Hans Maassen

Addition of freely independent random variables

Abstract A direct proof is given of Voiculescus addition theorem for freely independent real-valued random variables, using resolvents of self-adjoint operators. In contrast to the original proof, no assumption is made on the existence of moments above the second.

The essentially commutative dilations of dynamical semigroups onMn

AbstractFor identity and trace preserving one-parameter semigroups {Tt}t≧0 on then×n-matricesMn we obtain a complete description of their “essentially commutative” dilations, i.e., dilations, which can be constructed on a tensor product ofMn by a commutativeW*-algebra.We show that the existence of an essentially commutative dilation forTt is equivalent to the existence of a convolution semigroup of probability measures ρt on the group Aut(Mn) of automorphisms onMn such that

A SCATTERING THEORY FOR MARKOV CHAINS

T_t = \smallint _{Aut\left( {M_n } \right)} \alpha d\rho _t \left( \alpha \right)

Generalised Brownian Motion and Second Quantisation

, and this condition is then characterised in terms of the generator ofTt. There is a one-to-one correspondence between essentially commutative Markov dilations, weak*-continuous convolution semigroups of probability measures and certain forms of the generator ofTt. In particular, certain dynamical semigroups which do not satisfy the detailed balance condition are shown to admit a dilation. This provides the first example of a dilation for such a semigroup.

Quantum Poisson processes and dilations of dynamical semigroups

The q deformed commutation relation aa*−qa*a=1 for the harmonic oscillator is considered with q∈[−1,1]. An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of a+a* in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural q deformation of the Gaussian.

Symmetric Hilbert spaces arising from species of structures

In the operator algebraic formulation of probability theory Markov processes typically appear as perturbations of Bernoulli processes. We develop a scattering theory for this situation. This theory applies to the isomorphism problem between Markov processes and Bernoulli shifts as well as to the description of open quantum systems.

Quantum stochastic calculus and the dynamical Stark effect

If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravellings the time-averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states, the quantum trajectory converges in the mean to a random choice from these states.

An obstruction for q-deformation of the convolution product

A new approach to the generalised Brownian motion introduced by M. Bouzejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal’s notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor FV of the category of Hilbert spaces with contractions. A generalised Brownian motion is an algebra of creation and annihilation operators acting on FV (H) for arbitrary Hilbert spaces H and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions. The positivity is encoded by a � -semigroup of “broken pair partitions” whose representation space with respect to t is V . The existence of the second quantisation as functor t from Hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t . For a certain one parameter interpolation between the fermionic and the free Brownian motion it is shown that the “field algebras” ( K) are type II1 factors when K is infinite dimensional.