Commutation superoperator of a state and its applications to the noncommutative statistics

For a normal state on a von Neumann algebra the space of square-integrable operators is introduced. As distinct from the L² space in the classical probability theory, it possesses an additional skew-symmetric form and the associated superoperator, which is a convenient tool to describe commutation properties in L². It is shown that a state on the algebra of canonical commutation relations is Gaussian (quasi-free) iff the space of canonical observables is an invariant subspace of the corresponding commutation superoperator. Basing on these ideas a new approach to some problems in the noncommutative statistic is developed.