Infinite-dimensional structural decomposition of C*-extreme CCP maps

Investigate whether the finite-dimensional structural decomposition (equation (4.2)) for C*-extreme points of CCP(A,B(H)) extends to infinite-dimensional Hilbert spaces, potentially by proving that Φ(1) is a projection for every Φ ∈ CCP C*-ext(A,B(H)) and by extending Proposition 3.17(ii) to arbitrary H when P = Φ(1) is a projection.

Background

Theorem 4.12 gives a structural description (equation (4.2)) of C*-extreme points of CCP(A,B(H)) for finite-dimensional H. In Note 4.14, the authors explain that similar structure can be derived without invoking P-C*-extreme points in finite dimensions, but they are unsure whether such derivations can be carried out for infinite-dimensional H. They describe conditions (Φ(1) being a projection and a generalization of Proposition 3.17(ii)) that would suffice to obtain the structure in the infinite-dimensional setting.