Schur and Fourier multipliers of an amenable group acting on non-commutative $L^p$-spaces

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.