  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].
