Operator space Grothendieck inequalities for noncommutative $L_p$-spaces

Abstract

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 $q\in [p&#39;, p]$ ($p&#39;$ 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&#39;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.