@article{Kadison_Liu_2014, title={Derivations of Murray-von Neumann Algebras}, volume={115}, url={https://www.mscand.dk/article/view/19223}, DOI={10.7146/math.scand.a-19223}, abstractNote={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. }, number={2}, journal={MATHEMATICA SCANDINAVICA}, author={Kadison, Richard V. and Liu, Zhe}, year={2014}, month={Dec.}, pages={206–228} }