The 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 in [Noncommutative uncertainty principle, J. Funct. Anal. 270 (2016), 264-311.]

We sincerely thank Zhengwei Liu for useful exchange and suggestions that eventually lead to a substantial improvement of the paper. K.C.B acknowledges the support of INSPIRE Faculty grant DST/INSPIRE/04/2019/002754. S.G acknowledges the support of SERB grant MTR/2021/000818. SM was affiliated to CMI when the majority of the work was carried out. She would like to thank CMI for the institute postdoctoral fellowship.