Shlyakhtenko will study classes of operator algebras arising in connection with Voiculescu's free probability theory. The main emphasis of this research is placed on the study of operator-valued free random variables and operator algebras that they generate. Analysis of these von Neumann algebras is intimately tied with the goal of classification of free Araki-Woods factors, which are free probability analogs of ITPFI type III factors. Such analysis is also important for understanding of subfactors and automorphisms of amalgamated free product algebras. One of the goals of the present research is to develop free entropy-based techniques for dealing with operator-valued random variables.
Free probability theory is a highly non-commutative parallel to basic probability theory. Matrix-valued random variables (such as random matrices) naturally fit in the non-commutative probability framework; the asymptotic behavior of large random matrices is well-modeled by matrix-valued free random variables. Applications are in mathematics to the theory of operator algebras, subfactors, ergodic theory, as well as the theory of random matrices, which have connections with certain physical models.
