Lp-Approximation Properties, Multipliers, and Quantized Calculus

Approximation is a fundamental concept in mathematics, and a powerful tool in science and engineering, in which a core notion is that one may study an object of interest by modeling it with simpler objects and using a limiting process. The various approximation properties in mathematical analysis refer to the capacity of an object, such as an infinite-dimensional vector space, to be approximated in useful ways, for instance, by finite-dimensional subspaces. Such properties are central to a number of important fields, such as quantum probability theory, noncommutative geometry, and quantized calculus. This project will focus on Lp-approximation properties of von Neumann algebras and their applications in operator algebras, noncommutative analysis, and beyond. The project incorporates research opportunities for undergraduate and graduate students, as well as training for postdoctoral researchers. In this study, the principal investigator will explore Lp-approximation properties of group von Neumann algebras and their connections to the von Neumann rigidity property and Connes's quantized calculus. The major challenges of the proposed research include the absence of geometric/metric structure and a commutative product in the abstract setting. The proposed research necessitates a reinvention of classical theory on Fourier multipliers in a context with much less structure. Success in this proposed project will strengthen the existing connection between functional analysis and harmonic analysis and will enhance our understanding of the rigidity of discrete groups, the theory of noncommutative geometry, and its applications to quantum information theory. 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.