Applications of operator algebra theory to certain problems in analysis

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Abstract Junge The aim of this proposal is to apply methods from the operator of algebras to problems in noncommutative analysis. This includes developing a theory of singular integrals and Fourier multipliers for noncommutative function spaces. The new insights gained from this work should also reflect back on the noncommutative spaces and their differential geometric properties. Another field of applications of tools from the theory of operator spaces lies in quantum information theory. Here violation of Bell inequalities and quantum error correction seems to relate well to the theory of cb-maps from operator space theory. In real life the order in which certain operations are executed can make a big difference. For example first boiling water and the adding oil is very different from first boiling oil and then adding water. The mathematical community has now fully accepted that one should allow the main object of interest to be performed in a certain order. Noncommutative analysis is about functions and their properties in the realm of non-commuting variables. The most important example here are matrix-valued functions. The theory of classical analysis, such as Fourier analysis or harmonic analysis has a lot to say about functions, and as this proposal intends to show, also about matrix-valued functions. Another natural application of this circle of ideas (the mathematical theory called functional analysis) lies in quantum information theory and the theoretical analysis of channels. Channels are the operation performed signals used to transport data in quantum information theory.