Generalized C⁎-convexity in completely positive maps

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 $H$. These subsets are defined with in the set of all completely positive (CP) maps from a unital $C^{⁎}$-algebra $A$ into the algebra $B\left(H\right)$ of bounded linear maps on $H$. In particular, we focus on certain P-$C^{⁎}$-convex sets, denoted by ${\mathrm{CP}}^{\left(P\right)}(A,B(H\left)\right)$, 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 an abstract characterization and the structure of P-$C^{⁎}$-extreme points. Further, we discuss the connection between P-$C^{⁎}$-extreme points and linear extreme points of these convex sets, as well as Krein-Milman type theorems. Additionally, using these studies, we completely characterize the $C^{⁎}$-extreme points of the $C^{⁎}$-convex set of all contractive completely positive maps from $A$ into $B\left(H\right)$, where $H$ is finite-dimensional.