$\mathrm{C}^*$-exactness and property A for group actions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911108062773248 |
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| author | Nishikawa, Hiroto |
| author_facet | Nishikawa, Hiroto |
| contents | For an action of a discrete group $Γ$ on a set $X$, we show that the Schreier graph on $X$ has property A if and only if the permutation representation on $\ell_2X$ generates an exact $\mathrm{C}^*$-algebra. This is well known in the case of the left regular action on $X=Γ$ as the equivalence of $\mathrm{C}^*$-exactness and property A of its Cayley graph. This also generalizes Sako's theorem, which states that exactness of the uniform Roe algebra $\mathrm{C}^*_{\mathrm{u}}(X)$ characterizes property A of $X$ when $X$ is uniformly locally finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathrm{C}^*$-exactness and property A for group actions Nishikawa, Hiroto Operator Algebras 46L05, 51F99 For an action of a discrete group $Γ$ on a set $X$, we show that the Schreier graph on $X$ has property A if and only if the permutation representation on $\ell_2X$ generates an exact $\mathrm{C}^*$-algebra. This is well known in the case of the left regular action on $X=Γ$ as the equivalence of $\mathrm{C}^*$-exactness and property A of its Cayley graph. This also generalizes Sako's theorem, which states that exactness of the uniform Roe algebra $\mathrm{C}^*_{\mathrm{u}}(X)$ characterizes property A of $X$ when $X$ is uniformly locally finite. |
| title | $\mathrm{C}^*$-exactness and property A for group actions |
| topic | Operator Algebras 46L05, 51F99 |
| url | https://arxiv.org/abs/2407.16130 |