Dilations and Rigid Factorisations on Noncommutative L-spaces

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