Abstract. A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected partitions respectively. The proof relies on Möbius inversion on the partition lattice. 1.

Let A denote the reduced amalgamated free product of a family

ABSTRACT. Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with a stochastic process and using the stochastic measures machinery introduced by Rota and Wallstrom, we calculate and give an interpretation of linearization coefficients for a number of polynomial families. The processes involved may have independent, freely independent, or q-independent increments. The use of noncommutative stochastic processes extends the range of applications significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier, and Rogers and continuous big q-Hermite polynomials. We also show that the q-Poisson process is a Markov process. 1.

Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a q-commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest. 1.

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam–Chihara polynomials and a relation between these polynomials for different values of parameters. 1. Introduction. The

ABSTRACT. This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If a1,..., ak are ∗-free R-diagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −nt ai1 · · · ain defines a dilation semigroup on the non-self-adjoint operator algebra generated by a1,..., ak. We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1,..., ak. Moreover, we show that Dt satisfies an optimal ultracontractive property: �Dt: L 2 → L ∞ � ∼ t −1 for small t> 0. 1.

We show that classical processes corresponding to operators what satisfy a q-commutative relation have linear regressions and quadratic conditional variances. From this we deduce that Bell’s inequality for their covariances can be extended from q = −1 to the entire range −1 ≤ q < 1. 1

Abstract: We introduce a non-commutative extension of Tsirelson-Vershik’s noises [TV98, Tsi04], called (non-commutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, in particular, capable of producing Arveson’s product system of type I and type II [Arv03]. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalar-expected) continuous Bernoulli shift is either finite or of type III. The role of (‘classical’) G-stationary flows for Tsirelson-Vershik’s noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Lévy processes. They lead, in particular, to units and ‘logarithms ’ of units in Arveson’s product systems [Kös04a]. Furthermore, we introduce (non-commutative) white noises, which are operator algebraic versions of Tsirelson’s ‘classical ’ noises. We give examples coming from probability, quantum probability and from Voiculescu’s theory of free probability [VDN92]. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: non-commutative extensions of stochastic Itô integration, stochastic logarithms and exponentials.

Abstract. We consider the concepts of continuous Bernoulli systems and non-commutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum Lévy processes one can make quite precise statements about the time behaviour of their moments. 1.

Abstract. In this paper combinatorial aspects of normal ordering arbitrary words in the creation and annihilation operators of the q-deformed boson are discussed. In particular, it is shown how by introducing appropriate q-weights for the associated “Feynman diagrams ” the normally ordered form of a general expression in the creation and annihilation operators can be written as a sum over all q-weighted Feynman diagrams, representing Wick’s theorem in the present context.