Entropy Theory Methods in von Neumann Algebras

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Quantum mechanics is an area of physics with numerous applications including biology and engineering. A basic consequence of the theory is that observables cannot commute with each other, and so the mathematical basis for quantum mechanics must be within a noncommutative framework. Based on this constraint, in the 1930's Murray and von Neumann provided a rigorous background for quantum mechanics using algebras of operators on a Hilbert space now called von Neumann algebras. Recently, these objects have provided a precise framework in which to study quantum computing, which has enormous potential application to breakthroughs in cryptography. Another area where von Neumann algebras have given tremendous insight is in the study of random matrices. Random matrices themselves are used in nuclear physics to model the nuclei of heavy atoms, as well as in theoretical neuroscience to describe the connections between neurons in the brain. This project will also contribute to US workforce development through diversity initiatives and training of graduate students. The goal of this project is to study group von Neumann algebras through two different approaches: Voiculescu's free probability theory, and Popa's deformation/rigidity theory. In Voiculescu's free probability theory, the PI plans to use the theory of random matrices to attack a well-known conjecture of Peterson and Thom on the structure of subalgebras of group von Neumann algebras of free groups. The random matrix approach uses so-called strong convergence, which demands that eigenvalue distributions converge weak*, and that the spectrum converges in the Hausdorff metric. Building on the 1-bounded entropy theory due to the PI and Jung, the PI establishes that if a natural strong convergence results holds for tensors of random unitaries, then one is able to deduce the Peterson-Thom conjecture as a corollary. Importing insight from sofic entropy theory into Popa's deformation/rigidity theory, the PI recently showed with de Santaigo-Hoff-Sinclair that given a (s-malleable) deformation of a tracial von Neumann algebra, and a diffuse, rigid subalgebra, one can necessarily extend it to a maximal rigid subalgebra. A potential avenue for exploration is in generalizing these results to approximately rigid subalgebra. Such a generalization gives a second approach to the Peterson-Thom conjecture. It would also provide a way to show that nonamenable groups with nontrivial cohomology in the left regular representation do not have Cartan subalgebras, a problem in the subject that has been open for the last decade. The PI also plans to study the implications of this theory to the case of von Neumann algebras of equivalence relations.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.