Derivations with values in an ideal $F$-spaces of measurable functions

It is known that any derivation on a commutative von Neumann algebra $ \mathcal {L}_{\infty} (\Omega, \mu)$ is identically equal to zero. At the same time, the commutative algebra $\mathcal {L}_{0}(\Omega, \mu)$ of complex measurable functions defined on a non-atomic measure space $(\Omega,\mu)$ admits non-zero derivations. Besides, every derivation on $\mathcal{L}_{\infty}(\Omega, \mu)$ with the values in an ideal normed subspace $X \subset \mathcal{L}_{0}(\Omega,\mu)$ is equal to zero. The same remains true for an ideal quasi-normed subspace $X \subset\mathcal{L}_{0}(\Omega, \mu)$.
Naturally, there is the problem of describing the class of ideal $F$-normed spaces $X \subset \mathcal{L}_{0}(\Omega, \mu)$ for which there is a non-zero derivation on $\mathcal{L}_{\infty}(\Omega, \mu)$ with the values in $X $. We give necessary and sufficient conditions for a complete ideal $F$-normed spaces $X$ to be such that there is a non-zero derivation $\delta: \mathcal{L}_{\infty}(\Omega, \mu) \to X$. In particular, it is shown that if the $F$-norm on $X$ is order semicontinuous, each derivation $\delta: \mathcal{L}_{\infty}(\Omega, \mu) \to X$ is equal to zero. At the same time, existence of a non-atomic idempotent $0\neq e \in X$, $\mu(e) < \infty$ for which the measure topology in $e \cdot X$ coincides with the topology generated by the $F$-norm implies the existence of a non-zero derivation $\delta: \mathcal{L}_{\infty}(\Omega, \mu)\to X$. Examples of such ideal $F$-normed spaces are algebras $\mathcal{L}_{0}(\Omega, \mu)$ with non-atomic measure spaces $(\Omega, \mu)$ equipped with the $F$-norm $\| f\|_{\Omega} = \int_{\Omega} \frac {| f |} {1+ | f |} d\mu $. For such ideal $ F$-spaces there is at least a continuum of pairwise distinct non-zero derivations $\delta: \mathcal{L}_{\infty}(\Omega, \mu)\to (\mathcal{L}_{0}(\Omega, \mu), \|\cdot\|_{\Omega})$.