$C^*$-simplicity and boundary actions of discrete quantum groups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911444004503552 |
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| author | Anderson-Sackaney, Benjamin Vergnioux, Roland |
| author_facet | Anderson-Sackaney, Benjamin Vergnioux, Roland |
| contents | We introduce and investigate several quantum group dynamical notions for the purpose of studying $C^*$-simplicity of discrete quantum groups via the theory of boundary actions. In particular we define a quantum analogue of Powers' Averaging Property (PAP) and a quantum analogue of strongly faithful actions. We show that our quantum PAP implies $C^*$-simplicity and the uniqueness of $σ$-KMS states, and that the existence of a strongly $C^*$-faithful quantum boundary action also implies $C^*$-simplicity and, in the unimodular case, the quantum PAP. We illustrate these results in the case of the unitary free quantum groups $\mathbb{F} U_F$ by showing that they satisfy the quantum PAP and that they act strongly $C^*$-faithfully on their quantum Gromov boundary. Moreover we prove that this particular action of $\mathbb{F} U_F$ is a quantum boundary action. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05178 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $C^*$-simplicity and boundary actions of discrete quantum groups Anderson-Sackaney, Benjamin Vergnioux, Roland Operator Algebras Quantum Algebra 46L67, 46L05, 20G42, 37B05 We introduce and investigate several quantum group dynamical notions for the purpose of studying $C^*$-simplicity of discrete quantum groups via the theory of boundary actions. In particular we define a quantum analogue of Powers' Averaging Property (PAP) and a quantum analogue of strongly faithful actions. We show that our quantum PAP implies $C^*$-simplicity and the uniqueness of $σ$-KMS states, and that the existence of a strongly $C^*$-faithful quantum boundary action also implies $C^*$-simplicity and, in the unimodular case, the quantum PAP. We illustrate these results in the case of the unitary free quantum groups $\mathbb{F} U_F$ by showing that they satisfy the quantum PAP and that they act strongly $C^*$-faithfully on their quantum Gromov boundary. Moreover we prove that this particular action of $\mathbb{F} U_F$ is a quantum boundary action. |
| title | $C^*$-simplicity and boundary actions of discrete quantum groups |
| topic | Operator Algebras Quantum Algebra 46L67, 46L05, 20G42, 37B05 |
| url | https://arxiv.org/abs/2411.05178 |