Operator space Grothendieck inequalities for noncommutative $L_p$-spaces

We prove the operator space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded iff $u$ factors through a direct sum of a $p$-column space and a $p$-row space. We also obtain several operator space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative $L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in [p', p]$ ($p'$ being the index conjugate to $p$). These results are the $L_p$-space analogues of the recent works on the operator space Grothendieck theorems by Pisier and Shlyakhtenko. The key ingredient of our arguments is some Khintchine type inequalities for Shlyakhtenko's generalized circular systems. One of our main tools is a Haagerup type tensor norm, which turns out particularly fruitful when applied to subspaces of noncommutative $L_p$-spaces ($2<p<\infty$). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative $L_p$-spaces, is equal to the factorization norm through a $p$-row space.