论文标题
通过奇特不平等表征施瓦兹地图
Characterizing Schwarz maps by tracial inequalities
论文作者
论文摘要
令$ ϕ $为$ n \ times n $矩阵$ {\ mathcal m} _n $的线性地图到$ m \ times m $矩阵$ {\ MATHCAL M} _M $。众所周知，$ ϕ $是$ 2 $ - 启动的，并且仅当所有$ k \ in {\ mathcal m} _n $中，并且所有严格的正$ x \ in {\ Mathcal M} _n $，$ dectorme如果$ ϕ $仅仅是Schwarz地图，则这种不平等通常不正确。我们表明，相应的曲折不等式$ {\ rm tr} [ϕ（k^*x^{ - 1} k）] \ geq {\ rm tr} [ϕ（k）^*ϕ（x）^{ - x）^{ - 1} ϕ（k）] $在此处指定的正面映射。我们还评论了这种不平等与各种单调性的联系，这些单调性在数学物理学中广泛使用，并将其应用于近亲，以获得一些新的确定结果。
Let $ϕ$ be a linear map from the $n\times n$ matrices ${\mathcal M}_n$ to the $m\times m$ matrices ${\mathcal M}_m$. It is known that $ϕ$ is $2$-positive if and only if for all $K\in {\mathcal M}_n$ and all strictly positive $X\in {\mathcal M}_n$, $ϕ(K^*X^{-1}K) \geq ϕ(K)^*ϕ(X)^{-1}ϕ(K)$. This inequality is not generally true if $ϕ$ is merely a Schwarz map. We show that the corresponding tracial inequality ${\rm Tr}[ϕ(K^*X^{-1}K)] \geq {\rm Tr}[ϕ(K)^*ϕ(X)^{-1}ϕ(K)]$ holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.
