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

It is known that any derivation on a commutative von Neumann algebra L∞(Ω,μ) is identically equal to zero. At the same time, the commutative algebra L0(Ω,μ) of complex measurable functions defined on a non-atomic measure space (Ω,μ) admits non-zero derivations. Besides, every derivation on L∞(Ω,μ) with the values in an ideal normed subspace X⊂L0(Ω,μ) is equal to zero. The same remains true for an ideal quasi-normed subspace X⊂L0(Ω,μ). Naturally, there is the problem of describing the class of ideal F-normed spaces X⊂L0(Ω,μ) for which there is a non-zero derivation on L∞(Ω,μ) 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 δ:L∞(Ω,μ)→X. In particular, it is shown that if the F-norm on X is order semicontinuous, each derivation δ:L∞(Ω,μ)→X is equal to zero. At the same time, existence of a non-atomic idempotent 0≠e∈X, μ(e)<∞ for which the measure topology in e⋅X coincides with the topology generated by the F-norm implies the existence of a non-zero derivation δ:L∞(Ω,μ)→X. Examples of such ideal F-normed spaces are algebras L0(Ω,μ) with non-atomic measure spaces (Ω,μ) equipped with the F-norm ∥f∥Ω=∫Ω|f|1+|f|dμ. For such ideal F-spaces there is at least a continuum of pairwise distinct non-zero derivations δ:L∞(Ω,μ)→(L0(Ω,μ),∥⋅∥Ω).