Fundamental Decomposition in Finite von Neumann Algebras

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The study of operators on Hilbert space became important with the advent of Quantum Mechanics, but in addition, understanding of these operators has proven to be vital to progress in many areas of mathematics. Historically, a method of studying and understanding such operators is to break them down into simpler components, based on spectral decomposition. This consists of describing parts of the operator that behave like multiplication by certain numbers, and to explain how these parts assemble into the whole. One major goal of this project is to advance such understanding of large classes of operators. Another major goal is to study families of operators that arise in various quantum mechanical models, in light of certain deep mathematical conjectures regarding finite dimensional approximations of infinite dimensional objects.More specifically, the principal investigator, together with collaborators, has made advances in recent years on spectral decomposition results for non-selfadjoint elements of finite von Neumann algebras. These are centered around upper triangular forms, analogous to the classical results of Issai Schur for matrices, and both utilize and extend results about hyperinvariant subspaces found recently by Haagerup and Schultz. Particular proposed projects include (a) studying norm convergence properties of bounded operators and (b) extending spectral distribution and upper-triangular form results to unbounded affiliated operators. In related directions, the principal investigator will work on the hyperinvariant subspace problem for elements of tracial von Neumann algebras and to investigate the Murray--von Neumann puzzle, which is akin to the Heisenberg relations. A second area of proposed research concerns the notion of bi-freeness and bi-free products. A third area of proposed research concerns quantum correlations. Recent results of the principal investigator showing non-closedness of the set of quantum correlations for five inputs and two outputs open the door to new understanding of these small cases, which the principal investigator proposes to pursue.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.