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Let ${\mathcal M}$ be a semifinite von Neumann algebra with a faithfulsemifinite normal trace $\tau$. We show that the symmetrically $\Delta$-normedoperator space $E({\mathcal M},\tau)$ corresponding to an arbitrarysymmetrically $\Delta$-normed function space $E(0,\infty)$ is an interpolationspace between $L_0({\mathcal M},\tau)$ and ${\mathcal M}$, which is in contrastwith 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 valuefunction $\mu(X)$. Several applications are given, e.g., it is shown that thepair $(L_0({\mathcal M},\tau),{\mathcal M})$ is ${\mathcal K}$-monotone when${\mathcal M}$ is a non-atomic finite factor.

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