then proves all the results needed in the first sections on interpolation of non-commutative $L^p$-spaces.

Characterizing amenability by intertwining Schur and Fourier multipliers {#Sect-Impossible}
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In this section we prove the following characterization of the amenability of a discrete group. It takes away our hope to give a direct positive answer to Question \[Que-One\] for non-amenable discrete groups.

Recall that the Schur multiplier $M_\phi^p$ and the Fourier multiplier $T_\phi^p$ were defined in the introduction for discrete groups. We define them more generally in Section \[Sect-Preliminaries\].

\[Thm-CounterExample\] Let $\Gamma$ be a discrete group, let ${\mathcal{H}}$ be a Hilbert space and let $p\geq 4$ be an even integer. Let ${\mathcal{U}}$ be a non-trivial ultrafilter on some set. Assume that there exists an isometric embedding $$\label{Eq-IntertwinerCounter}
j_p: L^p({\mathcal{L}}\Gamma) \to \prod_{\mathcal U} {\mathcal{S}}_p(\ell^2 \Gamma \otimes
  {\mathcal{H}})$$ such that for every $\phi :\Gamma \to \C$ of finite support, $j_p \circ T_\phi^p = (\prod_{{\mathcal{U}}} M_\phi^{p} \otimes id) \circ j_p$. Then, $\Gamma$ is amenable.

Conversely, if $\Gamma$ is amenable such an embedding exists for all $1 \leq p \leq \infty$ and we can take ${\mathcal{H}}=\C$.

In fact the proof below shows more generally that the same conclusion holds if there is such an embedding of $L^p({\mathcal{L}}(\Gamma))$ into 