Operator Algebras and Geometry
About this Title
Hitoshi Moriyoshi, Keio University, Yokohama, Japan and Toshikazu Natsume, Nagoya Institute of Technology, Nagoya, Japan. Translated by Hitoshi Moriyoshi, Keio University, Yokohama, Japan and Toshikazu Natsume, Nagoya Institute of Technology, Nagoya, Japan
Publication: Translations of Mathematical Monographs
Publication Year: 2008; Volume 237
ISBNs: 978-0-8218-3947-8 (print); 978-1-4704-1623-2 (online)
MathSciNet review: MR2464268
MSC: Primary 46-01; Secondary 19K35, 46L05, 46L10, 46L87, 58J22
In the early 1980's topologists and geometers for the first time came across unfamiliar words like $C^*$-algebras and von Neumann algebras through the discovery of new knot invariants (by V. F. R. Jones) or through a remarkable result on the relationship between characteristic classes of foliations and the types of certain von Neumann algebras. During the following two decades, a great deal of progress was achieved in studying the interaction between geometry and analysis, in particular in noncommutative geometry and mathematical physics. The present book provides an overview of operator algebra theory and an introduction to basic tools used in noncommutative geometry. The book concludes with applications of operator algebras to Atiyah–Singer type index theorems. The purpose of the book is to convey an outline and general idea of operator algebra theory, to some extent focusing on examples.
The book is aimed at researchers and graduate students working in differential topology, differential geometry, and global analysis who are interested in learning about operator algebras.
Graduate students and research mathematicians interested in applications of functional analysis to geometry and topology.
Table of Contents
- Von Neumann algebras
- Cyclic cohomology
- Quantizations and index theory
- Foliation index theorems