This proposal concerns research topics at the interface between noncommutative probability, operator spaces and von Neumann algebra theory. Noncommutative conditional expectations and martingales arise in the setting of von Neumann algebras, which are the natural framework for noncommutative measure theory and integration. The proposed research will follow two main directions. First, the PI will continue her ongoing study of martingale transforms in noncommutative spaces of p-integrable functions/operators and related martingale inequalities. One of the goals of this research is the attempt of developing a satisfactory theory of vector-valued martingales in the noncommutative setting. The motivation lies in the fact that martingale theory in the classical setting provides a bridge between probability theory and harmonic analysis. In another direction, the PI is pursuing the investigation of problems concerning the relationship between the structure of von Neumann algebras (their type) and embedding properties of their preduals, considered as operator spaces. In this direction, new surprising results related to a classification of injective factors up to completely bounded isomorphism classes of their preduals have very recently been obtained in collaboration with U. Haagerup. In particular, we showed that there exist uncountably many mutually non-isomorphic injective factors of type III_0 whose preduals are not isomorphic as operator spaces. This is an interesting result, since by a theorem of E. Christensen and A. Sinclair the factors in question are all isomorphic as operator spaces. Techniques and tools provided by noncommutative probability have also proven to be of crucial importance in this direction of research. One such example is the (completely isomorphic) realization of the operator space OH inside the predual of the injective type III_1 factor, proved by M. Junge. Our work with U. Haagerup on the best constant in M. Junge's Khintchine-type inequality implies that new ideas are to be discovered in order to settle the fascinating-still open- question concerning the completely isometric embedding of OH.
The study of von Neumann algebras (suitable algebras of bounded linear operators on Hilbert spaces) has been motivated by the development of quantum physics, where the idea of replacing numbers (scalars) by not necessarily commuting operators (such as matrices) first appeared. The interplay and deep connections between noncommutative probability, operator spaces and von Neumann algebra theory have already proven to be very fruitful. The proposed research holds the promise of bringing new insights into these areas of mathematics, as well as a renewed interest in fundamental topics which are at the heart of von Neumann algebra theory, such as Tomita-Takesaki theory and the theory of ergodic flows. This project will enhance the exchange of ideas between these different fields of mathematics, and as such, it has the potential of increasing further collaboration between mathematicians working in these areas.
