Innerness of derivations into noncommutative symmetric spaces is determined commutatively

Let $E=E(0,\mathrm{\infty })$ be a symmetric function space and $E(\mathcal{M},\tau )$ be a symmetric operator space associated with a semifinite von Neumann algebra with a faithful normal semifinite trace. Our main result identifies the class of spaces $E$ for which every derivation $\delta :\mathcal{A}\to E(\mathcal{M},\tau )$ is necessarily inner for each $C^{\ast }$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.