  We study some factorisation and dilation properties of completely positive
maps on noncommutative L^p-spaces. We show that Akcoglu's dilation theorem for
positive contractions on classical (=commutative) L^p-spaces has no reasonable
analog in the noncommutative setting. Our study relies on non symmetric analogs
of Pisier's operator space valued noncommutative L^p-spaces that we investigate
in the first part of the paper.
