes, Nelson, and Sinclair. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upgraded free independence phenomena for random unitaries Jekel, David Elayavalli, Srivatsav Kunnawalkam Operator Algebras Probability 46L54, 46L10, 60B20, 94A17 We study upgraded free independence phenomena for unitary elements $u_1$, $u_2$, \dots representing the large-$n$ limit of Haar random unitaries, showing that free independence extends to several larger algebras containing $u_j$ in the ultraproduct of matrices $\prod_{n \to \mathcal{U}} M_n(\mathbb{C})$. Using a uniform asymptotic freeness argument and volumetric analysis, we prove free independence of the Pinsker algebras $\mathcal{P}_j$ containing $u_j$. The Pinsker algebra $\mathcal{P}_j$ is the maximal subalgebra containing $u_j$ with vanishing $1$-bounded entropy defined by Hayes; $\mathcal{P}_j$ in particular contains the relative commutant $\{u_j\}' \cap \prod_{n \to \mathcal{U}} M_n(\mathbb{C})$, more generally any unitary that can be connected to $u_j$ by a sequence of commuting pairs of Haar unitaries, and any unitary $v$ such that $v\mathcal{P}_j v^* \cap \mathcal{P}_j$ is diffuse. Through an embedding argument, we go back and deduce analogous free independence results for $\mathcal{M}^{\mathcal{U}}$ when $\mathcal{M}$ is a free product of Connes embeddable tracial von Neumann algebras $\mathcal{M}_i$, which thus yields (in the Connes-embeddable case) a generalization and a new proof of Houdayer--Ioana's results on free independence of approximate commutants. It also yields a new proof of the general absorption results for Connes-embeddable free products obtained by the first author, Hayes, Nelson, and Sinclair. |
| title | Upgraded free independence phenomena for random unitaries |
| topic | Operator Algebras Probability 46L54, 46L10, 60B20, 94A17 |
| url | https://arxiv.org/abs/2404.17114 |