Commutators of free random variables

Let A be a unital C∗-algebra, given together with a specified state φ : A → C. Consider two selfadjoint elements a, b of A, which are free with respect to φ (in the sense of the free probability theory of Voiculescu). Let us denote c := i(ab− ba), where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper [12].