Fourier theoretic inequalities for inclusion of simple C*-algebras

TL;DR

This paper develops non-commutative Fourier inequalities for simple C*-algebra inclusions, extending classical inequalities and uncertainty principles to a broader algebraic setting.

Contribution

It introduces Fourier theoretic inequalities for simple C*-algebras with conditional expectations, generalizing prior non-commutative uncertainty principles.

Findings

Proves Hausdorff-Young inequality in this setting

Establishes Young's inequality for simple C*-algebras

Derives non-commutative uncertainty principles

Abstract

This paper originates from a naive attempt to establish various non-commutative Fourier theoretic inequalities for an inclusion of simple C*-algebras equipped with a conditional expectation of index-finite type. In this setting, we discuss the Hausdorff-Young inequality and Young's inequality. As a consequence, we prove the Hirschman-Beckner uncertainty principle and Donoho-Stark uncertainty principle. Our results generalize some of the results of Jiang, Liu and Wu [Noncommutative uncertainty principle, J. Funct. Anal., 270(1): 264--311, 2016].