The PI will investigate two fundamental problems in the theory of operators that are contained in II_1 factors. The first is Connes' embedding problem. Recent work of Collins and Dykema has shown that this problem is equivalent to a question about sums of operators in finite von Neumann algebras, and other recent work of Bercovici, Collins, Dykema, Li and Timotin has positively answered the first part of this question, showing that all Horn inequalities hold in all finite von Neumann algebras. The second problem is the hyperinvariant subspace problem. In particular, the PI will focus on the remaining open part of this problem for elements of II_1-factors, namely, the case of quasi-nilpotent operators in II_1-factors.
Operators on infinite dimensional Hilbert space are used in mathematical models of quantum mechanics, and they are of significance in diverse areas of mathematics. We will work on two fundamental problems in operator theory: Connes' embedding problem and the hyperinvariant subspace problem. These concern different aspects of the structure of operators on infinite dimensional Hilbert spaces. We will focus on operators whose algebras possess traces. The first problem is about how well such operators can be approximated (in their mixed moments with respect to the trace) by operators on finite dimensional spaces. We will attack this problem by examining eigenvalues of sums of operators. The second problem is about the possibility of decomposing operators on infinite dimensional space by restricting them to invariant subspaces. In some recent progress, Haagerup and Schultz have proved the existence of such subspaces for a large class of operators, and we will focus on some specific operators for which this question is unresolved.
Operators on infinite dimensional spaces are ubiquitous in mathematics and areessential to our understanding of physical phenomena. This understanding underlies our technological civilization. The nicest infinite dimensional spaces (called Hilbert spaces) are like usual Euclidean space in that they have a notion of angle. Operators are maps from aspace to itself, and two operators A and B can fail to commute (meaning A followed B is not equal to B followed by A). But a modicum of commutativity can be present if there is a trace, namely, a linear map tau that satisfies tau(AB)=tau(BA). One of the fundamental questions about such operators is: if there is a trace, can operators on Hilbert space always be approximated by operators on finite dimensional space? A large part of this project was to work on this fundamental problem, and on related questions. Progress was made, by reformulating this problem in terms of finite matrices of moments of unitary operators. A related and also fundamental problem in group theory was considered (concerned with sofic groups) and several results were proved. New computational methods were developed that may lead to discovery of an example of a non-sofic group. In addition, several other topics about operators were investigated, and progress that has been published in several papers.
