{"title":"Relating the Roe Algebra of a Space to the Uniform Roe Algebras of Its Discretizations","authors":"V. Manuilov","doi":"10.1134/s199508022460122x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The Roe algebra <span>\\(C^{*}(X)\\)</span> is a non-commutative <span>\\(C^{*}\\)</span>-algebra reflecting metric properties of a space <span>\\(X\\)</span>, and it is interesting to understand relation between the Roe algebra of <span>\\(X\\)</span> and the uniform Roe algebra of its discretizations. Here we construct, for a simplicial space <span>\\(X\\)</span>, a continuous field of <span>\\(C^{*}\\)</span>-algebras over <span>\\(\\mathbb{N}\\cup\\{\\infty\\}\\)</span> with the fibers over finite points the uniform <span>\\(C^{*}\\)</span>-algebras of discretizations of <span>\\(X\\)</span>, and the fiber over <span>\\(\\infty\\)</span> the Roe algebra of <span>\\(X\\)</span>. We also construct the direct limit of the uniform Roe algebras of discretizations and its embedding into the Roe algebra of <span>\\(X\\)</span>.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lobachevskii Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s199508022460122x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The Roe algebra \(C^{*}(X)\) is a non-commutative \(C^{*}\)-algebra reflecting metric properties of a space \(X\), and it is interesting to understand relation between the Roe algebra of \(X\) and the uniform Roe algebra of its discretizations. Here we construct, for a simplicial space \(X\), a continuous field of \(C^{*}\)-algebras over \(\mathbb{N}\cup\{\infty\}\) with the fibers over finite points the uniform \(C^{*}\)-algebras of discretizations of \(X\), and the fiber over \(\infty\) the Roe algebra of \(X\). We also construct the direct limit of the uniform Roe algebras of discretizations and its embedding into the Roe algebra of \(X\).
Abstract The Roe algebra \(C^{*}(X)\) is a non-commutative \(C^{*}\)-algebra reflecting metric properties of a space \(X\), and it is interesting to understand relation between the Roe algebra of \(X\) and the uniform Roe algebra of its discretizations.在这里,我们为简单空间 \(X\)构造了一个在 \(\mathbb{N}\cup\{infty\)上的\(C^{*}\)-代数的连续域,其在有限点上的纤维是 \(X\)离散化的统一\(C^{*}\)-代数,而在\(\infty\)上的纤维是 \(X\)的罗伊代数。)我们还构造了离散化的统一 Roe 代数的直接极限以及它对\(X\)的 Roe 代数的嵌入。
期刊介绍:
Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.