{"title":"一个无扭代数$\\ mathm {C}^*$-唯一群","authors":"Eduardo Scarparo","doi":"10.1216/RMJ.2020.50.1813","DOIUrl":null,"url":null,"abstract":"Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $\\mathbb{Z}[\\frac{1}{pq}]\\rtimes\\mathbb{Z}^2$ admits a unique $\\mathrm{C}^*$-norm. The proof uses a characterization, due to Furstenberg, of closed $\\times p-$ and $\\times q-$invariant subsets of $\\mathbb{T}$.","PeriodicalId":351745,"journal":{"name":"arXiv: Operator Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"A torsion-free algebraically $\\\\mathrm{C}^*$-unique group\",\"authors\":\"Eduardo Scarparo\",\"doi\":\"10.1216/RMJ.2020.50.1813\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $\\\\mathbb{Z}[\\\\frac{1}{pq}]\\\\rtimes\\\\mathbb{Z}^2$ admits a unique $\\\\mathrm{C}^*$-norm. The proof uses a characterization, due to Furstenberg, of closed $\\\\times p-$ and $\\\\times q-$invariant subsets of $\\\\mathbb{T}$.\",\"PeriodicalId\":351745,\"journal\":{\"name\":\"arXiv: Operator Algebras\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-03-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Operator Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1216/RMJ.2020.50.1813\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Operator Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1216/RMJ.2020.50.1813","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A torsion-free algebraically $\mathrm{C}^*$-unique group
Let $p$ and $q$ be multiplicatively independent integers. We show that the complex group ring of $\mathbb{Z}[\frac{1}{pq}]\rtimes\mathbb{Z}^2$ admits a unique $\mathrm{C}^*$-norm. The proof uses a characterization, due to Furstenberg, of closed $\times p-$ and $\times q-$invariant subsets of $\mathbb{T}$.