Ring derivations of Murray–von Neumann algebras

Let M be a type II₁ von Neumann algebra, S(M) be the Murray–von Neumann algebra associated with M and let A be a ⁎-subalgebra in S(M) with M⊆A. We prove that any ring derivation D from A into S(M) is necessarily inner. Further, we prove that if A is an EW^⁎-algebra such that its bounded part A_b is a W^⁎-algebra without finite type I direct summands, then any ring derivation D from A into LS(A_b) — the algebra of all locally measurable operators affiliated with A_b, is an inner derivation. We also give an example showing that the condition M⊆A is essential. At the end of this paper, we provide several criteria for an abelian extended W^⁎-algebra such that all ring derivations on it are linear.