2001/12/31 by Alexandru Nica, Nica, Alexandru, Dimitri Shlyakhtenko +3
Mathematics · #46L54 #FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR) #math.OA #math.PR #msc:46L54
paper · pdf · doi:10.48550/arxiv.math/0201001
arxiv created 2001/12/31 · arxiv updated 2009/11/30
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 be in 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 factorization 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.