Isometric embeddings of noncommutative $L_p$-spaces into noncommutative symmetric spaces

Abstract

We establish a noncommutative version of a familiar Johnson--Maurey--Schechtman--Tzafriri Theorem, by showing that for any $0<p<2$ and a (not necessarily semifinite) von Neumann algebra $\mathcal{M}$ on a separable Hilbert space, if a symmetric quasi-Banach function space $E(0,1)$ containing the function $t\mapsto t^{-1/p}$, $0<t\le 1,$ then there exists a noncommutative probability space $(\mathcal{N},\sigma )$ such that $L_p(\mathcal{M})$ is isometric to a subspace of $E(\mathcal{N},\sigma )$. In particular, this answers two questions raised by Randrianantoanina in 2006.