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 subset of 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 Ab is a W*-algebra without finite type I direct summands, then any ring derivation D from A into LS(Ab) - the algebra of all locally measurable operators affiliated with Ab, is an inner derivation. We also give an example showing that the condition M subset of 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.

Automated summary

This paper proves that if A is an EW*-algebra and its bounded part Ab is a W*-algebra without finite type I direct summands, then any ring derivation from A into LS(Ab) is an inner derivation. An example that satisfies the condition M⊆A is also provided.