Uniform property $Γ$ and the small boundary property
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Kopsacheilis, Grigoris Liao, Hung-Chang Tikuisis, Aaron Vaccaro, Andrea |
| author_facet | Kopsacheilis, Grigoris Liao, Hung-Chang Tikuisis, Aaron Vaccaro, Andrea |
| contents | We prove that, for a free action $α\colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $Γ$ of the Cartan subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication has been demonstrated by Kerr and Szabó for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if $α$ is also minimal, then almost finiteness of $α$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $α\colon G \curvearrowright X$ and $β\colon H \curvearrowright Y$ are free actions and $α$ has the small boundary property, then $α\times β\colon G \times H \curvearrowright X \times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $α$ and $β$ are free minimal actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform property $Γ$ and the small boundary property Kopsacheilis, Grigoris Liao, Hung-Chang Tikuisis, Aaron Vaccaro, Andrea Operator Algebras Dynamical Systems We prove that, for a free action $α\colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $Γ$ of the Cartan subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication has been demonstrated by Kerr and Szabó for free actions, from which we obtain that these two conditions are equivalent. We moreover show that, if $α$ is also minimal, then almost finiteness of $α$ is implied by tracial $\mathcal{Z}$-stability of the subalgebra $(C(X) \subseteq C(X) \rtimes_αG)$. The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if $α\colon G \curvearrowright X$ and $β\colon H \curvearrowright Y$ are free actions and $α$ has the small boundary property, then $α\times β\colon G \times H \curvearrowright X \times Y$ has the small boundary property. An analogous permanence property is obtained for almost finiteness in case $α$ and $β$ are free minimal actions. |
| title | Uniform property $Γ$ and the small boundary property |
| topic | Operator Algebras Dynamical Systems |
| url | https://arxiv.org/abs/2406.09808 |