{"title":"关于Steinberg代数的一个子代数的扶正器的注记","authors":"R. Hazrat, Huanhuan Li","doi":"10.1090/spmj/1695","DOIUrl":null,"url":null,"abstract":"<p>For an ample Hausdorff groupoid <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper G\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {G}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and the Steinberg algebra <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript upper R Baseline left-parenthesis script upper G right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mi>R</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">A_R(\\mathcal {G})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with coefficients in the commutative ring <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\n <mml:semantics>\n <mml:mi>R</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with unit, the centralizer is described for the subalgebra <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript upper R Baseline left-parenthesis upper U right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mi>R</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">A_R(U)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\">\n <mml:semantics>\n <mml:mi>U</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">U</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> an open closed invariant subset of the unit space of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper G\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {G}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify several results in the literature, and the corresponding results for Leavitt path algebras follow.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"A note on the centralizer of a subalgebra of the Steinberg algebra\",\"authors\":\"R. Hazrat, Huanhuan Li\",\"doi\":\"10.1090/spmj/1695\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For an ample Hausdorff groupoid <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper G\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">G</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {G}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, and the Steinberg algebra <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A Subscript upper R Baseline left-parenthesis script upper G right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mi>R</mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">G</mml:mi>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_R(\\\\mathcal {G})</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with coefficients in the commutative ring <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper R\\\">\\n <mml:semantics>\\n <mml:mi>R</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">R</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with unit, the centralizer is described for the subalgebra <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A Subscript upper R Baseline left-parenthesis upper U right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mi>R</mml:mi>\\n </mml:msub>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>U</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_R(U)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> with <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U\\\">\\n <mml:semantics>\\n <mml:mi>U</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">U</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> an open closed invariant subset of the unit space of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper G\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">G</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {G}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify several results in the literature, and the corresponding results for Leavitt path algebras follow.</p>\",\"PeriodicalId\":51162,\"journal\":{\"name\":\"St Petersburg Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2021-12-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"St Petersburg Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1695\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1695","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
A note on the centralizer of a subalgebra of the Steinberg algebra
For an ample Hausdorff groupoid G\mathcal {G}, and the Steinberg algebra AR(G)A_R(\mathcal {G}) with coefficients in the commutative ring RR with unit, the centralizer is described for the subalgebra AR(U)A_R(U) with UU an open closed invariant subset of the unit space of G\mathcal {G}. In particular, it is shown that the algebra of the interior of the isotropy is indeed the centralizer of the diagonal subalgebra of the Steinberg algebra. This will unify several results in the literature, and the corresponding results for Leavitt path algebras follow.
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This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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