Dynamical McDuff-type properties for group actions on von Neumann algebras
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arXiv
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| Format: | Preprint |
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2023
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| author | Szabó, Gábor Wouters, Lise |
| author_facet | Szabó, Gábor Wouters, Lise |
| contents | We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II$_1$-factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C$^*$-dynamics. Given a countable discrete group $G$ and an amenable action $G\curvearrowright M$ on any separably acting semi-finite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing $G$-action is suitably absorbed at the level of each fibre in the direct integral decomposition of $M$, then it is tensorially absorbed by the action on $M$. As a direct application of Ocneanu's theorem, we deduce that if $M$ has the McDuff property, then every amenable $G$-action on $M$ has the equivariant McDuff property, regardless whether $M$ is assumed to be injective or not. By employing Tomita-Takesaki theory, we can extend the latter result to the general case where $M$ is not assumed to be semi-finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11748 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamical McDuff-type properties for group actions on von Neumann algebras Szabó, Gábor Wouters, Lise Operator Algebras 46L55 We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II$_1$-factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C$^*$-dynamics. Given a countable discrete group $G$ and an amenable action $G\curvearrowright M$ on any separably acting semi-finite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing $G$-action is suitably absorbed at the level of each fibre in the direct integral decomposition of $M$, then it is tensorially absorbed by the action on $M$. As a direct application of Ocneanu's theorem, we deduce that if $M$ has the McDuff property, then every amenable $G$-action on $M$ has the equivariant McDuff property, regardless whether $M$ is assumed to be injective or not. By employing Tomita-Takesaki theory, we can extend the latter result to the general case where $M$ is not assumed to be semi-finite. |
| title | Dynamical McDuff-type properties for group actions on von Neumann algebras |
| topic | Operator Algebras 46L55 |
| url | https://arxiv.org/abs/2301.11748 |