General Krein–Milman theorem for UCP(A,B(H))

Establish whether the Krein–Milman theorem holds in full generality for the C*-convex set UCP(A,B(H)); specifically, determine if UCP(A,B(H)) equals the bounded-weak closure of the C*-convex hull of its C*-extreme points for arbitrary unital C*-algebras A and arbitrary Hilbert spaces H.

Background

The paper surveys known cases where the quantum analogue of the Krein–Milman theorem holds for the generalized state space UCP(A,B(H)), namely when H is finite-dimensional; when A is commutative with arbitrary H; and when A is separable or a type I factor with H separable. Beyond these scenarios, the authors note that the general case is unresolved.