$C^\ast$-extreme points of unital completely positive maps invariant under group action

TL;DR

This paper characterizes the $C^*$-extreme points of the set of unital completely positive maps invariant under a group action, and establishes a Krein--Milman type theorem within this $C^*$-convex framework.

Contribution

It provides a characterization of $C^*$-extreme points and proves a Krein--Milman theorem for invariant unital completely positive maps under group actions.

Findings

Characterization of $C^*$-extreme points of invariant UCP maps

Proof of a Krein--Milman type theorem in $C^*$-convexity setting

Analysis of $C^*$-convex structure under group automorphisms

Abstract

In this work, we study a sub-collection of unital completely positive maps from a unital $C^\ast$-algebra $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$ in the setting of $C^\ast$-convexity. Let $\tau$ be an action of a group $G$ on the $C^\ast$-algebra $\mathcal{A}$ through $C^\ast$-automorphisms. We focus our attention to the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, which remain invariant under $\tau$. We denote this collection by the notation $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. This collection forms a $C^\ast$-convex set. We characterize the set of $C^\ast$-extreme points of $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^\ast$-convexity for the set $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$.