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

## TL;DR

This paper demonstrates that symmetrically Δ-normed operator spaces associated with semifinite von Neumann algebras serve as interpolation spaces between $L_0$ and the algebra itself, with applications to ${
m K}$-functional and monotonicity properties.

## Contribution

It establishes that these operator spaces are interpolation spaces between $L_0$ and the algebra, contrasting with classical results, and explores ${
m K}$-functional and monotonicity in this context.

## Key findings

- Symmetrically Δ-normed operator spaces are interpolation spaces between $L_0$ and the algebra.
- The ${
m K}$-functional of an operator coincides with that of its singular value function.
- The pair $(L_0( au), ext{algebra})$ is ${
m K}$-monotone in non-atomic finite factors.

## 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.