  Let A be a unital $C^*$-algebra, given together with a specified state
$\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with
respect to $\phi$ (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.
