$C^\ast$-extreme points of unital completely positive maps invariant under group action
Chaitanya J. Kulkarni
TL;DR
This work analyzes unital completely positive maps from a unital C*-algebra 𝔄 to ℬ(ℋ) that are invariant under a group action τ. Framing the invariant maps within the C*-convexity paradigm, the authors develop a Radon–Nikodym-type theory and obtain both sufficient and necessary conditions for C*-extremality, including a two-term decomposition and a phi-invertible operator criterion. They provide a complete characterization of the C*-extreme points and establish a Krein–Milman type theorem showing that the invariant CP maps are the BW-closure of the C*-convex hull of their C*-extreme points under suitable hypotheses. The results extend the noncommutative convexity theory for CP maps to the setting of group-invariant maps and yield structural insight into ergodic-type decompositions of invariant maps.
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 $τ$ 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 $τ$. We denote this collection by the notation $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. This collection forms a $C^\ast$-convex set. We characterize the set of $C^\ast$-extreme points of $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^\ast$-convexity for the set $\text{UCP}^{G_τ} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$.