Noncommutative Uncertainty Principles

TL;DR

This paper extends classical uncertainty principles to the setting of finite index subfactors, establishing key inequalities and characterizing minimizers using analytic and categorical methods.

Contribution

It introduces uncertainty principles for subfactors, proves fundamental inequalities, and characterizes minimizers, broadening the scope beyond abelian groups.

Findings

Proved Hausdorff-Young, Young's, Hirschman-Beckner, and Donoho-Stark inequalities for subfactors.

Characterized the minimizers of these uncertainty principles.

Showed the minimizer is uniquely determined by support conditions.

Abstract

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's $\lambda$-lattices, modular tensor categories etc.