In this paper we establish the Lp–Lq boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices 1<p≤2≤q<∞. 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 Hörmander's Fourier multiplier theorem on Rn and the corresponding known results for Fourier multipliers on compact Lie groups.