Algebras of Unbounded Operators

Algebraic and Topological Aspects of Murray–von Neumann Algebras

Aleksey Ber, Vladimir Chilin, Galina Levitina, Fedor Sukochev, Dmitriy Zanin

• Publisher: De Gruyter
• Year: 2025
• Topic: Mathematics
• eBook ISBN:
  9783111599687
• Subtopic: Algebra
• Index: Mathematics

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.

Table of contents

• Preface
• Contents
• General notation
• Introduction
• 1 Preliminaries on the theory of von Neumann algebras and general theory of linear operators in a Hilbert space
• 2 Classes of unbounded operators
• 3 Properties of locally measurable operators
• 4 Topologies on algebras of unbounded operators
• 5 Properties of derivations on algebras of locally measurable operators
• 6 Derivations on the algebras of measurable operators affiliated with a type Ifin von Neumann algebra
• 7 Complete description of derivations on algebras of locally measurable operators
• Bibliography
• Subject index
• Notation index