  We study the global regularity, for all time and all initial data in
$H^{1/2}$, of a recently introduced decimated version of the incompressible 3D
Navier-Stokes (dNS) equations. The model is based on a projection of the
dynamical evolution of Navier-Stokes (NS) equations into the subspace where
helicity (the $L^2-$scalar product of velocity and vorticity) is sign-definite.
The presence of a second (beside energy) sign-definite inviscid conserved
quadratic quantity, which is equivalent to the $H^{1/2}-$Sobolev norm, allows
us to demonstrate global existence and uniqueness, of space-periodic solutions,
together with continuity with respect to the initial conditions, for this
decimated 3D model. This is achieved thanks to the establishment of two new
estimates, for this 3D model, which show that the $H^{1/2}$ and the time
average of the square of the $H^{3/2}$ norms of the velocity field remain
finite. Such two additional bounds are known, in the spirit of the work of H.
Fujita and T. Kato \cite{kato1,kato2}, to be sufficient for showing
well-posedness for the 3D NS equations. Furthermore, they are directly linked
to the helicity evolution for the dNS model, and therefore with a clear
physical meaning and consequences.
