Smooth Fourier multipliers on group von Neumann algebras

We investigate Fourier multipliers on the compact dual of arbitrary discrete groups. Our main result is a Hörmander–Mihlin multiplier theorem for finite-dimensional cocycles with optimal smoothness condition. We also find Littlewood–Paley type inequalities in group von Neumann algebras, prove L_p estimates for noncommutative Riesz transforms and characterize L_∞ → BMO boundedness for radial Fourier multipliers. The key novelties of our approach are to exploit group cocycles and cross products in Fourier multiplier theory in conjunction with BMO spaces associated to semigroups of operators and a noncommutative generalization of Calderón–Zygmund theory.