{"title":"不变嵌入和加权排列","authors":"M. Mastnak, H. Radjavi","doi":"10.1090/proc/16835","DOIUrl":null,"url":null,"abstract":"<p>We prove that for any fixed unitary matrix <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\"application/x-tex\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, any abelian self-adjoint algebra of matrices that is invariant under conjugation by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\"application/x-tex\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\"application/x-tex\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We use this result to analyse the structure of matrices <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Superscript asterisk Baseline upper A\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A^*A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> commutes with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A upper A Superscript asterisk\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">AA^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and to characterize matrices that are unitarily equivalent to weighted permutations.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Invariant embeddings and weighted permutations\",\"authors\":\"M. Mastnak, H. Radjavi\",\"doi\":\"10.1090/proc/16835\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that for any fixed unitary matrix <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U\\\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, any abelian self-adjoint algebra of matrices that is invariant under conjugation by <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U\\\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U\\\"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We use this result to analyse the structure of matrices <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A\\\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A Superscript asterisk Baseline upper A\\\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">A^*A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> commutes with <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A upper A Superscript asterisk\\\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">AA^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and to characterize matrices that are unitarily equivalent to weighted permutations.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-03-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16835\",\"RegionNum\":3,\"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":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16835","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们证明，对于任何固定的单元矩阵 U U，任何在 U U 共轭下不变的矩阵无边自交代数都可以嵌入到一个在 U U 共轭下仍然不变的最大无边自交代数中。我们利用这一结果来分析 A ∗ A A^*A 与 A A ∗ AA^* 共轭的矩阵 A A 的结构，并描述与加权排列单元等价的矩阵的特征。
We prove that for any fixed unitary matrix UU, any abelian self-adjoint algebra of matrices that is invariant under conjugation by UU can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by UU. We use this result to analyse the structure of matrices AA for which A∗AA^*A commutes with AA∗AA^*, and to characterize matrices that are unitarily equivalent to weighted permutations.
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