  We introduce the notion of a regular mapping on a non-commutative $L_p$-space
associated to a hyperfinite von Neumann algebra for $1\le p\le \infty$. This is
a non-commutative generalization of the notion of regular or order bounded map
on a Banach lattice. This extension is based on our recent paper [P3], where we
introduce and study a non-commutative version of vector valued $L_p$-spaces. In
the extreme cases $p=1$ and $p=\infty$, our regular operators reduce to the
completely bounded ones and the regular norm coincides with the $cb$-norm. We
prove that a mapping is regular iff it is a linear combination of bounded,
completely positive mappings. We prove an extension theorem for regular
mappings defined on a subspace of a non-commutative $L_p$-space. Finally, let
$R_p$ be the space of all regular mappings on a given non-commutative
$L_p$-space equipped with the regular norm. We prove the isometric identity
$R_p=(R_\infty,R_1)^\theta$ where $\theta=1/p$ and where $(\ .\ ,\ .\ )^\theta$
is the dual variant of Calder\'on's complex interpolation method.
