A characterization of freeness by a factorization property of R-transforms

Alexandru Nica, Dimitri Shlyakhtenko, and Roland Speicher

Let $M$ be a $B$-probability space. Assume that $B$ itself is a $D$-probability space; then $M$ can be viewed as a $D$-probability space as well. Let $X\in M$. We characterize freeness of $X$ from $B$ with amalgamation over $D$ in terms of a certain factorization condition linking the $B$-valued and $D$-valued $R$-transforms of $X$. We give an application to random matrices.