$C^{\ast }$-extreme points of unital completely positive maps on real $C^{\ast }$-algebras

Abstract

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

A shorter proof of Theorem 3.18 is added. Proposition 4.3 and Lemma 4.4 are newly added. Modified the proof of Proposition 4.8. This work is accepted for publication in Studia Mathematica