{"title":"abelian <s:1> *代数的Banach空间理论表征","authors":"Ryotaro Tanaka","doi":"10.1090/bproc/175","DOIUrl":null,"url":null,"abstract":"<p>A Banach space theoretical characterization of abelian <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^*</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>- algebras among all <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^*</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-algebras is given. As an application, it is shown that if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\">\n <mml:semantics>\n <mml:mi>B</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">B</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are <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^*</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-algebras (nonlinearly) isomorphic to each other with respect to the structure of Birkhoff-James orthogonality, and if either <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> or <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\">\n <mml:semantics>\n <mml:mi>B</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">B</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is abelian, then they are <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"asterisk\">\n <mml:semantics>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n <mml:annotation encoding=\"application/x-tex\">*</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-isomorphic. Moreover, it is pointed out that the same kind of characterization is not valid for preduals of abelian von Neumann algebras.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A Banach space theoretical characterization of abelian 𝐶*-algebras\",\"authors\":\"Ryotaro Tanaka\",\"doi\":\"10.1090/bproc/175\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A Banach space theoretical characterization of abelian <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^*</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>- algebras among all <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^*</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-algebras is given. As an application, it is shown that if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A\\\">\\n <mml:semantics>\\n <mml:mi>A</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper B\\\">\\n <mml:semantics>\\n <mml:mi>B</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">B</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> are <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^*</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-algebras (nonlinearly) isomorphic to each other with respect to the structure of Birkhoff-James orthogonality, and if either <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A\\\">\\n <mml:semantics>\\n <mml:mi>A</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> or <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper B\\\">\\n <mml:semantics>\\n <mml:mi>B</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">B</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is abelian, then they are <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"asterisk\\\">\\n <mml:semantics>\\n <mml:mo>∗<!-- ∗ --></mml:mo>\\n <mml:annotation encoding=\\\"application/x-tex\\\">*</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-isomorphic. Moreover, it is pointed out that the same kind of characterization is not valid for preduals of abelian von Neumann algebras.</p>\",\"PeriodicalId\":106316,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society, Series B\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-06-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/bproc/175\",\"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/175","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A Banach space theoretical characterization of abelian 𝐶*-algebras
A Banach space theoretical characterization of abelian C∗C^*- algebras among all C∗C^*-algebras is given. As an application, it is shown that if AA and BB are C∗C^*-algebras (nonlinearly) isomorphic to each other with respect to the structure of Birkhoff-James orthogonality, and if either AA or BB is abelian, then they are ∗*-isomorphic. Moreover, it is pointed out that the same kind of characterization is not valid for preduals of abelian von Neumann algebras.
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