Operator Algebra Theory in Applications

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Order matters. In real life it is easy to observe that the order of certain operations greatly change the final outcome. As an example one may think of heating water and adding oil. After almost a century mathematicians have finally embraced the challenge from quantum mechanics and developed a theory which allows to study noncommuting operations. The work funded in this project will take this challenge literally and study classical operations from calculus and introduce an order dependence for space, time and all the operations performed in space and time. A particular range of applications of the theoretical work will gain new insights in quantum information theory. Quantum information theory modifies Shannon's theory of sending reliable information through noisy channels to quantum devices in the hope of taking advantage of "teleportation" or other non-traditional "spooky behaviour."More concretely, the proposed work aims to study the connection between noncommutative geometry and noncommutative harmonic analysis. The goal is to adapt some terminology from noncommutative geometry, for example curvature, and show that it an be used for estimates involving Laplace and Dirac operators on noncommutative spaces. The work in quantum information theory will develop inequalities for entropy related expressions and apply them to notions of capacity, asymmetry and time to equilibrium.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.