Do C*-extreme instruments remain extreme in infinite dimensions?

Determine whether, for the set I_H(X, A) of normalized completely positive (CP) instruments on a measurable space (X, O(X)) with values in CP(A, B(H)), every C*-extreme instrument is also an extreme point when the Hilbert space H is infinite-dimensional. This asks whether the implication “C*-extreme ⇒ extreme” that holds in finite dimensions extends to the infinite-dimensional setting.

Background

In the finite-dimensional setting, the authors prove that C*-extreme instruments are also extreme in the usual convex sense. The proof leverages properties specific to finite-dimensional operator spaces (e.g., Hilbert–Schmidt norm geometry) to show that any proper convex decomposition contradicts C*-extremity.

By contrast, the infinite-dimensional case may lack these geometric features, and it is not currently established whether the same implication holds. Resolving this question would clarify the relationship between quantum convexity (C*-convexity) and classical convexity for instruments in infinite-dimensional Hilbert spaces, paralleling known results for CP maps and POVMs.