#### Title

DERIVATIONS OF MURRAY-VON NEUMANN ALGEBRAS

#### Abbreviated Journal Title

Math. Scand.

#### Keywords

OPERATOR ALGEBRAS; RINGS; Mathematics

#### 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 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.

#### Journal Title

Mathematica Scandinavica

#### Volume

115

#### Issue/Number

2

#### Publication Date

1-1-2014

#### Document Type

Article

#### Language

English

#### First Page

206

#### Last Page

228

#### WOS Identifier

#### ISSN

0025-5521

#### Recommended Citation

"DERIVATIONS OF MURRAY-VON NEUMANN ALGEBRAS" (2014). *Faculty Bibliography 2010s*. 5539.

http://stars.library.ucf.edu/facultybib2010/5539

## Comments

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