Roland Speicher

Combinatorial theory of the free product with amalgamation and operator-valued free probability theory

Preliminaries on non-crossing partitions Operator-valued multiplicative functions on the lattice of non-crossing partitions Amalgamated free products Operator-valued free probability theory Operator-valued stochastic processes and stochastic differential equations Bibliography.

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].

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

A new example of ‘independence’ and ‘white noise’

\SSn

Second order freeness and fluctuations of random matrices: II. Unitary random matrices

(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

A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation

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

Stochastic integration on the Cuntz algebra O

, where the matrix

Wigner chaos and the fourth moment

t_{js}^{ir}

Free diffusions, free entropy and free Fisher information

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