No abstract

The paper is discussing infinite divisibility in the setting of operatorvalued boolean, free and, more general, c-free independences. Particularly, using Hilbert bimodules and non-commutative functions techniques, we obtain analogues of the Levy-Hincin integral representation for infinitely divisible real measures.1 Proposition 2.7. If, in the above setting, X and Y are boolean independent over φ, then, for all positive integers n, we have thatDefinition 2.4 can be reformulated in terms of Proposition 2.7. Namely, for µ ∈ Σ B:D we define the n-th boolean cumulant of µ as the multilinear map B n,µ : B n −→ D given by the recurrence:From Proposition 2.7, we have the following Remark 2.8. An element µ ∈ Σ B:D is -infinite divisible if for any positive integer n there exist µ n ∈ Σ B:D such that for all positive integers m we have that B m,µ = nB m,µn .

Abstract. We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier [r 1 , r 2 ].

We consider the notion of monotonic independence in a more general frame, similar to the construction of operator-valued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi-inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arcsine distribution as well as a connection to operator-valued conditional freeness. IntroductionAn important notion in noncommutative probability is monotonic independence, introduced by P. Y. Lu and Naofumi Muraki. Since its beginning, the study of this notion of independence was done by constructions, techniques and developments similar to the theory of free probability. R. Speicher [1998] developed an operatorvalued analogue of free independence. The present paper addresses problems similar to ones discussed in that work, but in the context of monotonic independence.Other motivation is that while for the free Fock space over a Hilbert space there is a straightforward analogous semi-inner-product bimodule construction, as illustrated in [Pimsner 1997;Speicher 1998], there are no similar constructions for its various deformations, such as the q-Fock spaces [Effros and Popa 2003]. As shown in Section 4, the monotone and weakly monotone Fock-like spaces, which are strongly connected to monotonic independence, admit analogous semi-innerproduct bimodules.The paper is structured in six sections. Section 2 presents the definition of the monotonic independence over an algebra. In Section 3 there are constructed maps with similar properties to the maps H and K from the theory of monotonic independence, as introduced in [Muraki 2000; Bercovici 2005b]. Section 4 deals with semi-inner-product bimodule analogues of the monotone and weakly monotone products of Hilbert spaces and algebras of annihilation operators, as introduced in MSC2000: primary 46L53; secondary 46L08.

Haiman and Schmitt showed that one can use the antipode S F of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode S H of an algebraically free analogue H of F may be used to invert non-commutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for S F are labeled by trees. By contrast, the noncommuting monomials of S H are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the Connes-Kreimer Hopf algebra.

We show that real second order freeness appears in the study of Haar unitary and unitarily invariant random matrices when transposes are also considered. In particular we obtain the unexpected result that a unitarily invariant random matrix will be asymptotically free from its transpose.

We use the theory of fully matricial, or noncommutative, functions to investigate infinite divisibility and limit theorems in operator-valued noncommutative probability. Our main result is an operator-valued analogue for the Bercovici-Pata bijection. An important tool is Voiculescu's subordination property for operator-valued free convolution.

It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute the q-Wick products and usual operator products in terms of each other.