Ergodic states on type III$_1$ factors and ergodic actions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910657478131712 |
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| author | Marrakchi, Amine Vaes, Stefaan |
| author_facet | Marrakchi, Amine Vaes, Stefaan |
| contents | Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra $M$ that admits a state $φ$ with trivial centralizer $M_φ$ must be a type III$_1$ factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense $G_δ$ set among all faithful normal states on any III$_1$ factor with separable predual. Through Connes' Radon-Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_14217 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ergodic states on type III$_1$ factors and ergodic actions Marrakchi, Amine Vaes, Stefaan Operator Algebras Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra $M$ that admits a state $φ$ with trivial centralizer $M_φ$ must be a type III$_1$ factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense $G_δ$ set among all faithful normal states on any III$_1$ factor with separable predual. Through Connes' Radon-Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper. |
| title | Ergodic states on type III$_1$ factors and ergodic actions |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2305.14217 |