[@ChenXuYin Section 7].

In [@NeuRic], Neuwirth and Ricard studied Question \[Que-One\] for discrete groups. They noted that the inequality $\geq$ always holds, and they proved the other inequality in the case $G$ is amenable. This is an $L^p$-version of the Bo[ż]{}ejko-Fendler result. Their proof relies on the fact that the amenability of $G$ allows them to construct a completely isometric embedding of $L^p({\mathcal{L}}G)$ into an ultrapower of ${\mathcal{S}}_p(\ell^2 G)$ that intertwines Fourier and Schur multipliers.

Marius Junge (personal communication) pointed out to the second-named author that a positive answer to Question \[Que-One\] would also follow from the existence of a completely isometric embedding of $L^p({\mathcal{L}}G)$ into an ultraproduct of $L^p({\mathcal{M}}_n) \otimes {\mathcal{S}}_p(L^2 G)$ for some net of von Neumann algebras ${\mathcal{M}}_n$, that intertwines Fourier and Schur multipliers. He suggested that some weaker approximation property (as exactness) might provide such an embedding. Our first result (Theorem \[Thm-CounterExample\]) partially answers his suggestion negatively: if $p\neq 2$ is an even integer, $G$ is a discrete group, and such an embedding exists, then $G$ is amenable. This bad news was the motivation to study Question \[Que-One\] for general locally compact groups.

The main part of the present paper is to extend the Neuwirth-Ricard result 