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| Format: | Preprint |
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2008
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| Online Access: | https://arxiv.org/abs/0812.1325 |
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| _version_ | 1866911059505315840 |
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| author | Houdayer, Cyril |
| author_facet | Houdayer, Cyril |
| contents | We show that for any type ${\rm III_1}$ free Araki-Woods factor $\mathcal{M} = Γ(H_\R, U_t)"$ associated with an orthogonal representation $(U_t)$ of $\R$ on a separable real Hilbert space $H_\R$, the continuous core $M = \mathcal{M} \rtimes_σ\R$ is a semisolid ${\rm II_\infty}$ factor, i.e. for any non-zero finite projection $q \in M$, the ${\rm II_1}$ factor $qMq$ is semisolid. If the representation $(U_t)$ is moreover assumed to be mixing, then we prove that the core $M$ is solid. As an application, we construct an example of a non-amenable solid ${\rm II_1}$ factor $N$ with full fundamental group, i.e. $\mathcal{F}(N) = \R^*_+$, which is not isomorphic to any interpolated free group factor $L(\F_t)$, for $1 < t \leq +\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0812_1325 |
| institution | arXiv |
| publishDate | 2008 |
| record_format | arxiv |
| spellingShingle | Structural results for free Araki-Woods factors and their continuous cores Houdayer, Cyril Operator Algebras 46L10, 46L54 We show that for any type ${\rm III_1}$ free Araki-Woods factor $\mathcal{M} = Γ(H_\R, U_t)"$ associated with an orthogonal representation $(U_t)$ of $\R$ on a separable real Hilbert space $H_\R$, the continuous core $M = \mathcal{M} \rtimes_σ\R$ is a semisolid ${\rm II_\infty}$ factor, i.e. for any non-zero finite projection $q \in M$, the ${\rm II_1}$ factor $qMq$ is semisolid. If the representation $(U_t)$ is moreover assumed to be mixing, then we prove that the core $M$ is solid. As an application, we construct an example of a non-amenable solid ${\rm II_1}$ factor $N$ with full fundamental group, i.e. $\mathcal{F}(N) = \R^*_+$, which is not isomorphic to any interpolated free group factor $L(\F_t)$, for $1 < t \leq +\infty$. |
| title | Structural results for free Araki-Woods factors and their continuous cores |
| topic | Operator Algebras 46L10, 46L54 |
| url | https://arxiv.org/abs/0812.1325 |