Abstract. We examine, for −1 < q < 1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) Xt = at + a ∗ t – where the at fulfill the q-commutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for p-norm of the eigenvalues for matrices with independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative Lp for 2 < p < ∞. 0. Introduction and Notation Martingale inequalities have a long tradition in probability. The applications of the work of Burkholder and his collaborators [B73,?, BDG72, B71a, B71b, BGS71, BG70, B66] ranges from classical harmonic analysis to stochastical differential equations and the geometry of Banach spaces. When proving the estimates for the ‘little square function ’ Burkholder

Abstract. It is shown that if one keeps track of crossings, Feynman diagrams can be used to compute q-Wick products and normal products in terms of each other. 1.

We prove that the von Neumann algebras generated by n q-Gaussian elements, are factors for n � 2. 1

Let A denote the reduced amalgamated free product of a family

Here we study dilations of q-commuting tuples. In [BBD] the authors gave the correspondence between the two standard dilations of commuting tuples and here these results have been extended to q-commuting tuples. We are able to do this when q-coefficients ‘qij ’ are of modulus one. We introduce ‘maximal q-commuting subspace ’ of a n-tuple of operators and ‘standard q-commuting dilation’. Our main result is that the maximal q-commuting subspace of the standard noncommuting dilation of q-commuting tuple is the ‘standard q-commuting dilation’. We also introduce q-commuting Fock space as the maximal q-commuting subspace of full Fock space and give a formula for projection operator onto this space. This formula for projection helps us in working with the completely positive maps arising in our study. The first version of the Main Theorem (Theorem 19) of the paper for normal tuples using some tricky norm estimates and then use it to prove the general version of this theorem.

It is shown that the kernel of the Fock representation of a certain Wick algebra with braided operator of coefficients T, ||T | | ≤ 1, coincides with the largest quadratic Wick ideal. Improved conditions on the operator T for the Fock inner product to be strictly positive are given. 1. Introduction. The problem of positivity of the Fock space inner product is central in the study of the Fock representation of Wick algebras (see [2], [3], [5], [6]). The paper [6] presents several conditions on the coefficients of the Wick algebra for the Fock inner product to be positive. If the operator of coefficients of

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.

We consider the concept of q-circular system, which is a deformation of the circular system from free probability, taking place in the framework of the so-called "qcommutation relations". We show that certain averages of random unitaries in noncommutative tori behave asymptotically like a q-circular system. More precisely: let q be in (-1, 1); let s, k be positive integers; let (# ij ) 1#i<j#ks be independent random variables with values in the unit circle, such that R # ij = q, # 1 # i < j # ks; and let U 1 , . . . , U ks be random unitaries such that U i U j = # ij U j U i , #1 # i < j # ks. If we set: X r := 1 # k ( U r + U r+s + + U r+(k-1)s ), 1 # r # s, then the family X 1 , . . . , X s behaves for k ## like a q-circular system with s elements. The above result generalizes to the case when instead of the hypothesis " R # ij = q" we start with " R # ij = z", where z is a complex number such that |z| < 1. In this case the limit distribution of X 1 ,...