  This paper attempts to develop a theory of sufficiency in the setting of
non-commutative algebras parallel to the ideas in classical mathematical
statistics. Sufficiency of a coarse-graining means that all information is
extracted about the mutual relation of a given family of states. In the paper
sufficient coarse-grainings are characterized in several equivalent ways and
the non-commutative analogue of the factorization theorem is obtained. Among
the applications the equality case for the strong subadditivity of the von
Neumann entropy, the Imoto-Koashi theorem and exponential families are treated.
The setting of the paper allows the underlying Hilbert space to be infinite
dimensional.
