{"title":"具有弱三元罗克林性质的有限群作用交叉积的稳定秩","authors":"Xiaochun Fang, Zhongli Wang","doi":"arxiv-2407.09867","DOIUrl":null,"url":null,"abstract":"Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra,\nlet $G$ be a finite group, and let $\\alpha\\colon G\\rightarrow \\mathrm{Aut}(A)$\nbe an action of $G$ on $A$ which has the weak tracial Rokhlin property. We\nprove that if $A$ has property (TM), then the crossed product $A\\rtimes_\\alpha\nG$ has property (TM). As a corollary, if $A$ is an infinite-dimensional\nseparable simple unital C*-algebra which has stable rank one and strict\ncomparison, $\\alpha\\colon G\\rightarrow \\mathrm{Aut}(A)$ is an action of a\nfinite group $G$ on $A$ with the weak tracial Rokhlin property, then\n$A\\rtimes_\\alpha G$ has stable rank one.","PeriodicalId":501114,"journal":{"name":"arXiv - MATH - Operator Algebras","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property\",\"authors\":\"Xiaochun Fang, Zhongli Wang\",\"doi\":\"arxiv-2407.09867\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra,\\nlet $G$ be a finite group, and let $\\\\alpha\\\\colon G\\\\rightarrow \\\\mathrm{Aut}(A)$\\nbe an action of $G$ on $A$ which has the weak tracial Rokhlin property. We\\nprove that if $A$ has property (TM), then the crossed product $A\\\\rtimes_\\\\alpha\\nG$ has property (TM). As a corollary, if $A$ is an infinite-dimensional\\nseparable simple unital C*-algebra which has stable rank one and strict\\ncomparison, $\\\\alpha\\\\colon G\\\\rightarrow \\\\mathrm{Aut}(A)$ is an action of a\\nfinite group $G$ on $A$ with the weak tracial Rokhlin property, then\\n$A\\\\rtimes_\\\\alpha G$ has stable rank one.\",\"PeriodicalId\":501114,\"journal\":{\"name\":\"arXiv - MATH - Operator Algebras\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-07-13\",\"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-2407.09867\",\"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 - MATH - Operator Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.09867","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property
Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra,
let $G$ be a finite group, and let $\alpha\colon G\rightarrow \mathrm{Aut}(A)$
be an action of $G$ on $A$ which has the weak tracial Rokhlin property. We
prove that if $A$ has property (TM), then the crossed product $A\rtimes_\alpha
G$ has property (TM). As a corollary, if $A$ is an infinite-dimensional
separable simple unital C*-algebra which has stable rank one and strict
comparison, $\alpha\colon G\rightarrow \mathrm{Aut}(A)$ is an action of a
finite group $G$ on $A$ with the weak tracial Rokhlin property, then
$A\rtimes_\alpha G$ has stable rank one.