{"title":"𝒪_{∞}稳定𝒞*-代数的分类","authors":"James Gabe","doi":"10.1090/memo/1461","DOIUrl":null,"url":null,"abstract":"<p>I present a proof of Kirchberg’s classification theorem: two separable, nuclear, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper O Subscript normal infinity\">\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:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathcal O_\\infty</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-stable <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript asterisk\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">C^\\ast</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-algebras are stably isomorphic if and only if they are ideal-related <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K upper K\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>K</mml:mi>\n <mml:mi>K</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">KK</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.</p>","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.0000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Classification of 𝒪_{∞}-Stable 𝒞*-Algebras\",\"authors\":\"James Gabe\",\"doi\":\"10.1090/memo/1461\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>I present a proof of Kirchberg’s classification theorem: two separable, nuclear, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script upper O Subscript normal infinity\\\">\\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:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathcal O_\\\\infty</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-stable <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript asterisk\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>C</mml:mi>\\n <mml:mo>∗<!-- ∗ --></mml:mo>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C^\\\\ast</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-algebras are stably isomorphic if and only if they are ideal-related <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper K upper K\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>K</mml:mi>\\n <mml:mi>K</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">KK</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.</p>\",\"PeriodicalId\":49828,\"journal\":{\"name\":\"Memoirs of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.0000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Memoirs of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/memo/1461\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Memoirs of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/memo/1461","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我提出了基希贝格分类定理的一个证明：当且仅当两个可分离的、核的、O ∞ \mathcal O_\infty -stable C ∗ C^\ast -gealbras 是理想相关的 K KK -equivalent 时，它们是稳定同构的。特别是，这为基希贝格-菲利普斯定理提供了一个更基本的证明，该定理在论文中被单独列出，以增加这一重要特例的可读性。
I present a proof of Kirchberg’s classification theorem: two separable, nuclear, O∞\mathcal O_\infty-stable C∗C^\ast-algebras are stably isomorphic if and only if they are ideal-related KKKK-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.
期刊介绍：
Memoirs of the American Mathematical Society is devoted to the publication of research in all areas of pure and applied mathematics. The Memoirs is designed particularly to publish long papers or groups of cognate papers in book form, and is under the supervision of the Editorial Committee of the AMS journal Transactions of the AMS. To be accepted by the editorial board, manuscripts must be correct, new, and significant. Further, they must be well written and of interest to a substantial number of mathematicians.
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