(\[asser=norm\_parties\_finies\]). The implication (\[asser=norm\_locCompact\]) implies (\[asser=norm\_parties\_semifinies\]) is trivial. Assume (\[asser=norm\_parties\_semifinies\]). Let $x \in {\mathcal{S}}_p(L^2(X,\mu))$. The supports of $x$ and $x^\ast$ are $\sigma$-finite projections in $B(L^2X)$. So, let $X_0 \subset X$ be $\sigma$-finite such that the support projections of $x$ and $x^\ast$ project onto spaces contained in $L^2(X_0, \mu)$. Then, the multiplier $\psi$ applied to $x$ is equal to the restriction of $\psi$ to $X_0 \times X_0$ applied to $x$. Hence, (\[asser=norm\_locCompact\]) follows.

Let $G$ be a locally compact group. Let $\phi \in {M_{\mathcal{CB}}A}(G)$ and set $\check{\phi} \in L^\infty(G \times G)$ by $\check{\phi}(s,t) = \phi(s t^{-1})$. Then, $\check{\phi}$ is a Schur multiplier acting on ${\mathcal{S}}_p(L^2G)$, see [@LafSal] for details. We will denote this map by $M_\phi^p$.

The von Neumann algebra of $G$. {#subsect=groupvna}
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Let $G$ be a locally compact group equipped with a left Haar measure, and denote by $\Delta:G\to (0,\infty)$ the modular function. Recall (see [@TakII p. 65]) that $\Delta$ is the group morphism satisfying the following equations for all compactly supported continuous function $f:G \to \C$ : $$\label{eq=modular1} \int f(ts) dt = \Delta(s)^{-1} \int f(t) dt,$$ $$\label{eq=modular2} \int f(t^{-1}) dt = \int f(t) \Delta(t)^{-1} dt.$$

Denote by $\lambda$ and $\rho$ the left and right regular representations of $G$ 