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

Abstract

Abstract. We study the operator space U M D property, introduced by Pisier in the context of noncommutative vector-valued Lp-spaces. It is unknown whether the property is independent of p in this setting. We prove that for 1 < p, q < ∞ , the Schatten q-classes Sq are OU M Dp . 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 Lq-spaces. Namely, we show that if M is a QW EP von Neumann algebra (i.e., a quotient of a C * -algebra with Lance's weak expectation property) equipped with a normal, faithful tracial state τ , then Lq(M, τ ) is OU M Dp for 1 < p, q < ∞ .

Pseudo-localization of singular integrals and noncommutative Calderón–Zygmund theory

“…Of course, this is closely related to the geometry of the operator space in question and in particular to the notion of UMD p operator spaces, also defined by Pisier. In this context a great variety of problems come into scene, like the independence of the UMD p condition with respect to p (see [42] for some advances) or the operator space analog of Burkholder's geometric characterization of the UMD property in terms of ζ -convexity [5]. Remark 6.9.…”

On the $\text{OUMD}$ property for the column Hilbert space $C$

“…He also thanks Quanhua Xu for the helpful suggestion during the preparation of this paper. He would like thank Javier Parcet for explaining carefully the gap in [8] to him.…”

“…In principle, this embedding theorem should solve the Ruan problem, since in [8], Musat presented a proof of the following: Statement 1.3. For any 1 < p, q < ∞, the operator space S q is OUMD p .…”

“…However, very recently, Javier Parcet found a gap in the proof of the above result. The main gap is in the proof of Proposition 4.10 in [8],…”

“…where the system (4.19) and (4.20) has solution only for 3 2 < p < 3. For the reader's convenience, we reproduce in the appendix the results in [8] which are not affected by this gap.…”