Commutators of free random variables

Abstract

Let A be a unital $C^*$-algebra, given together with a specified state $ϕ:A \to C$. 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 funct-an/9604012.