Sums of Hermitian Operators and Connections to Connes' Embedding Problem; Hyperinvariant Subspaces

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AbstractDykemaThe 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.