Theorem \[thm=contraction\_well\_defined\] that $P_F x P_F \in {\mathcal{S}}_p(L^2F)$ with norm at most $|F|^{1/p}\|x\|_{L^p({\mathcal{L}}G)}$. When $G$ is amenable, one then chooses a Følner net $F_n \subset G$ and consider the maps $x \mapsto |F_n|^{-1/p} P_{F_n} x P_{F_n}$.

In all the definitions and statements we give, $L^p({\mathcal{L}}G)$ will be the Connes-Hilsum space, but the theory of interpolation [@TerpII], [@Izumi] of non-commutative $L^p$-spaces associated to a weight plays an important role in our proofs. This is a bit technical. Therefore, to make the paper readable we collect (and prove when necessary) all the results we need from interpolation in Theorem \[thm=interpolation\_BB\], and in the proofs we use this theorem as an abstract black box.

The structure of the paper is as follows. Section \[Sect-Impossible\], which is independent from the rest of the paper, gives a sufficient (and necessary) condition for a discrete group to be amenable in terms of intertwiners of Schur and Fourier multipliers. Section \[Sect-Preliminaries\] recalls the necessary results and fixes the notation. In Section \[Sect=transference\], we prove that the inequality $\geq$ in Question \[Que-One\] holds for any locally compact group. Section \[Sect=amenable\] gives the affirmative answer to Question \[Que-One\] for amenable groups. The last Section \[Sect=interpolation\] on interpolation states and 