ON A GENERAL CONCEPT OF SUFFICIENCY IN VON NEUMANN
ALGEBRAS

Abstract: A general question about the sufficiency of a subalgebra of some bigger algebra in the
general operator algebra framework, under the assumption that the subalgebra in question is
complete with respect to a family of states, is considered. Two particular cases are dealt with:
sufficiency for Bayesian discrimination and sufficiency for unbiased estimation with minimal
variance. It turns out that in both cases sufficiency is equivalent to the existence of a
map from the bigger algebra into the smaller one having some specific properties.

2000 AMS Mathematics Subject Classification: Primary: 46L53; Secondary: 81P45,
62B15.

Keywords and phrases: Sufficient subalgebras, normal states, Bayesian discrimination,
unbiased estimation.