  Let H be a subgroup of some locally compact group G. Assume H is approximable
by discrete subgroups and G admits neighborhood bases which are
"almost-invariant" under conjugation by finite subsets of H. Let $m: G \to
\mathbb{C}$ be a bounded continuous symbol giving rise to an Lp-bounded Fourier
multiplier (not necessarily cb-bounded) on the group von Neumann algebra of G
for some $1 \le p \le \infty$. Then, $m_{\mid_H}$ yields an Lp-bounded Fourier
multiplier on the group von Neumann algebra of H provided the modular function
$\Delta_H$ coincides with $\Delta_G$ over H. This is a noncommutative form of
de Leeuw's restriction theorem for a large class of pairs (G,H), our
assumptions on H are quite natural and recover the classical result. The main
difference with de Leeuw's original proof is that we replace dilations of
gaussians by other approximations of the identity for which certain new
estimates on almost multiplicative maps are crucial. Compactification via
lattice approximation and periodization theorems are also investigated.
