Dynamical propagation and Roe algebras of warped spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866918374311723008 |
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| author | de Laat, Tim Vigolo, Federico Winkel, Jeroen |
| author_facet | de Laat, Tim Vigolo, Federico Winkel, Jeroen |
| contents | Given a non-singular action $Γ\curvearrowright (X,μ)$, we define the $*$-algebra $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of $μ$. We prove that the algebraic crossed product $L^{\infty}X \rtimes_{\rm alg} Γ$ surjects onto $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ and that this surjection is a $\ast$-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07006 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamical propagation and Roe algebras of warped spaces de Laat, Tim Vigolo, Federico Winkel, Jeroen Operator Algebras Dynamical Systems Metric Geometry 37A15, 37A55, 47L65, 51F30 Given a non-singular action $Γ\curvearrowright (X,μ)$, we define the $*$-algebra $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of $μ$. We prove that the algebraic crossed product $L^{\infty}X \rtimes_{\rm alg} Γ$ surjects onto $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ and that this surjection is a $\ast$-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of $\mathbb C_{\rm fp}[Γ\curvearrowright X]$ and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones. |
| title | Dynamical propagation and Roe algebras of warped spaces |
| topic | Operator Algebras Dynamical Systems Metric Geometry 37A15, 37A55, 47L65, 51F30 |
| url | https://arxiv.org/abs/2308.07006 |