Derivations with values in ideals of semifinite von Neumann algebras

Let M be a semifinite von Neumann algebra with a faithful semifinite normal trace τ and let A be an arbitrary C^⁎-subalgebra of M. Assume that E is a fully symmetric function space on (0,∞) having Fatou property and order continuous norm and E(M,τ) is the corresponding symmetric operator space. We prove that every derivation δ:A→E(M,τ):=E(M,τ)∩M is inner, strengthening earlier results by Kaftal and Weiss [28]. In the case when M is a semifinite non-finite factor, we show that our assumptions on E(0,∞) are sharp.