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.

F. Math. Z. (2019). arXiv admin note: text overlap with arXiv:1808.10557