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

TL;DR

This paper characterizes and classifies $C^*$-extreme points of unital completely positive maps on real $C^*$-algebras, highlighting differences from complex cases and providing a complete classification in certain finite-dimensional settings.

Contribution

It provides a complete classification of $C^*$-extreme points for unital commutative real $C^*$-algebras and compares their structure with complex cases.

Findings

Necessary and sufficient conditions for $C^*$-extreme points are identical in real and complex matrix algebras.

Significant structural differences exist between real and complex commutative $C^*$-algebras.

Complete classification of $C^*$-extreme points for contractive skew-symmetric real matrices.

Abstract

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 $M_n(\mathbb{R})$.