  We investigate Fourier multipliers with smooth symbols defined over locally
compact Hausdorff groups. Our main results in this paper establish new
H\"ormander-Mikhlin criteria for spectral and non-spectral multipliers. The key
novelties which shape our approach are three. First, we control a broad class
of Fourier multipliers by certain maximal operators in noncommutative $L_p$
spaces. This general principle ---exploited in Euclidean harmonic analysis
during the last 40 years--- is of independent interest and might admit further
applications. Second, we replace the formerly used cocycle dimension by the
Sobolev dimension. This is based on a noncommutative form of the Sobolev
embedding theory for Markov semigroups initiated by Varopoulos, and yields more
flexibility to measure the smoothness of the symbol. Third, we introduce a dual
notion of polynomial growth to further exploit our maximal principle for
non-spectral Fourier multipliers. The combination of these ingredients yields
new $L_p$ estimates for smooth Fourier multipliers in group algebras.
