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Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. (English) Zbl 0597.46067
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”].

MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L30 States of selfadjoint operator algebras
62B05 Sufficient statistics and fields
46L55 Noncommutative dynamical systems
81Q99 General mathematical topics and methods in quantum theory
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