Infinitesimal non-crossing cumulants and free probability of type B

Abstract

Free probabilistic considerations of type B first appeared in the paper of Biane, Goodman and Nica [P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003Soc. 355 ( ) 2263Soc. 355 ( -2303. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko [S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at www.arxiv.org under reference arXiv:0903.2721]. The interplay between "type B" and "infinitesimal" is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A 1 , . . . , A k in an infinitesimal noncommutative probability space (A, ϕ, ϕ ) and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, ϕ, ϕ ), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A 1 , . . . , A k is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal counterpart for a theorem of Speicher from "usual" free probability. We show that the lattices NC (B) (n) of non-crossing partitions of type B appear in the combinatorial study of (A, ϕ, ϕ ), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimally free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A 1 , . . . , A k in (A, ϕ)

A construction which relates c-freeness to infinitesimal freeness

“…, A k . One has, as found in [11], a natural way of defining a family (κ ′ n : A n → C) ∞ n=1 of infinitesimal free cumulant functionals associated to (A, ϕ, ϕ ′ ), obtained essentially by "taking a formal derivative with respect to ϕ" in the formulas describing the free cumulants of ϕ. In terms of cumulants, the infinitesimal free independence of a family of unital subalgebras A 1 , .…”

“…Another way of thinking about Equations (3.6) and (3.7) goes by treating the double sums on their right-hand sides as single sums over certain sets of non-crossing partitions "of type B" (cf. [11], also [10]; a brief review of this point of view is shown in Section 5.1 below).…”

“…In the papers [3] and then [11] it was pointed out that, when used on the lattices N C (B) (n), the general machinery of Rota leads to the infinitesimal free cumulant functionals κ ′ n : A n → C associated to an I-ncps (A, ϕ, ϕ ′ ). The same considerations apply, without any changes, to the framework of (V, ϕ, ϕ ′ ) used in Section 3.3.…”

“…(2) The existence of µ ′ is a special case of the general construction of a free product of I-ncps spaces which is e.g. discussed in Section 2 of [11]. Uniqueness follows from Lemma 8.1(3).…”

Fock space associated to Coxeter groups of type B

“…Our construction of type B Fock space uses the geometry and length function of Coxeter groups based on simple roots, but the main idea of the paper [9] is to use all roots and another length function. Recently, connections between type B and infinitesimal free probability were put into evidence in [7,32,28,27].…”