OPERATOR SPACE Lp EMBEDDING THEORY I

Let X 1 and X 2 be subspaces of quotients of R ⊕ OH and C ⊕ OH respectively. We use new free probability techniques to construct a completely isomorphic embedding of the Haagerup tensor product X 1 ⊗ h X 2 into the predual of a sufficiently large QWEP von Neumann algebra. As an immediate application, given any 1 < q ≤ 2, our result produces a completely isomorphic embedding of ℓq (equipped with its natural operator space structure) into L 1 (A) with A a QWEP von Neumann algebra.