2412.05008Dec 6, 2024To appear in Journal of Mathematical Analysis and Applications with the title "Generalized $C^*$-convexity in Completely Positive Maps". Removed the closed range assumption in Lemma 4.10 and subsequent results are improved accordingly
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
Nearby in the stack
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.
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