9. Operator-Valued Free Probability Theory and Block Random Matrices

Gaussian random matrices fit quite well into the framework of free probability theory, asymptotically they are semi-circular elements, and they have also nice freeness properties with other (e.g. non-random) matrices. Gaussian random matrices are used as input in many basic models in many different mathematical, physical, or engineering areas. Free probability theory provides then useful tools for the calculation of the asymptotic eigenvalue distribution for such models. However, in many situations, Gaussian random matrices are only the first approximation to the considered phenomena, and one would also like to consider more general kinds of such random matrices. Such generalizations often do not fit into the framework of our usual free probability theory. However, there exists an extension, operator-valued free probability theory, which still shares the basic properties of free probability but is much more powerful because of its wider domain of applicability. In this chapter, we will first motivate the operator-valued version of a semi-circular element and then present the general operator-valued theory. Here we will mainly work on a formal level; the analytic description of the theory, as well as its powerful consequences, will be dealt with in the following chapter.