Amenable actions of compact and discrete quantum groups on von Neumann algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916680818491392 |
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| author | De Commer, K. De Ro, J. |
| author_facet | De Commer, K. De Ro, J. |
| contents | Let $\mathbb{G}$ be a compact quantum group and $A\subseteq B$ an inclusion of $σ$-finite $\mathbb{G}$-dynamical von Neumann algebras. We prove that the $\mathbb{G}$-inclusion $A\subseteq B$ is strongly equivariantly amenable if and only if it is equivariantly amenable, using techniques from the theory of non-commutative $L^p$-spaces. In particular, if $(A, α)$ is a $\mathbb{G}$-dynamical von Neumann algebra with $A$ $σ$-finite, the action $α: A \curvearrowleft \mathbb{G}$ is strongly (inner) amenable if and only if the action $α: A \curvearrowleft \mathbb{G}$ is (inner) amenable. By duality, we also obtain the same result for $\mathbb{G}$ a discrete quantum group, so that, in particular, a discrete quantum group is inner amenable if and only it is strongly inner amenable. This result can be seen as a dynamical generalization of Tomatsu's result on the amenability/co-amenability duality. We also provide the first explicit examples of amenable discrete quantum groups that act non-amenably on a von Neumann algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_05571 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Amenable actions of compact and discrete quantum groups on von Neumann algebras De Commer, K. De Ro, J. Operator Algebras Quantum Algebra Let $\mathbb{G}$ be a compact quantum group and $A\subseteq B$ an inclusion of $σ$-finite $\mathbb{G}$-dynamical von Neumann algebras. We prove that the $\mathbb{G}$-inclusion $A\subseteq B$ is strongly equivariantly amenable if and only if it is equivariantly amenable, using techniques from the theory of non-commutative $L^p$-spaces. In particular, if $(A, α)$ is a $\mathbb{G}$-dynamical von Neumann algebra with $A$ $σ$-finite, the action $α: A \curvearrowleft \mathbb{G}$ is strongly (inner) amenable if and only if the action $α: A \curvearrowleft \mathbb{G}$ is (inner) amenable. By duality, we also obtain the same result for $\mathbb{G}$ a discrete quantum group, so that, in particular, a discrete quantum group is inner amenable if and only it is strongly inner amenable. This result can be seen as a dynamical generalization of Tomatsu's result on the amenability/co-amenability duality. We also provide the first explicit examples of amenable discrete quantum groups that act non-amenably on a von Neumann algebra. |
| title | Amenable actions of compact and discrete quantum groups on von Neumann algebras |
| topic | Operator Algebras Quantum Algebra |
| url | https://arxiv.org/abs/2408.05571 |