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Algebras of Unbounded Operators

Algebraic and Topological Aspects of Murray–von Neumann Algebras

De Gruyter

ISBN 978-3-11-159791-1

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Bibliografische Daten

Fachbuch

Buch. Softcover

2025

In englischer Sprache

Umfang: 420 S.

Format (B x L): 17 x 24 cm

Gewicht: 682

Verlag: De Gruyter

ISBN: 978-3-11-159791-1

Weiterführende bibliografische Daten

Das Werk ist Teil der Reihe: de Gruyter Textbook

Produktbeschreibung

Derivations on von Neumann algebras are well understood and are always inner, meaning that they act as commutators with a fixed element from the algebra itself. The purpose of this book is to provide a complete description of derivations on algebras of operators affiliated with a von Neumann algebra. The book is designed to serve as an introductory graduate level to various measurable operators affiliated with a von Neumann algebras and their properties. These classes of operators form their respective algebras and the problem of describing derivations on these algebras was raised by Ayupov, and later by Kadison and Liu. A principal aim of the book is to fully resolve the Ayupov-Kadison-Liu problem by proving a necessary and sufficient condition of the existence of non-inner derivation of algebras of measurable operators. It turns out that only for a finite type I von Neumann algebra M may there exist a non-inner derivation on the algebra of operators affiliated with M. In particular, it is established that the classical derivation d/dt of functions of real variables can be extended up to a derivation on the algebra of all measurable functions. This resolves a long-standing problem in classical analysis.

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