  We give an alternative proof of a Marcinkiewicz interpolation theorem for non
commutative maximal functions and positive maps, slightly refining earlier
versions of the statement. The main novelty is that it provides a substitute
for the maximal function of a martingale in $L_p$, $1<p\leq \infty$, losing
very little on numerical constants. For non positive maps, the above mentioned
theorem fails but we can still obtain some interpolation results by weakening
the maximal norms that we consider.
