Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC ^*-algebra andH be a Hilbert space. Let π be an element ofH ^∞ (CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ?₁ ∈₁ H ^∞ (B(H,K)) and ∈₂ H ^∞ (B(K,H)) such that ‖ε₁‖∞‖∈₂‖∞ ≤ ‖∈‖∞ and: $\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).$ We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.