Non-Tracial Derivations and Distributions

Michigan State University, East Lansing MI

Investigators

Abstract

Von Neumann algebras are mathematical objects that offer a rigorous framework for the study of quantum physics, and can be thought of as infinite-dimensional generalizations of matrix algebras. The theory was initiated by Francis J. Murray and John von Neumann in the 1930s, and since then researchers have discovered a vast number of applications to mathematics as well as biology, physics, and engineering. In particular, one of the modern fields that studies von Neumann algebras is free probability, wherein one views the theory of von Neumann algebras as a generalization of probability theory. This perspective yields connections to random matrix theory, allows one to model the statistical behavior of large-scale data sets, and notably has been used to understand the fundamental limits of wireless communications. Furthermore, connections with the mathematical disciplines of complex analysis, dynamical systems, group theory, and harmonic analysis arise from the study of derivations on von Neumann algebras, which are analogues of the derivative in Leibniz and Newton's theory of calculus. The so-called "non-tracial" von Neumann algebras admit additional connections to physics: they have an inherit dynamical system that satisfies the Kubo-Martin-Schwinger condition from quantum statistical mechanics; and they arise naturally in conformal field theory. This project is part of the principal investigator's research efforts to better understand non-tracial von Neumann algebras through the use of free probability and an analysis of derivations. The goal of this project is to study the structure of non-tracial (type III) von Neumann algebras through three approaches, which are interconnected via free probabilistic distributions and derivations. The first approach concerns derivations on a von Neumann algebra, which lie within the scope of two major methodologies: Voiculescu's free probability theory, and Popa's deformation/rigidity theory. The former through free Fisher information and non-microstates free entropy, and the latter through deformations arising from the completely positive semigroups associated to closable derivations. The principal investigator seeks to show that derivations on non-tracial von Neumann algebras give rise to completely positive semigroups, and then use free probability and deformation/rigidity to extract structural information in the spirit of Peterson as well as Dabrowski and Ioana. The second approach concerns non-microstates regularity conditions on finite generating sets of von Neumann algebras equipped with non-tracial states, which can be regarded as non-commutative probability distributions and therefore analyzed with free probability. The principal investigator has shown that when these generators have finite free Fisher information and are well-behaved under the modular automorphism group associated to the state, then the von Neumann algebra they generate is a full factor. In recent work with Charlesworth on free Stein information, the principal investigator has found a weaker regularity condition than having finite free Fisher information that is more easily adapted to the non-tracial case. He will explore the structural consequences of this regularity condition for non-tracial von Neumann algebras, and attempt to adapt other existing regularity conditions to the non-tracial case. The third approach concerns random matrix models for non-tracial von Neumann algebras. In the tracial case, understanding the connection between the Gaussian Unitary Ensemble (GUE) and the free group factors has benefited random matrix theory by providing a clear picture of the large N limit, and has benefited free probability theory by providing finite dimensional approximations and informing the definition of microstates free entropy. The principal investigator has found a non-tracial version of the Dyson-Schwinger equation, which is a non-commutative PDE that (in the tracial case) makes explicit the connection between the GUE and the free group factors. A non-tracial Dyson-Schwinger equation gives rise to a free Araki-Woods factor, which were defined by Shlyakhtenko as the non-tracial analogue of the free group factors. The principal investigator aims to use non-tracial Dyson-Schwinger equations to develop random matrix models for free Araki-Woods factors, and more generally to develop microstates free entropy for non-tracial von Neumann algebras. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.