{"title":"Direct finiteness of representable regular rings with involution: A counterexample","authors":"Christian Herrmann","doi":"arxiv-2408.16437","DOIUrl":null,"url":null,"abstract":"Bruns and Roddy constructed a $3$-generated modular ortholattice $L$ which\ncannot be embedded into any complete modular ortholattice. Motivated by their\napproach, we use shift operators to construct a $*$-regular $*$-ring $R$ of\nendomorphisms of an inner product space (which can be chosen as the Hilbert\nspace $\\ell^2$) such that direct finiteness fails for $R$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"53 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.16437","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
Bruns and Roddy constructed a $3$-generated modular ortholattice $L$ which
cannot be embedded into any complete modular ortholattice. Motivated by their
approach, we use shift operators to construct a $*$-regular $*$-ring $R$ of
endomorphisms of an inner product space (which can be chosen as the Hilbert
space $\ell^2$) such that direct finiteness fails for $R$.