{"title":"A strong subadditivity-like inequality for quantum entropy in semifinite von Neumann algebras","authors":"Andrzej Łuczak","doi":"10.1002/mana.202300383","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math> be a semifinite von Neumann algebra with a normal faithful semifinite trace <span></span><math>\n <semantics>\n <mi>τ</mi>\n <annotation>$\\tau$</annotation>\n </semantics></math>, and let <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$\\mathcal {A}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$\\mathcal {B}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$\\mathcal {R}$</annotation>\n </semantics></math> be its subalgebras such that <span></span><math>\n <semantics>\n <mrow>\n <mi>R</mi>\n <mo>⊂</mo>\n <mi>A</mi>\n <mo>∩</mo>\n <mi>B</mi>\n </mrow>\n <annotation>$\\mathcal {R}\\subset \\mathcal {A}\\cap \\mathcal {B}$</annotation>\n </semantics></math> and that <span></span><math>\n <semantics>\n <mi>τ</mi>\n <annotation>$\\tau$</annotation>\n </semantics></math> restricted to any of these subalgebras is semifinite. Denote by <span></span><math>\n <semantics>\n <msub>\n <mi>E</mi>\n <mi>A</mi>\n </msub>\n <annotation>$\\mathbb {E}_\\mathcal {A}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <msub>\n <mi>E</mi>\n <mi>B</mi>\n </msub>\n <annotation>$\\mathbb {E}_\\mathcal {B}$</annotation>\n </semantics></math>, and <span></span><math>\n <semantics>\n <msub>\n <mi>E</mi>\n <mi>R</mi>\n </msub>\n <annotation>$\\mathbb {E}_\\mathcal {R}$</annotation>\n </semantics></math> the normal conditional expectations from <span></span><math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math> onto <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$\\mathcal {A}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$\\mathcal {B}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$\\mathcal {R}$</annotation>\n </semantics></math>, respectively, such that <span></span><math>\n <semantics>\n <mi>τ</mi>\n <annotation>$\\tau$</annotation>\n </semantics></math> is invariant with respect to any of them. The quadruple <span></span><math>\n <semantics>\n <mi>M</mi>\n <annotation>$\\mathcal {M}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$\\mathcal {A}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>B</mi>\n <annotation>$\\mathcal {B}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$\\mathcal {R}$</annotation>\n </semantics></math> is said to be a commuting square if\n\n </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 9","pages":"3192-3206"},"PeriodicalIF":0.8000,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300383","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
Abstract
Let be a semifinite von Neumann algebra with a normal faithful semifinite trace , and let , , be its subalgebras such that and that restricted to any of these subalgebras is semifinite. Denote by , , and the normal conditional expectations from onto , and , respectively, such that is invariant with respect to any of them. The quadruple , , , is said to be a commuting square if
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Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index
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