We introduce a non-commutative analog of Banach space valued Lp-spaces in the category of operator spaces. Thus, given a von Neumann algebra M equipped with a faithful normal semi-finite trace and an operator space E, we introduce the space Lp(M, ; E), which is an E-valued version of non-commutative Lp, and we prove the basic properties one should expect of such an extension (e.g. Fubini, duality, . . .). There are two important restrictions for the theory to be satisfactory: first M should be injective, secondly E cannot be just a Banach space, it should be given with an operator space structure and all the stability properties (e.g. duality) should be formulated in the category of operator spaces. This leads naturally to a theory of "completely p-summing maps" between operator spaces, analogous to the Grothendieck-Pietsch-Kwapie theory (i.e. "absolutely p-summing maps") for Banach spaces. As an application, we obtain a characterization of maps factoring through the operator space version of Hilbert space. More generally, we study the mappings between operator spaces which factor through a non-commutative Lp-space (or through an ultraproduct of them), using completely p-summing maps. In this setting, we also discuss the factorization through subspaces, or through quotients of subspaces of Lp-spaces. Astrisque 247, SMF 1998.
