Project in Operator Algebra

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AbstractRadulescuThe aim of this project is to investigate a range of problems in the structure theory of von Neumann algebras, concerning the harmonic analysis of such algebras arising in connection with discrete groups and in deformation quantization theory and on the convex analysis of these von Neumann algebras. The goal is to determine the structure of the von Neumann algebras that reflect the properties of (quantum) non commutative probability, and to relate this structure to the representation theory of Lie groups and their discrete subgroups. An invariant for such von Neumann algebras, which will be investigated in this project, comes from the analysis of the structure of convex sets, consisting of noncommutative moments associated with these von Neumann algebras. This approach is related to Voiculescu's free entropy, where the (renormalized) asymptotic measure of such sets plays an essential role. The operator algebras studied in this project are related to the atom model introduced by Wigner. His approach, while philosophically relying on Heisenberg's Uncertaneity Principle, was based on the theory of random matrices. The random matrices have also numerous other applications in prediction theory or atmospheric science. A surprising, recent, discovery, by Voiculescu, shows that the random matrices are, the asymptotic generators for the von Neumann algebras associated to quantum probability. The aim of this project is to develop a conceptual relation between the random matrices in von Neumann algebras and the quantization theory. Such a development would probably explain the complementarity between the two theories (random matrices versus infinite matrices) and hence would hopefully give new tools to solve a number of long-standing problems in these areas.