Differential operators on von Neumann algebras

Washington University, Saint Louis MO

Investigators

Abstract

Differential operators on von Neumann algebras Nikolai Weaver Project Summary Differential operators on operator algebras have been studied extensively as a model for the infinitesimal time development of a quantum-mechanical system, either isolated or interacting with a macroscopic system. The proposed research involves a study of differential operators on von Neumann algebras building on the investigator's prior work on the special case of first-order differential operators. Operators on a Hilbert space are infinite analogs of finite matrices. They have applications throughout mathematics, but the proposed research relates specifically to mathematical physics. Von Neumann algebras of operators have been used to model certain quantum mechanical systems, particularly those arising in quantum statistical mechanics. The proposal involves using ideas of Fields medalist Alain Connes to study the time-evolution of these algebras, with the double aim of clarifying the general theory and giving insight into specific examples.