$C^*$-extreme contractive completely positive maps
Anand O. R, K. Sumesh
TL;DR
The article develops $P$-$C^*$-convexity for completely positive maps $oldsymbol{\Phi:\mathcal{A} o ext{ obreakspace} ext{ obreakspace} ext{ obreakspace} ext{B}({\mathcal{H}})}$ with $oldsymbol{\Phi(1)=P}$ and introduces the associated $P$-$C^*$-extreme points, extending classical $C^*$-convexity and Krein–Milman theory to this generalized setting. In finite-dimensional $ ext{ obreakspace} ext{H}$ with invertible $P$, the $P$-$C^*$-extreme points are precisely invertible conjugates of $C^*$-extreme UCP maps, providing a complete structural picture when combined with established results for $ ext{UCP}(\mathcal{A}, ext{B}({\mathcal{H}}))$ and nested compressions. The paper also proves a Krein–Milman type theorem for the BW-topology and characterizes the $C^*$-extreme points of contractive CP maps $ ext{CCP}(\mathcal{A}, ext{B}({\mathcal{H}}))$, revealing a block-structural decomposition tied to the support projection $P= ext{CP}(1)$. Further, in the CCP$^ imes$ setting, $C^*$-extremality coincides with $UCP$-extremality, and in commutative cases the extremal maps reduce to $*$-homomorphisms, highlighting a coherent unification of extreme-point structures across noncommutative and commutative contexts.
Abstract
In this paper we generalize a specific quantized convexity structure of the generalized state space of a $C^*$-algebra and examine the associated extreme points. We introduce the notion of $P$-$C^*$-convex subsets, where $P$ is any positive operator on a Hilbert space $\mathcal{H}$. These subsets are defined with in the set of all completely positive (CP) maps from a unital $C^*$-algebra $\mathcal{A}$ into the algebra $B(\mathcal{H})$ of bounded linear maps on $\mathcal{H}$. In particular, we focus on certain $P$-$C^*$-convex sets, denoted by $\mathrm{CP}^{(P)}(\mathcal{A},B(\mathcal{H}))$, and analyze their extreme points with respect to this new convexity structure. This generalizes the existing notions of $C^*$-convex subsets and $C^*$-extreme points of unital completely positive maps. We significantly extend many of the known results regarding the $C^*$-extreme points of unital completely positive maps into the context of $P$-$C^*$-convex sets we are considering. This includes abstract characterization and structure of $P$-$C^*$-extreme points. Further, using these studies, we completely characterize the $C^*$-extreme points of the $C^*$-convex set of all contractive completely positive maps from $\mathcal{A}$ into $B(\mathcal{H})$, where $\mathcal{H}$ is finite-dimensional. Additionally, we discuss the connection between $P$-$C^*$-extreme points and linear extreme points of these convex sets, as well as Krein-Milman type theorems.