Completely bounded norms of k $k$ -positive maps

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2024-05-25 DOI:10.1112/jlms.12936

Guillaume Aubrun, Kenneth R. Davidson, Alexander Müller-Hermes, Vern I. Paulsen, Mizanur Rahaman

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引用次数: 0

Abstract

Given an operator system $S$ , we define the parameters $r_{k} (S)$ (resp. $d_{k} (S)$ ) defined as the maximal value of the completely bounded norm of a unital $k$ -positive map from an arbitrary operator system into $S$ (resp. from $S$ into an arbitrary operator system). In the case of the matrix algebras $M_{n}$ , for $1 ⩽ k ⩽ n$ , we compute the exact value $r_{k} (M_{n}) = \frac{2 n - k}{k}$ and show upper and lower bounds on the parameters $d_{k} (M_{n})$ . Moreover, when $S$ 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 $(r_{k} (S))$ tends to 1 if and only if $S$ is exact and that the sequence $(d_{k} (S))$ tends to 1 if and only if $S$ has the lifting property.

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k $k$ 正映射的完全有界规范

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.