Free Evolution on Algebras with Two States

ABSTRACT. The key result in the paper concerns two transformations, Φ : (ρ, ψ) 7→ φ and Bt : ψ 7→ φ, where ρ, ψ, φ are states on the algebra of non-commutative polynomials, or equivalently joint distributions of d-tuples of non-commuting operators. These transformations are related to free probability: if ⊞ is the free convolution operation, and {ρt} is a free convolution semigroup, we show that Φ [ρ, ψ ⊞ ρt] = Bt[Φ [ρ, ψ]]. The maps {Bt} were introduced by Belinschi and Nica as a semigroup of transformations such that B1 is the bijection between infinitely divisible distributions in Boolean and free probability theories. They showed that for γt the free heat semigroup and Φ the Boolean version of the Kolmogorov representation for infinitely divisible measures,