Revisiting the Marcinkiewicz theorem for non commutative maximal functions

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 $L_p$, $1<p\le \mathrm{\infty }$, 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.