Noncommutative Uncertainty Principles

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.Comment: 41 pages, 71 figure

Equality cases for the uncertainty principle in finite Abelian groups

We consider the families of finite Abelian groups \ZZ/p\ZZ\times \ZZ/p\ZZ, \ZZ/p^2\ZZ and \ZZ/p\ZZ\times \ZZ/q\ZZ for $p,q$ two distinct prime numbers. For the two first families we give a simple characterization of all functions whose support has cardinality $k$ while the size of the spectrum satisfies a minimality condition. We do it for a large number of values of $k$ in the third case. Such equality cases were previously known when $k$ divides the cardinality of the group, or for groups \ZZ/p\ZZ.Comment: Mistakes have been corrected. This paper has been accepted for publication in Acta Sci. Math. (Szeged