Free harmonic analysis

Abstract

In this chapter we shall present an approach to free probability based on analytic functions. At the end of the previous chapter, we defined the Cauchy transform of a random variable a in an algebra A with a state φ to be the formal power series G(z)=1zM(1z) where M(z) = 1 + ∑n ≥ 1αnzn and αn = φ(an) are the moments of a. Then R(z), the R-transform of a, was defined to be the formal power series R(z) = ∑n ≥ 1κnzn−1 determined by the moment-cumulant relation which we have shown to be equivalent to the equations G(R(z)+1/z)=z=1/G(z)+R(G(z)).