I. Characterizations of Freeness

Let M be a B-probability space. Assume that B itself is a D-probability space; then M can be viewed as D-probability space as well. Let X ∈ M. We look at the question of relating the properties of X as B-valued random variable to its properties as D-valued random variable. We characterize freeness of X from B with amalgamation over D: (a) in terms of a certain fac-torization condition linking the B-valued and D-valued cumulants of X, and (b) for D finite-dimensional, in terms of linking the B-valued and the D-valued Fisher information of X. We give an application to random matrices. For the second characterization we derive a new operator-valued description of the conjugate variable and introduce an operator-valued version of the liberation gradient.