Innerness of derivations into noncommutative symmetric spaces is determined commutatively

Abstract: Let $E=E(0,\infty)$ 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 $C^*$-subalgebra $\mathcal{A}$ in the class of all semifinite von Neumann algebras $\mathcal{M}$ as those with the Levi property.