Commutators of free random variables

Let A be a unital C -algebra, given together with a specified state ' : A ! C. Consider two selfadjoint elements a; b of A, which are free with respect to ' (in the sense of the free probability theory of Voiculescu). Let us denote c := i(ab \Γ 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]. 1. Introduction and presentation of the results In this paper we show how the combinatorial description of freeness can be used to determine the distribution of the commutator of two free random variables. The concept of freene...