Subordination in free probability. I

Free probability theory is a mathematical field discovered by Dan-Virgil Voiculescu where the discoverer himself and his successors have being actively working during the last decades. Being formally a part of the operator algebras theory, free probability in many aspects should be compared with the classical probability (more general concepts are considered in the framework of quantum or noncommutative probability). In particular, there exists a nontrivial parllelism between free limit theorems and free decomposition theory of probability laws, on the one hand, and corresponding classical results, on the other hand.
Voiculescu introduced in free probability certain analytic tools which substitute the classical apparatus of characteristic functions. The concept of subordination related to the analogous concept in the analytic functions theory was proposed by several mathematicians to remove some “technical assumptions” made in first Voiculescu's publications. This notion made it possible to prove limit theorems under natural assumptions. Moreover, the subordination turned out to be a concept of independent interest deserving further investigation. This will be sketched in the talk.