in $M_n \otimes B({\mathcal{H}})$ is $1 \otimes {\mathcal{M}}'$, and is therefore equipped with the weight $1 \otimes \psi$. This naturally identifies ${\mathcal{S}}_p^n \otimes L^p({\mathcal{M}},\psi)$ with $L^p(M_n \otimes {\mathcal{M}}, 1 \otimes \psi)$. We use this identification to define a Banach space structure on ${\mathcal{S}}_p^n \otimes L^p({\mathcal{M}},\psi)$.

If ${\mathcal{M}}\subset B({\mathcal{H}}),{\mathcal{N}}\subset B({\mathcal{H}}')$ are von Neumann algebras and $\psi,\psi'$ are weights on ${\mathcal{M}}'$, ${\mathcal{N}}'$, a bounded map $u:L^p({\mathcal{M}},\psi) \to L^p({\mathcal{N}},\psi')$ between the non-commutative $L^p$-spaces (Connes-Hilsum construction) is called completely bounded if $\|u\|_{cb}:=\sup_n \|u_n\|<\infty$. Here $u_n = id \otimes u : {\mathcal{S}}_p^n \otimes L^p({\mathcal{M}},\psi)\to {\mathcal{S}}_p^n \otimes L^p({\mathcal{N}},\psi')$. When ${\mathcal{M}}$ and ${\mathcal{N}}$ are semifinte, one can check that this definition agrees with the natural operator space structure on non-commutative $L^p$-spaces given in [@MR1648908].

Definition of $L^p$ Fourier multipliers {#subsection=Fourier}
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Recall that the Fourier algebra $A(G) = \left\{ \varphi\colon s\mapsto \langle \lambda_s \xi,\eta\rangle, \xi,\eta \in L^2 G\right\}$ coincides with the predual of ${\mathcal{L}}G$ through the pairing $\langle \varphi,\lambda(f) \rangle = \int \varphi(s) f(s) ds$ for every $f \in L^1(G)$. We record here the explicit isomorphism between the Fourier algebra $A(G)$ and $L^1({\mathcal{L}}G)$.

\[lemma=isom\_AG\_L1\] An element $\varphi \in A(G)$ corresponds (isometrically) to $x_\varphi \in L^1({\mathcal{L}}G)$ that satisfies $P_F x_\varphi P_F = (\varphi(ts^{-1}))_{s,t \in F}$.

Start by considering 