Hiai Fumio. Lectures on Selected Topics in von Neumann Algebras

European Mathematical Society, 2021. — 252 p. — (EMS Series of Lectures in Mathematic). — ISBN 978-3-98547-004-4.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.Von Neumann algebras – An overview
Tomita–Takesaki modular theory
Standard form
Measurable operators
Conditional expectations and generalized conditional expectations
Connes’ cocycle derivatives
Operator-valued weights
Takesaki duality and structure theory
Haagerup’s Lp-spaces
Relative modular operators and Connes’ cocycle derivatives (continued)
Spatial derivatives and spatial Lp-spacesPositive self-adjoint operators and positive quadratic forms