Abstract:
The paper establishes a connection between two recent combinatorial developments in free probability: the non-crossing linked partitions introduced by Dykema in 2007 to study the S-transform, and the partial order << on NC(n) introduced by Belinschi and Nica in 2008 in order to study relations between free and Boolean probability. More precisely, one has a canonical bijection between NCL(n) (the set of all non-crossing linked partitions of {1, ..., n}) and the set {(p,q) | p,q in NC(n), p<

Abstract:
We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid construction, very similar to the one of a transformation group. We discuss examples related to Toeplitz algebras on subsemigroups of discrete groups,to Cuntz-Krieger algebras, and to crossed-products by partial automorphisms in the sense of Exel.

Abstract:
Let k be a positive integer and let D_k denote the space of joint distributions for k-tuples of selfadjoint elements in C*-probability space. The paper studies the concept of "subordination distribution of \mu \boxplus \nu with respect to \nu" for \mu, \nu \in D_k, where \boxplus is the operation of free additive convolution on D_k. The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space Multi-variable subordination turns out to have nice relations to a process of evolution towards \boxplus-infinite divisibility on D_k that was recently found by Belinschi and Nica (arXiv:0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion.

Abstract:
We consider the concept of irreducible meandric system introduced by Lando and Zvonkin. We place this concept in the lattice framework of NC(n). As a consequence, we show that the even generating function for irreducible meandric systems is the R-transform of XY, where X and Y are classically (commuting) independent random variables, and each of X,Y has centred semicircular distribution of variance 1. Following this point of view, we make some observations about the symmetric linear functional on polynomials which has R-transform given by the even generating function for meanders.

Abstract:
We study the set $\sncb (p,q)$ of annular non-crossing permutations of type B, and we introduce a corresponding set $\ncb (p,q)$ of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between $\sncb (p,q)$ and $\ncb (p,q)$ is a poset isomorphism, where the partial order on $\sncb (p,q)$ is induced from the hyperoctahedral group $B_{p+q}$, while $\ncb (p,q)$ is partially ordered by reverse refinement. In the case when $q=1$, we prove that $\ncb (p,1)$ is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between $\sncd (p,q)$ and $\ncd (p,q)$. For $q=1$, the poset $\ncd (p,1)$ coincides with a poset ``$NC^{(D)} (p+1)$'' constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper.

Abstract:
Free probabilistic considerations of type B first appeared in a paper by Biane, Goodman and Nica in 2003. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shlyakhtenko (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, phi, phi'), and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, phi, phi'), 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 of non-crossing partitions of type B appear in the combinatorial study of (A, phi, phi'), 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, phi) can be naturally upgraded to infinitesimal freeness in (A, phi, phi'), for a suitable choice of a "companion functional" phi'.

Abstract:
We show that for all q in the interval (-1,1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C*-algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (-1,1).

Abstract:
We study joint moments of a (2d)-tuple A_1, ..., A_d, B_1, ..., B_d of canonical operators on the full Fock space over a d-dimensional space, where A_1, ..., A_d act on the left and B_1, ..., B_d act on the right. The joint action of the A_i's and B_i's can be described in terms of the concept of a double-ended queue. The description which results for joint moments of the (2d)-tuple suggests that, for a general noncommutative probability space (A, phi) one should consider a certain family of "(l,r)-cumulant functionals", which enlarges the family of free cumulant functionals of (A, phi). The main result of the paper can then be phrased by saying that one has a simple formula for a relevant family of joint (l,r)-cumulants of A_1, ..., A_d, B_1, ..., B_d. This formula extends a known formula for joint free cumulants of A_1, ..., A_d or of B_1, ..., B_d (with the two d-tuples considered separately), which is used in one of the operator models for the R-transform of free probability.

Abstract:
In the free probability theory of Voiculescu two of the most frequently used *-distributions are those of a Haar unitary and of a circular element. We define an $R$-diagonal pair as a generalization of these distributions by the requirement that their two-dimensional $R$-transform (or free cumulants) have a special diagonal form. We show that the class of such $R$-diagonal pairs has an absorption property under nested multiplication of free pairs. This implies that in the polar decomposition of such an element the polar part and the absolute value are free. Our calculations are based on combinatorial statements about non-crossing partitions, in particular on a canonical bijection between the set of intervals of NC(n) and the set of 2-divisible partitions in NC(2n). In a forthcoming paper the theory of $R$-diagonal pairs will be used to solve the problem of the free commutator.