{"title":"无限连通域上有界全纯函数的射影自由代数","authors":"A. Brudnyi","doi":"10.1090/spmj/1718","DOIUrl":null,"url":null,"abstract":"<p>The algebra <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>H</mml:mi>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>D</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^\\infty (D)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of bounded holomorphic functions on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D subset-of double-struck upper C\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>D</mml:mi>\n <mml:mo>⊂<!-- ⊂ --></mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">D\\subset \\mathbb {C}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is projective free for a wide class of infinitely connected domains. In particular, for such <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\n <mml:semantics>\n <mml:mi>D</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> every rectangular left-invertible matrix with entries in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>H</mml:mi>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>D</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^\\infty (D)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>H</mml:mi>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>D</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^\\infty (D)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> asserting that its covering dimension is <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and the second Čech cohomology group is trivial.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Projective free algebras of bounded holomorphic functions on infinitely connected domains\",\"authors\":\"A. Brudnyi\",\"doi\":\"10.1090/spmj/1718\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The algebra <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msup>\\n <mml:mi>H</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>D</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H^\\\\infty (D)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of bounded holomorphic functions on <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D subset-of double-struck upper C\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>D</mml:mi>\\n <mml:mo>⊂<!-- ⊂ --></mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">C</mml:mi>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">D\\\\subset \\\\mathbb {C}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is projective free for a wide class of infinitely connected domains. In particular, for such <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper D\\\">\\n <mml:semantics>\\n <mml:mi>D</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">D</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> every rectangular left-invertible matrix with entries in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msup>\\n <mml:mi>H</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>D</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H^\\\\infty (D)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H Superscript normal infinity Baseline left-parenthesis upper D right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msup>\\n <mml:mi>H</mml:mi>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>D</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H^\\\\infty (D)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> asserting that its covering dimension is <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2\\\">\\n <mml:semantics>\\n <mml:mn>2</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and the second Čech cohomology group is trivial.</p>\",\"PeriodicalId\":51162,\"journal\":{\"name\":\"St Petersburg Mathematical Journal\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2022-06-27\",\"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/1718\",\"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/1718","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Projective free algebras of bounded holomorphic functions on infinitely connected domains
The algebra H∞(D)H^\infty (D) of bounded holomorphic functions on D⊂CD\subset \mathbb {C} is projective free for a wide class of infinitely connected domains. In particular, for such DD every rectangular left-invertible matrix with entries in H∞(D)H^\infty (D) can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of H∞(D)H^\infty (D) asserting that its covering dimension is 22 and the second Čech cohomology group is trivial.
期刊介绍：
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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