Sufficient subalgebras and the relative entropy of states of a von Neumann algebra (Q1079160)

Let M be a von Neumann algebra with faithful normal states \(\phi\) and \(\omega\). If the relative entropy S(\(\phi\),\(\omega)\) (introduced by Araki) is finite then a subalgebra \(M_ 0\) of M is called weakly sufficient with respect to \(\phi\) and \(\omega\) if \(S(\phi,\omega)=S(\phi | M_ 0,\omega | M_ 0)\). A noncommutative version of the Halmos-Savage theorem says that weak sufficiency of \(M_ 0\) is equivalent with the condition: \([D\phi,D\omega]_ t\in M_ 0\) for all \(t\in {\mathbb{R}}\). Other characterizations are given in terms of conditional expectations. Most of the results have been generalized in another paper of the author [''Sufficiency of channels over von Neumann algebras''].