关于一族从$M_{n}(\mathbb{C})$到$M_{n^{2}}(\mathbb{C})$的线性映射
On a family of a linear maps from $M_{n}(\mathbb{C})$ to $M_{n^{2}}(\mathbb{C})$
摘要：
Bhat刻画了定义在$B(\mathcal{H})$上的保持酉共轭的线性映射族.我们推广了这一思想，研究了有限维矩阵代数上具有类似等方差性质的映射。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线性映射的$k$-正性是有意义的。Choi证明了由$n$矩阵代数定义在$n$上的线性映射的$n$-正性与$(n-1)$-正性是不同的.本文提出了一个参数线性映射族$Φ_{α, β,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$，研究了它的正性、完全正性、可分解性等性质，确定了映射族$Φ_{α, β,n}$对任意自然数$n \geq 3$为正的参数$α$和$β$的值.我们主要研究了$n=3,$的情形，即$Φ_{α, β,3}$，并研究了$2$的性质-正性、完全正性和可分解性.特别是，我们给值的参数$α$和$β$的家庭的地图$Φ_{α, β,3}$是$2$-积极的，不完全积极的。
英文摘要：
Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ 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. 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 $Φ_{α, β,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$ 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 $n \geq 3$. 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.
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