Marek Bożejko

An Example of a Generalized Brownian Motion

We present an example of a generalized Brownian motion. It is given by creation and annihilation operators on a “twisted” Fock space ofL2(ℝ). These operators fulfill (for a fixed −1≦μ≦1) the relationsc(f)c*(g)−μc*(g)c(f)=〈f,g〉1 (f, g ∈L2(ℝ)). We show that the distribution of these operators with respect to the vacuum expectation is a generalized Gaussian distribution, in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula. We also indicate that our Brownian motion is one component of ann-dimensional Brownian motion which is invariant under the quantum groupSνU(n) of Woronowicz (withμ =v2).

q-Gaussian Processes: Non-commutative and Classical Aspects

Abstract: We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) – where the at fulfill the q-commutation relations for some covariance function – 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 possesses 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].

INTERACTING FOCK SPACES AND GAUSSIANIZATION OF PROBABILITY MEASURES

We prove that any probability measure on ℝ, with moments of all orders, is the vacuum distribution, in an appropriate interacting Fock space, of the field operator plus (in the nonsymmetric case) a function of the number operator. This follows from a canonical isomorphism between the L2-space of the measure and the interacting Fock space in which the number vectors go into the orthogonal polynomials of the measure and the modified field operator into the multiplication operator by the x-coordinate. A corollary of this is that all the momenta of such a measure are expressible in terms of the Szego–Jacobi parameters, associated to its orthogonal polynomials, by means of diagrams involving only noncrossing pair partitions (and singletons, in the nonsymmetric case). This means that, with our construction, the combinatorics of the momenta of any probability measure (with all moments) is reduced to that of a generalized Gaussian. This phenomenon we call Gaussianization. Finally we define, in terms of the Szego–...

Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces.

In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group

On a class of free Lévy laws related to a regression problem

\SSn

Remarks on t-transformations of measures and convolutions

(or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations

Meixner Class of Non-Commutative Generalized Stochastic Processes with Freely Independent Values I. A Characterization

d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=\delta_{ij}\id

Weakly amenable groups and amalgamated products

, where the matrix

Positive definite bounded matrices and a characterization of amenable groups

t_{js}^{ir}

The normal distribution is -infinitely divisible

is given by a self-adjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by