The notion of freeness in a noncommutative probability theory is a replacement of independence in the classical counterpart, see e.g. D. Shlyakhtenko [Int. Math. Res. Not. 1996, No. 20, 1013-1025 (1996; Zbl 0872.15018)]. Let \(D\) and \(B\) be some subalgebras of an algebra \(A\) such that \(D \subset B\). Actually, the problem here that the authors have in mind is to know what are possible general theorems about closing the gap between \(B\) and \(D\), under the circumstances where only the information on the \(B\)-valued distribution of a random variable \(X\) is available instead of its \(D\)-valued distribution that is really required. When \(X\) is free from \(B\) over \(D\), then the authors show that this freeness condition can equivalently be characterized in terms of cumulants and also in terms of free Fisher information [cf. A. Nica, D. Shlyakhtenko and the third author, Adv. Math. 141, No. 2, 282-321 (1999; Zbl 0929.46052)].
More precisely, the first main result asserts that freeness from a subalgebra \(B\) can be characterized by a factorization property of the \(B\)-valued cumulants. The secondly treated question is whether knowing that the free entropy of a random variable \(X\) conditioned on random variables \(Y\) and \(Z\) is the same as the free entropy of \(X\) conditioned on \(Z\) implies that \(X\) and \(Y\) are conditionally free over \(Z\). As a matter of fact, this is related to the previous questions by setting \(B=W^*(Y,Z)\) and \(D=W^*(Z)\). For the case \(D=\mathbb{C}\), the above question was solved in the affirmative by D. Voiculescu [Adv. Math. 146, No. 2, 101-166 (1999; Zbl 0956.46045)].
In the present paper, the authors extend the affirmative solution of Voiculescu to the case where \(D\) is finite-dimensional. However, the general case is still an open problem. The next main result is about a reformation of the characterizing equations for conjugate variables in terms of operator-valued cumulants, in connection with the concept of relative Fisher information.