Title

Derivations Of Murray-Von Neumann Algebras

Abstract

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 II1 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 vonNeumann 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.

Publication Date

1-1-2014

Publication Title

Mathematica Scandinavica

Volume

115

Issue

2

Number of Pages

206-228

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.7146/math.scand.a-19223

Socpus ID

84917697753 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84917697753

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