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

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 ${\text{UCP}}^{G_{\tau }}(\mathcal{A},\mathcal{B}(\mathcal{H}))$. This collection forms a $C^{\ast }$-convex set. We characterize the set of $C^{\ast }$-extreme points of ${\text{UCP}}^{G_{\tau }}(\mathcal{A},\mathcal{B}(\mathcal{H}))$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^{\ast }$-convexity for the set ${\text{UCP}}^{G_{\tau }}(\mathcal{A},\mathcal{B}(\mathcal{H}))$.

16 pages