Research Projects

The idea of replacing functions by linear operators (bounded or unbounded), that is the process of quantisation, can be traced back to the early foundations of quantum mechanics and has a profound impact in both pure and applied mathematics. This applies to for example representation theory, operator algebra, non-commutative geometry, quantum and free probability, operator system or operator space theory. The quantisation of the classical measure theory leads to the theory of L_p-spaces defined over an arbitrary von Neumann algebra, so called non-commutative L_p-spaces. This theory was initiated by Segal, Dixmier and Kunze in the fifties and was continued years later by Haagerup, Fack-Kosaki and many others. Though some basic properties of these spaces were explored, very little progress was made in the direction of harmonic analysis, ergodic theory or probability theory in this new set-up. In this project, we will consider various norm estimates on non-commutative L_p-spaces for operators coming from harmonic analysis. We will mainly focus on establishing non-commutative analogue of the Calderon-Zygmund operators and other kinds of averaging operators on exponential growth measure spaces. One of these kinds of results will be a non-commutative analogue of Calderon-Zygmund theory developed by Hebisch-Steger. One can also consider recent Cotlar type inequalities proved by Srivasatava and Guo et al. We will also like to consider the operator-valued analogue of other kinds of operators coming from harmonic analysis, for example Hilbert transforms defined on curves, maximal operators coming from ball and spherical averages etc. Another part of the project will be focused on isometries on noncommutative L_p-spaces. Isometries on non-commutative L_p-spaces have been studied a lot. However, there are still some important problems which have been left open. For example no one knows if there exists a complete isometry from nc L_p into L_q. One more important problem in this direction is to study two dimensional subspaces of non-commutative L_p-spaces. There has been a recent breakthrough by Heinävaara where he investigated it for Schatten classes. Any advancement on these problems will have a lot of potential applications in structure theory of von Neumann algebras, geometry of non-euclidean spaces and geometry of non-commutative Banach spaces. All the problems which we will consider are fundamental in nature and many times very easy to state. On the other hand, the topic related to which the project is proposed has a lot of applications in mathematical physics. So a better understanding of these problems can possibly influence some development in mathematical physics as well.