---
abstract: |
    Consider a completely bounded Fourier multiplier $\phi$ of a locally compact group $G$, and take $1 \leq p \leq \infty$. One can associate to $\phi$ a Schur multiplier on the Schatten classes ${\mathcal{S}}_p(L^2 G)$, as well as a Fourier multiplier on $L^p({\mathcal{L}}G)$, the non-commutative $L^p$-space of the group von Neumann algebra of $G$. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the $L^p$-Fourier multiplier. When $G$ is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups.

    For a discrete group $G$ and in the special case when $p\neq 2$ is an even integer, we show the following. If there exists a map between $L^p({\mathcal{L}}G)$ and an ultraproduct of $L^p({\mathcal{M}}) \otimes {\mathcal{S}}_p(L^2G)$ that intertwines the Fourier multiplier with the Schur multiplier, then $G$ must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups.
address:
- 'M. Caspers, Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France'
- 'M. de la Salle, CNRS, Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France'
author:
- Martijn Caspers
- 