Sufficient subalgebras and the relative entropy of states of a von Neumann algebra

A superalgebra M0 of a von Neumann algebra M is called weakly sufficient with respect to a pair (phi,ω) of states if the relative entropy of phi and ω coincides with the relative entropy of their restrictions to M0. The main result says that M0 is weakly sufficient for (phi,ω) if and only if M0 contains the Radon-Nikodym cocycle [Dphi, Dω]sub(t). Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian. (orig.)