The UMD property of symmetric operator spaces

Abstract

We prove that if $E$ is a UMD symmetric Banach function space on $(0,\infty)$, then $E(\mathcal{M},\tau)$ is UMD for every semifinite von Neumann algebra $\mathcal{M}$ equipped with a faithful normal semifinite trace $\tau$. This resolves in the affirmative an open problem that has circulated in the non-commutative world for more than four decades.