$\mathrm{C}^*$-exactness and property A for group actions

Hiroto Nishikawa
{"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.
$mathrm{C}^*$-actness 和群作用的属性 A
对于离散群 $\Gamma$ 在集合 $X$ 上的作用,我们证明了当且仅当 $\ell_2X$ 上的置换表示产生一个精确的 $\mathrm{C}^*$ 代数时,$X$ 上的施赖尔图是属性 A。这在 $X=\Gamma$ 上的左规则作用的情况下是众所周知的。该定理指出,当 $X$ 是均匀局部有限时,均匀罗厄代数 $\mathrm{C}^*_{m\mathrm{u}}(X)$ 的精确性表征了 $X$ 的性质 A。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信