Mikael de la Salle
date: |
    . [*MSC2010*]{}: 43A15, 46B08, 46B28, 46B70. [*Keywords*]{}: non-commutative $L^p$-spaces, Schur multiplier, Fourier multipliers, amenability.\
    \
    The first author was supported by the ANR project: ANR-2011-BS01-008-01.\
    The scond author was partially supported by the ANR projects NEUMANN and OSQPI. 
title: 'Schur and Fourier multipliers of an amenable group acting on non-commutative $L^p$-spaces'
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Introduction
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This paper studies the close connection between Schur and Fourier multipliers of locally compact groups $G$. In particular, we are interested in the relation between the completely bounded norms of such multipliers.

Given a bounded function $\phi:G \to \C$, the associated Fourier multiplier $T_\phi$, when it exists, is the unique weak-\* continuous map on the von Neumann algebra ${\mathcal{L}}G$ of $G$ extending $\lambda_g \mapsto \phi(g)\lambda_g$. The associated Schur multiplier $M_\phi$, when it exists, is the unique weak-\* continuous map on $B(L^2G)$ extending $(a_{s,t})_{s,t \in G} \in {\mathcal{S}}_2(L^2 G) \mapsto (\phi(s t^{-1}) a_{s,t})_{s,t \in G}$. Bo[ż]{}ejko and Fendler proved [@BozFen84] that $M_\phi$ indeed determines a bounded map if and only if $T_\phi$ defines a completely bounded map, and in this case the completely bounded norms coincide. The function $\phi$ is then called a completely bounded Fourier multiplier and the space of 