On the operator space $UMD$ property for noncommutative $L_p$-spaces Magdalena Musat 46L5146L5347L2560G46noncommutative $L_p$ spacesnoncommutative martingalesoperator space UMD property We study the operator space UMD property, introduced by Pisier in the context of noncommutative vector-valued $L_{p}$-spaces. It is unknown whether the property is independent of $p$ in this setting. We prove that for $1 < p$, $q < \infty$, the Schatten $q$-classes $S_{q}$ are OUMD${}_{p}$. The proof relies on properties of the Haagerup tensor product and complex interpolation. Using ultraproduct techniques, we extend this result to a large class of noncommutative $L_{q}$-spaces. Namely, we show that if $\mathcal{M}$ is a QWEP von Neumann algebra (i.e., a quotient of a $C^{*}$-algebra with Lance\'s weak expectation property) equipped with a normal, faithful tracial state $\tau$, then $L_{q}(\mathcal{M},\tau)$ is OUMD${}_{p}$ for $1 < p$, $q < \infty$. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2778 10.1512/iumj.2006.55.2778 en Indiana Univ. Math. J. 55 (2006) 1857 - 1892 state-of-the-art mathematics http://iumj.org/access/