  We study analogues of classical Hilbert transforms as fourier multipliers on
free groups. We prove their complete boundedness on non commutative $L^p$
spaces associated with the free group von Neumann algebras for all
$1<p<\infty$. This implies that the decomposition of the free group $\F_\infty$
into reduced words starting with distinct free generators is completely
unconditional in $L^p$. We study the case of Voiculescu's amalgamated free
products of von Neumann algebras as well. As by-products, we obtain a positive
answer to a compactness-problem posed by Ozawa, a length independent estimate
for Junge-Parcet-Xu's free Rosenthal inequality, a Littlewood-Paley-Stein type
inequality for geodesic paths of free groups, and a length reduction formula
for $L^p$-norms of free group von Neumann algebras.
