Sufficiency and strong commutants in quantum probability theory

A probability algebra (A, *, ω) consisting of a^*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a^*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.