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

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Consider a completely bounded Fourier multiplier ϕ \phi of a locally compact group G G , and take 1 ≤ p ≤ ∞ 1 \leq p \leq \infty . One can associate to ϕ \phi a Schur multiplier on the Schatten classes S p ( L 2 G ) \mathcal {S}_p(L^2 G) , as well as a Fourier multiplier on L p ( L G ) L^p(\mathcal {L} G) , the non-commutative L p L^p -space of the group von Neumann algebra of G G . We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the L p L^p -Fourier multiplier. When G G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups. For a discrete group G G and in the special case when p ≠ 2 p\neq 2 is an even integer, we show the following. If there exists a map between L p ( L G ) L^p(\mathcal {L} G) and an ultraproduct of L p ( M ) ⊗ S p ( L 2 G ) L^p(\mathcal {M}) \otimes \mathcal {S}_p(L^2G) that intertwines the Fourier multiplier with the Schur multiplier, then G G must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups.