Subordination Problems Related to Free Probability

Our research project is in the area of noncommutative probability. Noncommutative probability emerged in the early ’80s as a very powerful tool for the study of finite operator algebras. The fundamental idea is to view a pair (A, τ), where A is a unital algebra over the complex numbers (usually endowed with a suitable norm topology), and τ is a linear, unit preserving, functional on A, as a noncommutative probability space, in which the role of integration is taken by τ , while A plays the role of a function algebra over a probability space from the commutative case. There are several notions of independence specific only to the noncommutative setup. From an operator algebraic perspective, one can think of a noncommutative independence as a rule on how to extend τ from a family of independent subalgebras to the algebra they generate together. Our project is mainly related to the free independence. Free independence was introduced by Voiculescu [10] in the eighties with the intention of studying in a probabilistic framework the free group factors L(Fn) generated by the left regular representation of the free group with n generators Fn. Since then, free probability became a powerful tool in several other areas of mathematics beyond operator algebras, especially in random matrix theory (the paper [11] initiated this direction of investigation, which since then has known a spectacular growth). The subordination property (in the sense of Littlewood) for free convolutions, which forms the main subject of our research project, has been first noted in [12] by Voiculescu for free additive convolution: it has been shown in this paper (under an easily removable genericity condition) that the Cauchy-Stieltjes transform