Asymptotic structure of free Araki-Woods factors
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2014
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| _version_ | 1866915392274825216 |
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| author | Houdayer, Cyril Raum, Sven |
| author_facet | Houdayer, Cyril Raum, Sven |
| contents | The purpose of this paper is to investigate the structure of Shlyakhtenko's free Araki-Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ are $ω$-solid in the following sense: for every von Neumann subalgebra $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that is the range of a faithful normal conditional expectation and such that the relative commutant $Q' \cap M^ω$ is diffuse, we have that $Q$ is amenable. Next, we prove that the continuous cores of the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ associated with mixing orthogonal representations $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ are $ω$-solid type ${\rm II_\infty}$ factors. Finally, when the orthogonal representation $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that are globally invariant under the modular automorphism group $(σ_t^{φ_U})$ of the free quasi-free state $φ_U$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1406_6160 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Asymptotic structure of free Araki-Woods factors Houdayer, Cyril Raum, Sven Operator Algebras 46L10, 46L54, 46L55 The purpose of this paper is to investigate the structure of Shlyakhtenko's free Araki-Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ are $ω$-solid in the following sense: for every von Neumann subalgebra $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that is the range of a faithful normal conditional expectation and such that the relative commutant $Q' \cap M^ω$ is diffuse, we have that $Q$ is amenable. Next, we prove that the continuous cores of the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ associated with mixing orthogonal representations $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ are $ω$-solid type ${\rm II_\infty}$ factors. Finally, when the orthogonal representation $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that are globally invariant under the modular automorphism group $(σ_t^{φ_U})$ of the free quasi-free state $φ_U$. |
| title | Asymptotic structure of free Araki-Woods factors |
| topic | Operator Algebras 46L10, 46L54, 46L55 |
| url | https://arxiv.org/abs/1406.6160 |