Closed range of Φ(1) for C*-extreme points of CCP(A,B(H))

Ascertain whether, for any C*-extreme point Φ of the C*-convex set CCP(A,B(H)) of contractive completely positive maps, the operator Φ(1) necessarily has closed range; equivalently, determine whether the assumption “range(Φ(1)) is closed” in Lemma 4.10 holds automatically for such Φ.

Background

Lemma 4.10 proves that if Φ ∈ CCP C*-ext(A,B(H)) and range(Φ(1)) is closed, then Φ(1) is a projection. The authors note uncertainty about whether the closed-range condition is automatic for C*-extreme points, which would immediately imply the projection property and facilitate structural results.