Models in Quantum Probability

A noncommutative probability space is a quantum analog of the classical Kolmogorov construction. Unlike its classical counterpart, a noncommutative probability space has an important feature: quantum correlations. Our aim will be to construct some concrete examples of such objects and to describe their properties. In particular, a proper tools for description of quantum correlations should be constructed. It seems that methods of recently growing operator spaces theory should be applied for this task.

A further step in this research would be a description of evolution of noncommutative systems as well as their symmetries. Evolution is analyzed in terms of quantum dynamical semigroups which are composed of positive maps. One of our goals will be to construct some concrete examples of quantum semigroups and to investigate properties of positive maps involved in them. As regards quantum symmetries, we will construct concrete examples of quantum groups acting on noncommutative spaces.

The main tools in this research will come from linear algebra, functional analysis, measure theory and the theory of operator algebras.