When do pieces determine the whole? Extremal marginals of a completely positive map

We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extremal point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extremal, then the whole map is extremal. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.