{"title":"$\\mathrm{C}^*$-exactness and property A for group actions","authors":"Hiroto Nishikawa","doi":"arxiv-2407.16130","DOIUrl":null,"url":null,"abstract":"For an action of a discrete group $\\Gamma$ on a set $X$, we show that the\nSchreier graph on $X$ is property A if and only if the permutation\nrepresentation on $\\ell_2X$ generates an exact $\\mathrm{C}^*$-algebra. This is\nwell known in the case of the left regular action on $X=\\Gamma$. This also\ngeneralizes Sako's theorem, which states that exactness of the uniform Roe\nalgebra $\\mathrm{C}^*_{\\mathrm{u}}(X)$ characterizes property A of $X$ when $X$\nis uniformly locally finite.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Operator Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.16130","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
For an action of a discrete group $\Gamma$ on a set $X$, we show that the
Schreier graph on $X$ is 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 $X=\Gamma$. 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.