The theory of von Neumann algebras, originating with the work of F. J. Murray and J. von Neumann in the late 1930s, has grown into a rich discipline with connections to different branches of mathematics and physics. Following the breakthrough of Tomita–Takesaki theory, many great advances were made throughout the 1970s by H. Araki, A. Connes, U. Haagerup, M. Takesaki and others.
These lecture notes aim to present a fast-track study of some important topics in classical parts of von Neumann algebra theory that were developed in the 1970s. Starting with Tomita–Takesaki theory, this book covers topics such as the standard form, Connes’ cocycle derivatives, operator-valued weights, type III structure theory and non-commutative integration theory.
The self-contained presentation of the material makes this book useful not only to graduate students and researchers who want to know the fundamentals of von Neumann algebras, but also to interested undergraduates who have a basic knowledge of functional analysis and measure theory.
Keywords: von Neumann algebra, Tomita–Takesaki theory, modular operator, standard form, Connes’ cocycle derivative, operator-valued weight, relative modular operator, crossed product, KMS condition, Takesaki’s duality theorem,
Author(s): Fumio Hiai
Series: EMS Series of Lectures in Mathematic
Year: 2021
Preface
Von Neumann algebras – An overview
Preliminaries
Basics of von Neumann algebras
States, weights and traces
Classification of von Neumann algebras
Tomita–Takesaki theory
Classification of factors of type III
Crossed products and type III structure theory
Classification of AFD factors
Standard form and natural positive cone
Developments since the 1980s
Tomita–Takesaki modular theory
Tomita's fundamental theorem
KMS condition
Standard form
Definition and basic properties
Uniqueness theorem
tau-Measurable operators
tau-Measurable operators
Generalized s-numbers
L^p-spaces with respect to a trace
Conditional expectations and generalized conditional expectations
Conditional expectations
Generalized conditional expectations
Connes' cocycle derivatives
Basics of faithful semifinite normal weights
Connes' cocycle derivatives
Operator-valued weights
Generalized positive operators
Operator-valued weights
Pedersen–Takesaki construction
Takesaki duality and structure theory
Takesaki's duality theorem
Structure of von Neumann algebras of type III
Haagerup's L^p-spaces
Description of L^1(M)
Haagerup's L^p-spaces
Kosaki's interpolation L^p-spaces
Relative modular operators and Connes' cocycle derivatives (continued)
Relative modular operators
Connes' cocycle derivatives (continued)
Spatial derivatives and spatial L^p-spaces
Spatial derivatives
Proofs of theorems
Spatial L^p-spaces
Positive self-adjoint operators and positive quadratic forms
Positive self-adjoint operators
Positive quadratic forms
Bibliography
Index
Leere Seite
