2024/12/06 by Anand O. R, R, Anand O., K. Sumesh +1 · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2412.05008
openalex publication_date 2024/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(H) of bounded linear maps on H. In particular, we focus on certain P-C^*-convex sets, denoted by CP(P)(A,B(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 A into B(H), where 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.