A Murray-von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we study derivations of Murray-von Neumann algebras and their properties. We show that the "extended derivations" of a Murray-von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray-von Neumann algebra associated with a von Neumann algebra of type ${\rm II}_1$ into that von Neumann algebra is 0. This result is an extension, in two ways, of Singer's seminal result answering a question of Kaplansky, as applied to von Neumann algebras: the algebra may be non-commutative and contain unbounded elements. In another sense, as we indicate in the introduction, all the derivation results including ours extend what Singer's result says, that the derivation is the 0-mapping, numerically in our main theorem and cohomologically in our theorem on extended derivations. The cohomology in this case is the Hochschild cohomology for associative algebras.
Hochschild cohomology and moduli spaces of strongly homotopy associative algebras