Operator Space Embedding of Schatten p-Classes Into Von Neumann Algebra Preduals

Abstract.

Let X₁ and X₂ be subspaces of quotients of R $\oplus$ OH and C $\oplus$ OH respectively. We use new free probability techniques to construct a completely isomorphic embedding of the Haagerup tensor product ${\rm X}_1 \oplus_h {\rm X}_2$ 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 $\ell_q$ (equipped with its natural operator space structure) into $L_1({\mathcal{A}})$ with ${\mathcal{A}}$ a QWEP von Neumann algebra.