satisfies $P_F T_\phi^p(x) P_F = M_\phi (P_F x P_F)$ by and again.

The fact that $T_\phi^p$ is completely bounded follows from the argument in [@CanHaa Theorem 1.6] (see the proof of Theorem \[thm=embedding\_in\_ultraproduct\] for details): apply the preceding to the function $\phi(s,k) = \phi(s)$ on $G \times SU(2)$ and use that ${\mathcal{L}}SU(2) \simeq \oplus_{n \geq 1} M_n$.

Transference {#Sect=transference}
============

This section is devoted to the equality $\geq$ of Question \[Que-One\]. The proof relies on transference techniques [@NeuRic Lemma 2.4], which we adapt to non-discrete groups.

\[lem=recover\_phi\_from\_Iphi\] Let $G$ be a locally compact group and let $1 < p < \infty$ be such that the conjugate exponent $q$ satisfies $p/q \in \Q$. Then there is a net $x_\alpha \in L^p({\mathcal{L}}G)$ and $y_\alpha \in L^q({\mathcal{L}}G)$ such that $\|x_\alpha\|_p = \|y_\alpha\|_{q} = 1$ and for all $\phi \in {M_{\mathcal{CB}}A}(G)$, $$\lim_\alpha \langle T_\phi^p x_\alpha,y_\alpha^*\rangle = \phi(e).$$

The proof relies on Theorem \[thm=interpolation\_BB\]. Since $p/q \in \Q$ there exists $r<\infty$ and $k,k' \in \N$ such that $\frac 1 p = \frac k r$ and $\frac 1{q} = \frac{k'}{r}$.

Take a net of $\R^+$-valued fuctions $h_\alpha\in A_c(G)$ such that the support $S_\alpha$ of $h_\alpha$ converge to $\{e\}$. This means that every neighbourhood of $e$ in 