Fourier multipliers and group von Neumann algebras

In this paper we establish the L-p-L-q boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices 1 < p <= 2 <= q < infinity. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding Hormander's Fourier multiplier theorem on R-n and the corresponding known results for Fourier multipliers on compact Lie groups. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS.