Extreme points of unital completely positive maps invariant under partial action

TL;DR

This paper extends the classical Choquet theorem to characterize the extreme points of the set of unital completely positive maps invariant under a partial group action on a C*-algebra, revealing their structure in a convex analysis framework.

Contribution

It provides a characterization of extreme points of invariant unital completely positive maps under partial actions, extending barycentric decomposition concepts to this setting.

Findings

Characterization of extreme points of invariant maps.

Extension of Choquet theorem to partial actions.

Identification of the structure of invariant map sets.

Abstract

The classical Choquet theorem establishes a barycentric decomposition for elements in a compact convex subset of a locally convex topological vector space. This decomposition is achieved through a probability measure that is supported on the set of extreme points of the subset. In this work, we consider a partial action $\tau$ of a group $G$ on a $C^\ast$-algebra $\mathcal{A}$. For a fixed Hilbert space $\mathcal{H}$, we consider the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$ that are invariant under the partial action $\tau$. This set forms a compact convex subset of a locally convex topological vector space. To complete the picture of the barycentric decomposition provided by the classical Choquet theorem, we characterize the set of extreme points of this set.