Title: Unconditional basic sequences and homogeneous Hilbertian subspaces of non-commutative $L_p$ spaces

Authors: Marius Junge and Timur Oikhberg

Issue: Volume 56 (2007), Issue 2, 733-766

Abstract: Suppose $A$ is a von Neumann algebra with a normal faithful normalized trace $\tau$. We prove that if $E$ is a homogeneous Hilbertian subspace of $L_p(\tau)$ ($1 \leq p < \infty$) such that the norms induced on $E$ by $L_p(\tau)$ and $L_2(\tau)$ are equivalent, then $E$ is completely isomorphic to the subspace of $L_p([0,1])$ spanned by Rademacher functions. Consequently, any homogeneous subspace of $L_p(\tau)$ is completely isomorphic to the span of Rademacher functions in $L_p([0,1])$. In particular, this applies to the linear span of operators satisfying the canonical anti-commutation relations. We also show that the real interpolation space $(R,C)_{\theta,p}$ embeds completely isomorphically into $L_p({\mathcal{R}})$ (${\mathcal{R}}$ is the hyperfinite $II_1$ factor) for any $1 \leq p < 2$ and $\theta \in (0,1)$.