Rigid $\mathcal{OL}_p$structures of non-commutative $L_p$-spaces associated with hyperfinite von Neumann algebras - Junge, Marius; Ruan, Zhong-Jin; Xu, Quanhua (MATHEMATICA SCANDINAVICA, 2005)
This paper is devoted to the study of rigid local operator space structures on non-commutative $L_p$-spaces. We show that for $1\le p eq 2 < ∞$, a non-commutative $L_p$-space $L_p(\mathcal M)$ is a rigid $\mathcal{OL}_p$ space (equivalently, a rigid $\mathcal{COL}_p$ space) if and only if it is a matrix orderly rigid $\mathcal{OL}_p$ space (equivalently, a matrix orderly rigid $\mathcal{COL}_p$ space). We also show that $L_p(\mathcal M)$ has these local properties if and only if the associated von Neumann algebra $\mathcal M$ is hyperfinite. Therefore, these local operator space properties on non-commutative $L_p$-spaces characterize hyperfinite von Neumann algebras.