Interpolation between $L_0({\mathcal M},\tau)$ and $L_\infty({\mathcal M},\tau)$

Abstract

Let ${\mathcal M}$ be a semifinite von Neumann algebra with a faithful semifinite normal trace $\tau$. We show that the symmetrically $\Delta$-normed operator space $E({\mathcal M},\tau)$ corresponding to an arbitrary symmetrically $\Delta$-normed function space $E(0,\infty)$ is an interpolation space between $L_0({\mathcal M},\tau)$ and ${\mathcal M}$, which is in contrast with the classical result that there exist symmetric operator spaces $E({\mathcal M},\tau)$ which are not interpolation spaces between $L_1({\mathcal M},\tau)$ and ${\mathcal M}$. Besides, we show that the ${\mathcal K}$-functional of every $X\in L_0({\mathcal M},\tau)+ {\mathcal M} $ coincides with the ${\mathcal K}$-functional of its generalized singular value function $\mu(X)$. Several applications are given, e.g., it is shown that the pair $(L_0({\mathcal M},\tau),{\mathcal M})$ is ${\mathcal K}$-monotone when ${\mathcal M}$ is a non-atomic finite factor.