On a family of linear maps fromMn(C)toMn2(C)
On a family of linear maps fromMn(C)toMn2(C)
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在从 Mn(C) 到 Mn2(C) 的一系列线性映射上
DOI：
10.1016/j.laa.2018.06.011
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发表时间：
2018
影响因子：
1.1
通讯作者：
Sapra Gunjan
中科院分区：
文献类型：
--
作者：
Collins Benoit;Osaka Hiroyuki;Sapra Gunjan
Bhat [1] 描述了 B (H) 上定义的保持酉共轭的线性映射族。我们推广了这个想法，并研究了有限维矩阵代数上具有类似等方差性质的映射。我们证明了具有等方差性质的映射对于研究有限维矩阵代数上定义的线性映射的 k 正性具有重要意义。在[5]中，Choi 表明，对于在 n×n 矩阵代数上定义的线性映射，n-正性不同于 (n− 1)-正性。在本文中，我们提出了一个参数族线性映射 Φ α, β, n: M n (C)→ M n 2 (C)，并研究了正性、完全正性、可分解性等性质。我们确定了参数 α 和 β 的值，其中映射族 Φ α, β, n 对于任何自然数 n≥ 3 都为正。我们关注 n= 3 的情况，即 Φ α, β, 3、研究2-正性、完全正性和可分解性的性质。特别地，我们给出参数 α 和 β 的值，其中映射族 Φ α, β, 3 是 2-正的且不完全正的。
Bhat [1] characterizes the family of linear maps defined on B (H) which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study k-positivity of linear maps defined on finite-dimensional matrix algebras. In [5] Choi showed that n-positivity is different from (n− 1)-positivity for the linear maps defined on n by n matrix algebras. In this paper, we present a parametric family of linear maps Φ α, β, n: M n (C)→ M n 2 (C) and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters α and β for which the family of maps Φ α, β, n is positive for any natural number n≥ 3. We focus on the case of n= 3, that is, Φ α, β, 3 and study the properties of 2-positivity, completely positivity and decomposability. In particular, we give values of parameters α and β for which the family of maps Φ α, β, 3 is 2-positive and not completely positive.