Dilations and rigid factorisations on noncommutative L^p-spaces

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 nonsymmetric analogs of Pisier's operator space valued noncommutative L^p-spaces that we investigate in the first part of the paper.