Mohammad Rouzbehani, Massoud Amini, Mohammad B. Asadi
{"title":"Goldie dimension for C*-algebras","authors":"Mohammad Rouzbehani, Massoud Amini, Mohammad B. Asadi","doi":"10.1017/s0013091523000767","DOIUrl":null,"url":null,"abstract":"<p>In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra <span>A</span> has Goldie dimension <span>n</span> if and only if the dimension of the center of its local multiplier algebra is <span>n</span>. In this case, <span>A</span> has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of <span>n</span> prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.</p>","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"16 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Edinburgh Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0013091523000767","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.
期刊介绍:
The Edinburgh Mathematical Society was founded in 1883 and over the years, has evolved into the principal society for the promotion of mathematics research in Scotland. The Society has published its Proceedings since 1884. This journal contains research papers on topics in a broad range of pure and applied mathematics, together with a number of topical book reviews.