Approximation properties of groups and operator algebras

Vanderbilt University, Nashville TN

Investigators

Abstract

This project addresses several open problems related to approximations of groups, graphs, and operator algebras by finite structures. The project aims to advance the technique that is used to prove amenability of the full topological group of a Cantor minimal system to other classes. The Connes embedding problem (CEP) concerns a fundamental approximation property of tracial states on von Neumann algebras. The principal investigator plans to develop an algebraic approach to CEP that she started in several recent papers. This approach is based on certain algebraic properties of noncommutative polynomials over unitary variables. One of the rapidly developing areas in group theory is a study of approximations of discrete groups by finite groups, in particular, so-called sofic approximations. Several long-standing conjectures are known to be true for sofic groups. Among them are CEP for group von Neumann algebras, Gottschalk's conjecture, the determinant conjecture, and Kaplansky's direct finiteness conjecture. Moreover, there are currently no known examples of nonsofic groups. One of the aims of the project is to study properties of sofic approximations for several classes of groups in the hope of shedding greater light on this state of affairs. Groups and operator algebras arise naturally in all areas of mathematics and also in certain parts of physics and chemistry. In particular, the methods of group theory lie at the roots of many parts of algebra. The notion of an "amenable group" was introduced by John von Neumann in 1929, and in recent years many famous conjectures have been shown to be true for the class of amenable groups. The core part of the proposed project is to develop recent results of the principal investigator in order to establish that certain classes of groups are amenable or to prove that they exhibit certain weaker forms of this property. In the process, the project is expected to strengthen the connections between several areas of mathematics. Finally, the principal investigator will seek to attract young researchers to the field and organize conferences and seminars on the subject.