Operator algebras between theory and application

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Order matters. The order in which certain operations are performed can dramatically change the outcome in real life and in the sciences. However, in the usual multiplication of numbers the order is irrelevant. Since the product AB is the same as BA one says that the factors A and B commute. Inspired by the fundamentals of quantum mechanics, mathematicians have investigated a new type of multiplication that respects the order of operations. During the last century this has led to spectacular new discoveries in mathematical theories embracing noncommutativity (i.e., allowing AB to be different from BA) such as noncommutative geometry or quantum (i.e., noncommutative) probability. In this line of research a similar program is applied to fundamental concepts in classical harmonic analysis such as Fourier series and estimates for solving differential equations in noncommutative spaces. Quite surprisingly, the abstract tools developed in this investigation are also useful in other disciplines. The research of the principal investigator will include a thorough analysis of quantum channels in quantum information theory. Research in quantum information theory usually takes place in computer science and physic departments. However, as long as quantum computers are not available in large numbers, the limitations and advantages of quantum computers can be understood only using theoretical, mathematical tools. The same applies for the capacities of devices transmitting information through the use of quantum mechanics. Interdisciplinary research in this work will also include mathematical aspects of big data and compressed sensing. All aspects of this research will also serve to enhance the teaching mission of the university, and in particular the formation of students who are familiar with pure mathematics and certain applications alike. The theory of operator algebras provides many important tools that are essential in understanding noncommutative aspects of classical objects, such as Brownian motion, derivatives and derivations, tangent and cotangent spaces, Laplace-Beltrami operators, singular integral kernels, quantum channels, and capacity of quantum channels. The project will aim to connect theoretical aspects of the theory of completely positive maps with more applied aspects in quantum information theory and harmonic analysis, in particular those analytic properties of operators that have a geometric or metric flavor. The proposed work on the Grothendieck program for triple-tensor norms belongs to the core subject in operator space theory but is also motivated by quantum information and compressed sensing. Previous research of the principal investigator related to quantum information theory has already demonstrated the potential to connect to topics in computer science and physics. The proposed new research on private capacity of channels may even have an impact beyond science.