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Trigonometric chaos and $\mathrm{X}_p$ inequalities II -- $\mathrm{X}_p$ inequalities in group von Neumann algebras

In the line of previous work by Naor, we establish new forms of metric $\mathrm{X}_p$ inequalities in group algebras under very general assumptions. Our results' applicability goes beyond the previously known setting in two directions. In first place, we find continuous forms of the $\mathrm{X}_p$ inequality in the $n$-dimensional torus. Second, we consider transferred forms of the sharp scalar valued metric $\mathrm{X}_p$ inequality in the von Neumann algebra $\mathcal{L}(\mathrm{G})$ of a discrete group $\mathrm{G}$. As a byproduct of our results, some metric consequences and their relation with bi-Lipschitz nonembeddability of Banach spaces are explored in the context of noncommutative $L_p$ spaces.