to arbitrary locally compact groups: we first prove in Theorem \[Thm-TransferenceEstimate\] by a transference technique that the inequality $\geq$ always holds. We then prove the inequality $\leq$ for $G$ amenable in Theorem \[thm=embedding\_in\_ultraproduct\] and its Corollary \[Cor-MainEquality\]. Our proof is close to the proof given in [@NeuRic Section 3], but we encounter several technical issues that we have to overcome, mainly in the case of non-unimodular groups.

Let us explain a bit more in details the technical complications that we meet. Recall [@PisXu] that to a von Neumann algebra ${\mathcal{M}}$ one can associate non-commutative $L^p$ spaces $L^p({\mathcal{M}})$ for $1 \leq p \leq \infty$. There are several different ways to define $L^p({\mathcal{M}})$, but they all yield (completely) isometric spaces. When ${\mathcal{M}}= B({\mathcal{H}})$, $L^p({\mathcal{M}})$ is most naturally realized as the Schatten class ${\mathcal{S}}_p({\mathcal{H}})$ (which is contained in $B({\mathcal{H}})$). When ${\mathcal{M}}$ carries a normal faithful finite trace $\tau$ (for example ${\mathcal{M}}= {\mathcal{L}}G$ for a discrete group $G$), the natural choice is to see $L^p({\mathcal{M}})$ as the completion of ${\mathcal{M}}$ for the norm $\|x\|_p = \tau(|x|^p)^{1/p}$ ($L^p({\mathcal{M}})$ contains ${\mathcal{M}}$). When $G$ is a locally compact group, there is a natural weight on ${\mathcal{L}}G$, the Plancherel weight, which is a trace if and only if 