Multilinear Fourier multipliers of a locally compact group

Abstract

In his paper "Group C*-algebras without the completely bounded approximation property", Haagerup proves several important results about the weak amenability of locally compact groups. Among these, is the result that a lattice in a second-countable, unimodular, locally compact group is weakly amenable if and only if the surrounding group itself is weakly amenable. A key ingredient in his proof is a method of using (linear) completely bounded Fourier multipliers on the lattice to construct (linear) completely bounded Fourier multipliers on the surrounding group. We use a similar approach to construct multilinear completely bounded Fourier multipliers on the group from multilinear completely bounded Fourier multipliers on the lattice. Our construction is both bounded in the norm of completely bounded Fourier multipliers and preserves uniform convergence on compact sets for bounded nets. We also prove an equivalent characterization of weak amenability where the Fourier algebra is replaced by the space of continuous and compactly supported $n$-linear Fourier multiplier symbols.