Derivations on Banach *-ideals in von Neumann algebras

It is known that any derivation δ:M→M on the von Neumann algebra M is an inner, i.e. δ(x):=δa(x)=[a,x]=ax-xa, x∈M, for some a∈M. If H is a separable infinite-dimensional complex Hilbert space and K(H) is a C∗-subalgebra of compact operators in C∗-algebra B(H) of all bounded linear operators acting in H, then any derivation δ:K(H)→K(H) is a spatial derivation, i.e. there exists an operator a∈B(H) such that δ(x)=[x,a] for all x∈K(H). In addition, it has recently been established by Ber A. F., Chilin V. I., Levitina G. B. and Sukochev F. A. (JMAA, 2013) that any derivation δ:E→E on Banach symmetric ideal of compact operators E⊆K(H) is a spatial derivation. We show that the same result is also true for an arbitrary Banach ∗-ideal in every von Neumann algebra M. More precisely: If M is an arbitrary von Neumann algebra, E be a Banach ∗-ideal in M and δ:E→E is a derivation on E, then there exists an element a∈M such that δ(x)=[x,a] for all x∈E, i.e. δ is a spatial derivation.