标题： 实数$C^*$-代数上的单元完全正映射的$C^*$-极点 显示英文标题

标题： $C^*$-extreme points of unital completely positive maps on real $C^*$-algebras

摘要： 在本文中，我们研究了在实酉$C^*$-代数$\mathcal{A}$到实希尔伯特空间$\mathcal{H}$上所有有界实线性映射的代数$B(\mathcal{H})$中的所有单位完全正（UCP）映射的$C^*$-凸集$\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$内的$C^*$-极点的一般性质和结构。 我们分析了实数和复数$C^*$-代数情况下$C^*$-极值点结构的差异。 特别是，我们证明了在矩阵代数之间的一个UCP映射成为$C^*$-极值点的充要条件在实数和复数矩阵代数情况下是相同的。 我们还观察到，当$\mathcal{A}$是一个交换的实$C^*$-代数时，$C^*$-极值点的结构与当$\mathcal{A}$是一个交换的复$C^*$-代数时存在显著差异。 我们对$\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$的$C^*$-极值点进行了完整的分类，其中$\mathcal{A}$是单位交换实$C^*$-代数，$\mathcal{H}$是有限维实希尔伯特空间。 作为应用，我们分类了所有$C^*$-极值点，在$C^*$-凸集中的所有收缩反对称实矩阵的$M_n(\mathbb{R})$中。 显示英文摘要

摘要： In this paper, we investigate the general properties and structure of $C^*$-extreme points within the $C^*$-convex set $\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$ of all unital completely positive (UCP) maps from a unital real $C^*$-algebra $\mathcal{A}$ to the algebra $B(\mathcal{H})$ of all bounded real linear maps on a real Hilbert space $\mathcal{H}$. We analyze the differences in the structure of $C^*$-extreme points between the real and complex $C^*$-algebra cases. In particular, we show that the necessary and sufficient conditions for a UCP map between matrix algebras to be a $C^*$-extreme point are identical in both the real and complex matrix algebra cases. We also observe significant differences in the structure of $C^*$-extreme points when $\mathcal{A}$ is a commutative real $C^*$-algebra compared to when $\mathcal{A}$ is a commutative complex $C^*$-algebra. We provide a complete classification of the $C^*$-extreme points of $\mathrm{UCP}(\mathcal{A},B(\mathcal{H}))$, where $\mathcal{A}$ is a unital commutative real $C^*$-algebra and $\mathcal{H}$ is a finite-dimensional real Hilbert space. As an application, we classify all $C^*$-extreme points in the $C^*$-convex set of all contractive skew-symmetric real matrices in $M_n(\mathbb{R})$.