Embedding of C _q and R _q into noncommutative L _p -spaces, 1 ≤ p<q ≤ 2

We prove that a quotient of a subspace of C _p ⊕ _p R _p (1 ≤ p <2) embeds completely isomorphically into a noncommutative L _p -space, where C _p and R _p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C _q and R _q (p<q ≤ 2) as quotients of subspaces of C _p ⊕ _p R _p . Consequently, C _q and R _q embed completely isomorphically into a noncommutative L _p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.