Cocycle perturbations and ergodicity for actions on type III factors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911318931406848 |
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| author | Isono, Yusuke |
| author_facet | Isono, Yusuke |
| contents | We study cocycle perturbations of state preserving actions on type $\mathrm{III}_1$ factors. Extending the theorem of Marrakchi and Vaes for type $\mathrm{II}_1$ factors, we show that a state preserving outer $\mathbb Z$-action on a type $\mathrm{III}_1$ factor with trivial bicentralizer admits a unitary cocycle whose perturbation becomes an ergodic action. This partially answers a question of Marrakchi and Vaes. A major difference from the type $\mathrm{II}$ case is that the modular automorphism group naturally appears as part of the action, making the construction of the required cocycle more delicate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cocycle perturbations and ergodicity for actions on type III factors Isono, Yusuke Operator Algebras We study cocycle perturbations of state preserving actions on type $\mathrm{III}_1$ factors. Extending the theorem of Marrakchi and Vaes for type $\mathrm{II}_1$ factors, we show that a state preserving outer $\mathbb Z$-action on a type $\mathrm{III}_1$ factor with trivial bicentralizer admits a unitary cocycle whose perturbation becomes an ergodic action. This partially answers a question of Marrakchi and Vaes. A major difference from the type $\mathrm{II}$ case is that the modular automorphism group naturally appears as part of the action, making the construction of the required cocycle more delicate. |
| title | Cocycle perturbations and ergodicity for actions on type III factors |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2512.12931 |