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.