Project in Operator Algebra

University Of Iowa, Iowa City IA

Investigators

Abstract

Abstract Radulescu Radulescu will investigate in this project a range of problems in the structure theory of von Neumann algebras. He will concentrate on the harmonic analysis of the von Neumann algebras arising in connection with discrete groups, free probability and in deformation quantization theory. The goal is to determine the structure of the von Neumann algebras that reflect the properties of non commutative probability, and to relate this structure to the representation theory of Lie groups and their discrete subgroups. Radulescu intends to investigate the problem of characterizing the factors of discrete groups by computing their invariants, using ideas of Murray and von Neumann, related to the fundamental groups of such an algebra, the set of all index values for subfactors or the structure of convex sets of non-commutative moments. He will investigate also Connes embedding conjecture via tensor products on operator algebras. The aim of this research is to discover non-commutative aspects of the real word hidden in "quantum" structures. Non-commutativity is a true aspect of the nature: for example two matrices do not necessary commute when multiplied. The famous Heisenberg's Uncertaneity principle reveals the same non-commutative aspect of nature. Von Neumann has developed a class of "continuous" matrices, that, while keeping some of the properties of usual matrices, are subtle enough to encode many of the aspects of quantum mechanics. Recent developments by Jones have proven that aspects of such von Neumann algebras are intimately related to aspects of topology, knot theory and ultimately biology. Likewise recent work by Voiculescu has proven that the there is a non-commutative probability theory encoded by the same type of objects. The aim of this research project is to contribute to the understanding and classification of some of the structural aspects of von Neumann algebras. _________________________________________ > --