Free evolution on algebras with two states

TL;DR

This paper explores transformations on states in non-commutative probability, revealing their properties, fixed points, and connections to free and Boolean probability theories, with implications for free Meixner families.

Contribution

It introduces a two-variable map Phi from two-state algebra theory, extending previous single-variable results and analyzing its properties and fixed points.

Findings

Phi is positive and well-behaved under free Meixner families.

The paper characterizes fixed points and the range of Phi.

Connections between c-free Appell polynomials and the map Phi are established.

Abstract

The key result in the paper concerns two transformations, Phi(rho, psi) and B_t(psi) on states on the algebra of non-commutative polynomials, or equivalently on joint distributions of d-tuples of non-commuting operators. These transformations are related to free probability: Phi intertwines the action of B_t and the free convolution with the semigroup {rho_t}. The maps {B_t} were introduced by Belinschi and Nica as a semigroup of transformations such that B_1 is the bijection between infinitely divisible distributions in the Boolean and free probability theories. They proved the intertwining property above for a single-variable version of the map Phi and the particular case of the free heat semigroup. The more general two-variable map Phi comes, not from free probability, but from the theory of two-state algebras, also called the conditionally free probability theory, introduced by Bozejko, Leinert, and Speicher. Orthogonality of the c-free versions of the Appell polynomials, investigated in arXiv:0803.4279, is closely related to the single-variable map Phi. On the other hand, more general free Meixner families behave well under all the transformations above, and provide clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Phi and descriptions of its fixed points and range.