Almost almost periodic type $\mathrm{III}_1$ factors and their 3-cohomology obstructions
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866910811097661440 |
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| author | Marrakchi, Amine |
| author_facet | Marrakchi, Amine |
| contents | We construct an exemple of a full factor $M$ such that its canonical outer modular flow $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic but $M$ has no almost periodic state. This can only happen if the discrete spectrum of $σ^M$ contains a nontrivial integral quadratic relation. We show how such a nontrivial relation can produce a 3-cohomological obstruction to the existence of an almost periodic state. To obtain our main theorem, we first strengthen a recent result of Bischoff and Karmakar by showing that for any compact connected abelian group $K$, every cohomology class in $ H^3(K,\mathbb{T})$ can be realized as an obstruction of a $K$-kernel on the hyperfinite $\mathrm{II}_1$ factor. We also prove a positive result : if for a full factor $M$ the outer modular flow $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic, then $M \otimes R$ has an almost periodic state, where $R$ is the hyperfinite $\mathrm{II}_1$ factor. Finally, we prove a positive result for crossed product factors associated to strongly ergodic actions of hyperbolic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01516 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost almost periodic type $\mathrm{III}_1$ factors and their 3-cohomology obstructions Marrakchi, Amine Operator Algebras 46L10, 46L36, 20J06, 22B05 We construct an exemple of a full factor $M$ such that its canonical outer modular flow $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic but $M$ has no almost periodic state. This can only happen if the discrete spectrum of $σ^M$ contains a nontrivial integral quadratic relation. We show how such a nontrivial relation can produce a 3-cohomological obstruction to the existence of an almost periodic state. To obtain our main theorem, we first strengthen a recent result of Bischoff and Karmakar by showing that for any compact connected abelian group $K$, every cohomology class in $ H^3(K,\mathbb{T})$ can be realized as an obstruction of a $K$-kernel on the hyperfinite $\mathrm{II}_1$ factor. We also prove a positive result : if for a full factor $M$ the outer modular flow $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic, then $M \otimes R$ has an almost periodic state, where $R$ is the hyperfinite $\mathrm{II}_1$ factor. Finally, we prove a positive result for crossed product factors associated to strongly ergodic actions of hyperbolic groups. |
| title | Almost almost periodic type $\mathrm{III}_1$ factors and their 3-cohomology obstructions |
| topic | Operator Algebras 46L10, 46L36, 20J06, 22B05 |
| url | https://arxiv.org/abs/2502.01516 |