$C^*$-extreme points of unital completely positive maps on real $C^*$-algebras
Streszczenie
We investigate the general properties and structure of $C^*$-extreme points within the $C^*$-convex set $\mathrm{UCP}(\mathcal {A},\mathscr {B}({\mathcal {H}}))$ of all unital completely positive (UCP) maps from a unital real $C^*$-algebra $\mathcal {A}$ to the algebra $\mathscr {B}({\mathcal {H}})$ 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 ${\rm UCP} (\mathcal {A},\mathscr {B}(\mathcal {H}))$, 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 matrices in $M_n(\mathbb {R})$.
