{"title":"Maximal UHF subalgebras of certain C*-algebras","authors":"Nasser Golestani, Saeid Maleki Oche","doi":"arxiv-2407.17004","DOIUrl":null,"url":null,"abstract":"A well-known result in dynamical systems asserts that any Cantor minimal\nsystem $(X,T)$ has a maximal rational equicontinuous factor $(Y,S)$ which is in\nfact an odometer, and realizes the rational subgroup of the $K_0$-group of\n$(X,T)$, that is, $\\mathbb{Q}(K_0(X,T), 1) \\cong K^0(Y,S)$. We introduce the\nnotion of a maximal UHF subalgebra and use it to obtain the C*-algebraic alonog\nof this result. We say a UHF subalgebra $B$ of a unital C*-algebra $A$ is a\nmaximal UHF subalgebra if it contains the unit of $A$ any other such\nC*-subalgebra embeds unitaly into $B$. We prove that if $K_0(A)$ is\nunperforated and has a certain $K_0$-lifting property, then $B$ exists and is\nunique up to isomorphism, in particular, all simple separable unital\nC*-algebras with tracial rank zero and all unital Kirchberg algebras whose\n$K_0$-groups are unperforated, have a maximal UHF subalgebra. Not every unital\nC*-algebra has a maximal UHF subalgebra, for instance, the unital universal\nfree product $\\mathrm{M}_2 \\ast_{r} \\mathrm{M}_3$. As an application, we give a\nC*-algebraic realization of the rational subgroup $\\mathbb{Q}(G,u)$ of any\ndimension group $G$ with order unit $u$, that is, there is a simple unital AF\nalgebra (and a unital Kirchberg algebra) $A$ with a maximal UHF subalgebra $B$\nsuch that $(G,u)\\cong (K_0(A), [1]_0)$ and and $\\mathbb{Q}(G,u)\\cong K_0(B)$.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-24","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-2407.17004","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
A well-known result in dynamical systems asserts that any Cantor minimal
system $(X,T)$ has a maximal rational equicontinuous factor $(Y,S)$ which is in
fact an odometer, and realizes the rational subgroup of the $K_0$-group of
$(X,T)$, that is, $\mathbb{Q}(K_0(X,T), 1) \cong K^0(Y,S)$. We introduce the
notion of a maximal UHF subalgebra and use it to obtain the C*-algebraic alonog
of this result. We say a UHF subalgebra $B$ of a unital C*-algebra $A$ is a
maximal UHF subalgebra if it contains the unit of $A$ any other such
C*-subalgebra embeds unitaly into $B$. We prove that if $K_0(A)$ is
unperforated and has a certain $K_0$-lifting property, then $B$ exists and is
unique up to isomorphism, in particular, all simple separable unital
C*-algebras with tracial rank zero and all unital Kirchberg algebras whose
$K_0$-groups are unperforated, have a maximal UHF subalgebra. Not every unital
C*-algebra has a maximal UHF subalgebra, for instance, the unital universal
free product $\mathrm{M}_2 \ast_{r} \mathrm{M}_3$. As an application, we give a
C*-algebraic realization of the rational subgroup $\mathbb{Q}(G,u)$ of any
dimension group $G$ with order unit $u$, that is, there is a simple unital AF
algebra (and a unital Kirchberg algebra) $A$ with a maximal UHF subalgebra $B$
such that $(G,u)\cong (K_0(A), [1]_0)$ and and $\mathbb{Q}(G,u)\cong K_0(B)$.