On the operator space $UMD$ property for noncommutative $L_p$-spaces

Abstract

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 < oo, the Schatten q-classes Sq are OUMDp. The proof relies on properties of the Haagerup tensor product and complex interpolation. Using ultra-product techniques, we extend this result to a large class of noncommutative Lq-spaces. Namely, we show that if 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 T, then L q (M, τ) is OUMDp for 1 < p, q < oo.