Non-commutative Analysis and Symmetry in Operator Algebra

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Abstract Effros/Popa/Takesaki The three PI's intend to continue their investigations on a broad range of problems in operator algebra theory. Effros and Ruan are continuing their collaboration on operator spaces. Effros and Ruan are particularly interested in studying the local theory of von Neumann algebraic preduals. Building on their earlier result with Junge that that all von Neumann preduals are locally reflexive, and that the injective preduals have a simple characterization, Effros and Ruan hope to prove that the general architecture of a von Neumann algebra can be described in terms of the local structure of the preduals. Effros also intends to look at the quantized analogues of rotundity. A number of Popa's research projects will be concerned with his axiomatization of the standard invariant for subfactors. In particular he plans to work on the most challenging problem in subfactor theory: finding techniques that would be applicable to the theory of subfactors of hyperfinite factors. He intends to continue his studies with Bisch of property T in the context of subfactors. In a very different direction, he will investigate the structure theory of C*-algebras based in part on his earlier work on the theory of local approximation by finite dimensional algebras, and of the relative Dixmier property for C*-algebras. Takesaki plans to continue his development of a canonical approach to the theory of type III factors. In the next stage of his research program, Takesaki and his collaborators intend to complete their studies of outer automorphism actions of a discrete amenable group on an approximately finite dimensional (or hyperfinite) factor of type III-lambda (lambda larger than zero) and he expects that the type III-zero case will yield to this analysis. He will continue his investigation into the most difficult problem in the area: the classfication of one-parameter groups of automorphisms. Takesaki expects to apply classfication principles learned from factor theory to the classification of certain classes of C*-algebras. Operator algebraists study the mathematics of quantum physics. In 1926 Heisenberg discovered that the paradoxes of atomic particles could be resolved with a modified version of Newtonian physics. He showed that the equations of the classical theory were still valid, provided one reinterpreted their symbols. In the classical theory these variables stand for functions. Heisenberg showed one can predict the behaviour of atomic particles if one instead regarded the variables as representing possibly infinite arrays or ``matrices'' of numbers. A few years later, von Neumann gave a mathematically precise formulation of these quantum variable in terms of Hilbert space operators. He went on to suggest that since the classical notions of measurement and geometry that underlie so much of mathematics no longer correspond to our understanding of the real world, it was necessary to seek quantized versions of mathematics. As in physics, one must begin by replacing functions by operators. In the last fifty years, operator algebraists have succeeded in quantizing a remarkable number of areas of mathematics, including analysis, topology, differential and Riemanian geometry, probability theory, and the theory of symmetry. As in quantum physics, the quantum world of mathematics is remarkable in the completely new phenomena that occur. The theory has had profound applications to various areas, including knot theory and low-dimensional topology, index theory on foliated manifolds, the classification of dynamical systems, and most recently, mathematical frameworks for both the standard model of quantum field theory (Connes) and renormalization theory (Connes and Kreimer). In this broad framework, Effros is one of the founders of quantized functional analysis (operator space theory), Popa is a leading figure in the theory of quantum symmetries (subfactor theory), and Takesaki is internationally recognized for his work on the modern theory of non-commutative integration and its use in studying the structure of von Neumann algebras and their automorphism groups