Guillaume Aubrun, Kenneth R. Davidson, Alexander Müller-Hermes, Vern I. Paulsen, Mizanur Rahaman
{"title":"k $k$ 正映射的完全有界规范","authors":"Guillaume Aubrun, Kenneth R. Davidson, Alexander Müller-Hermes, Vern I. Paulsen, Mizanur Rahaman","doi":"10.1112/jlms.12936","DOIUrl":null,"url":null,"abstract":"<p>Given an operator system <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$\\mathcal {S}$</annotation>\n </semantics></math>, we define the parameters <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>S</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$r_k(\\mathcal {S})$</annotation>\n </semantics></math> (resp. <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>d</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>S</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$d_k(\\mathcal {S})$</annotation>\n </semantics></math>) defined as the maximal value of the completely bounded norm of a unital <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>$k$</annotation>\n </semantics></math>-positive map from an arbitrary operator system into <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$\\mathcal {S}$</annotation>\n </semantics></math> (resp. from <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$\\mathcal {S}$</annotation>\n </semantics></math> into an arbitrary operator system). In the case of the matrix algebras <span></span><math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mi>n</mi>\n </msub>\n <annotation>$\\mathsf {M}_n$</annotation>\n </semantics></math>, for <span></span><math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>⩽</mo>\n <mi>k</mi>\n <mo>⩽</mo>\n <mi>n</mi>\n </mrow>\n <annotation>$1 \\leqslant k \\leqslant n$</annotation>\n </semantics></math>, we compute the exact value <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>M</mi>\n <mi>n</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mfrac>\n <mrow>\n <mn>2</mn>\n <mi>n</mi>\n <mo>−</mo>\n <mi>k</mi>\n </mrow>\n <mi>k</mi>\n </mfrac>\n </mrow>\n <annotation>$r_k(\\mathsf {M}_n) = \\frac{2n-k}{k}$</annotation>\n </semantics></math> and show upper and lower bounds on the parameters <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>d</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>M</mi>\n <mi>n</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$d_k(\\mathsf {M}_n)$</annotation>\n </semantics></math>. Moreover, when <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$\\mathcal {S}$</annotation>\n </semantics></math> is a finite-dimensional operator system, adapting results of Passer and the fourth author [J. Operator Theory 85 (2021), no. 2, 547–568], we show that the sequence <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>r</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>S</mi>\n <mo>)</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$(r_k(\\mathcal {S}))$</annotation>\n </semantics></math> tends to 1 if and only if <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$\\mathcal {S}$</annotation>\n </semantics></math> is exact and that the sequence <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>d</mi>\n <mi>k</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>S</mi>\n <mo>)</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation>$(d_k(\\mathcal {S}))$</annotation>\n </semantics></math> tends to 1 if and only if <span></span><math>\n <semantics>\n <mi>S</mi>\n <annotation>$\\mathcal {S}$</annotation>\n </semantics></math> has the lifting property.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"109 6","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Completely bounded norms of \\n \\n k\\n $k$\\n -positive maps\",\"authors\":\"Guillaume Aubrun, Kenneth R. Davidson, Alexander Müller-Hermes, Vern I. Paulsen, Mizanur Rahaman\",\"doi\":\"10.1112/jlms.12936\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given an operator system <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$\\\\mathcal {S}$</annotation>\\n </semantics></math>, we define the parameters <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>S</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$r_k(\\\\mathcal {S})$</annotation>\\n </semantics></math> (resp. <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>d</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>S</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$d_k(\\\\mathcal {S})$</annotation>\\n </semantics></math>) defined as the maximal value of the completely bounded norm of a unital <span></span><math>\\n <semantics>\\n <mi>k</mi>\\n <annotation>$k$</annotation>\\n </semantics></math>-positive map from an arbitrary operator system into <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$\\\\mathcal {S}$</annotation>\\n </semantics></math> (resp. from <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$\\\\mathcal {S}$</annotation>\\n </semantics></math> into an arbitrary operator system). In the case of the matrix algebras <span></span><math>\\n <semantics>\\n <msub>\\n <mi>M</mi>\\n <mi>n</mi>\\n </msub>\\n <annotation>$\\\\mathsf {M}_n$</annotation>\\n </semantics></math>, for <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>⩽</mo>\\n <mi>k</mi>\\n <mo>⩽</mo>\\n <mi>n</mi>\\n </mrow>\\n <annotation>$1 \\\\leqslant k \\\\leqslant n$</annotation>\\n </semantics></math>, we compute the exact value <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>M</mi>\\n <mi>n</mi>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <mfrac>\\n <mrow>\\n <mn>2</mn>\\n <mi>n</mi>\\n <mo>−</mo>\\n <mi>k</mi>\\n </mrow>\\n <mi>k</mi>\\n </mfrac>\\n </mrow>\\n <annotation>$r_k(\\\\mathsf {M}_n) = \\\\frac{2n-k}{k}$</annotation>\\n </semantics></math> and show upper and lower bounds on the parameters <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>d</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>M</mi>\\n <mi>n</mi>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$d_k(\\\\mathsf {M}_n)$</annotation>\\n </semantics></math>. Moreover, when <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$\\\\mathcal {S}$</annotation>\\n </semantics></math> is a finite-dimensional operator system, adapting results of Passer and the fourth author [J. Operator Theory 85 (2021), no. 2, 547–568], we show that the sequence <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>r</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>S</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(r_k(\\\\mathcal {S}))$</annotation>\\n </semantics></math> tends to 1 if and only if <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$\\\\mathcal {S}$</annotation>\\n </semantics></math> is exact and that the sequence <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>d</mi>\\n <mi>k</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>S</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(d_k(\\\\mathcal {S}))$</annotation>\\n </semantics></math> tends to 1 if and only if <span></span><math>\\n <semantics>\\n <mi>S</mi>\\n <annotation>$\\\\mathcal {S}$</annotation>\\n </semantics></math> has the lifting property.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"109 6\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-05-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12936\",\"RegionNum\":2,\"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":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12936","RegionNum":2,"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
给定一个算子系统 S $\mathcal {S}$ ，我们定义参数 r k ( S ) $r_k(\mathcal {S})$ （或者 d k ( S ) $d_k(\mathcal {S})$ ），定义为从一个任意算子系统到 S $\mathcal {S}$ （或者从 S $\mathcal {S}$ 到一个任意算子系统）的单一 k $k$ 正映射的完全有界规范的最大值。在矩阵代数 M n $\mathsf {M}_n$ 的情况下，对于 1 ⩽ k ⩽ n $1 \leqslant k \leqslant n$ ，我们计算了精确值 r k ( M n ) = 2 n - k k $r_k(\mathsf {M}_n) = \frac{2n-k}{k}$ 并给出了参数 d k ( M n ) $d_k(\mathsf {M}_n)$ 的上界和下界。此外，当 S $\mathcal {S}$ 是有限维算子系统时，根据 Passer 和第四作者的结果 [J. Operator Theory 85 (2021), no.2, 547-568] 时，我们证明只有当 S $\mathcal {S}$ 是精确的，序列 ( r k ( S ) ) $(r_k(\mathcal {S}))$ 才会趋向于 1；只有当 S $\mathcal {S}$ 具有提升性质时，序列 ( d k ( S ) ) $(d_k(\mathcal {S}))$ 才会趋向于 1。
Given an operator system , we define the parameters (resp. ) defined as the maximal value of the completely bounded norm of a unital -positive map from an arbitrary operator system into (resp. from into an arbitrary operator system). In the case of the matrix algebras , for , we compute the exact value and show upper and lower bounds on the parameters . Moreover, when is a finite-dimensional operator system, adapting results of Passer and the fourth author [J. Operator Theory 85 (2021), no. 2, 547–568], we show that the sequence tends to 1 if and only if is exact and that the sequence tends to 1 if and only if has the lifting property.
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