  We introduce a noncommutative analogue of the absolute value of a regular
operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that
two classical operator norms, the regular norm and the decomposable norm are
identical. We also describe precisely the regular norm of several classes of
regular multipliers. This includes Schur multipliers and Fourier multipliers on
some unimodular locally compact groups which can be approximated by discrete
groups in various senses. A main ingredient is to show the existence of a
bounded projection from the space of completely bounded $\mathrm{L}^p$
operators onto the subspace of Schur or Fourier multipliers, preserving
complete positivity. On the other hand, we show the existence of bounded
Fourier multipliers which cannot be approximated by regular operators, on large
classes of locally compact groups, including all infinite abelian locally
compact groups. We finish by introducing a general procedure for proving
positive results on selfadjoint contractively decomposable Fourier multipliers,
beyond the amenable case.
