{"title":"关于Neretin群的一个开子群的von Neumann代数的类型","authors":"Ryoya Arimoto","doi":"10.1090/bproc/133","DOIUrl":null,"url":null,"abstract":"<p>The Neretin group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper N Subscript d comma k\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">N</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>d</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {N}_{d, k}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the totally disconnected locally compact group consisting of almost automorphisms of the tree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper T Subscript d comma k\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">T</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>d</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {T}_{d, k}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. This group has a distinguished open subgroup <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript d comma k\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>d</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {O}_{d, k}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript d comma k\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>d</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>k</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal {O}_{d, k}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is of type I or not.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"279 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On the type of the von Neumann algebra of an open subgroup of the Neretin group\",\"authors\":\"Ryoya Arimoto\",\"doi\":\"10.1090/bproc/133\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The Neretin group <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper N Subscript d comma k\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">N</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>d</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {N}_{d, k}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is the totally disconnected locally compact group consisting of almost automorphisms of the tree <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper T Subscript d comma k\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">T</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>d</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {T}_{d, k}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. This group has a distinguished open subgroup <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper O Subscript d comma k\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">O</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>d</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {O}_{d, k}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper O Subscript d comma k\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi class=\\\"MJX-tex-caligraphic\\\" mathvariant=\\\"script\\\">O</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>d</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>k</mml:mi>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal {O}_{d, k}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is of type I or not.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"279 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-01-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/133\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/133","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
摘要
Neretin群N d, k \mathcal {N}_{d, k}是由树T d, k \mathcal {T}_{d, k}的几乎自同构组成的完全不连通的局部紧群。这个群有一个开放的子群O d, k \mathcal {O}_{d, k}。我们证明了这个开放子群不是i型的，给出了p - e最近结果的另一种证明。Caprace, A. Le Boudec和N. Matte Bon指出Neretin群不是I型，并回答了他们的问题O d, k \mathcal {O}_{d, k}是否属于I型。
On the type of the von Neumann algebra of an open subgroup of the Neretin group
The Neretin group Nd,k\mathcal {N}_{d, k} is the totally disconnected locally compact group consisting of almost automorphisms of the tree Td,k\mathcal {T}_{d, k}. This group has a distinguished open subgroup Od,k\mathcal {O}_{d, k}. We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether Od,k\mathcal {O}_{d, k} is of type I or not.
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