Xp inequalities in group von Neumann algebras

Naor and Schechtman recently introduced the so-called metric Xp  inequalities, an obstruction for embeddings of $Lq$ into $Lp$ whenever $2 < q < p < \infty$. This invariant was refined by Naor via a fundamental inequality in the Hamming cube which strongly relies on Fourier analysis. In this talk, we will show that this latter result can be understood within the frame of noncommutative harmonic analysis, providing a general realization in the context of von Neumann algebras associated to discrete groups. Joint work with Jose Conde-Alonso and Javier Parcet.