Abstract
A subalgebraM 0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM 0. The main result says thatM 0 is weakly sufficient for (φ,ω) if and only ifM 0 contains the Radon-Nikodym cocycle [Dφ,Dω] t . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.
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