This book resulted from the lectures held at The Fields Institute (Waterloo, ON, Canada). Leading international experts presented current results on the theory of \(C^*\)-algebras and von Neumann algebras, together with recent work on the classification of \(C^*\)-algebras. Much of the material in the book is appearing here for the first time and is not available elsewhere in the literature.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Readership

Graduate students and research mathematicians interested in operator theory.

Reviews

"Contains ... a nice illustration of Elliott's classification techniques for inductive limits ... richly illustrated article ... on paths on Coxeter diagrams and sub-factors ... particularly welcome ... Overall this is a very nicely and surprisingly uniformly written book which is of interest both for the novice and the expert in operator algebras ... It may be hoped that the book will inspire some young researcher to new invention."

-- CMS Notes

Table of Contents

• C*-algebras: Definitions and examples
• C*-algebras: Constructions
• Positivity in C*-algebras
• K-theory I
• Tensor products of C*-algebras
• Crossed products I
• Crossed products II: Examples
• Free products
• K-theory II: Roots in topology and index theory
• C*-algebraic K-theory made concrete, or trick or treat with \(2 \times 2\) matrix algebras
• Dilation theory
• C*-algebras and mathematical physics
• C*-algebras and several complex variables

• Basic structure of von Neumann algebras
• von Neumann algebras (Type \(II_1\) factors)
• The equivalence between injectivity and hyperfiniteness, part I
• The equivalence between injectivity and hyperfiniteness, part II
• On the Jones index
• Introductory topics on subfactors
• The Tomita-Takesaki theory explained
• Free products of von Neumann algebras
• Semigroups of endomorphisms of \(\mathcal{B}(H)\)
• Classification of C*-algebras
• AF-algebras and Bratteli diagrams
• Classification of amenable C*-algebras I
• Classification of amenable C*-algebras II
• Simple AI-algebras and the range of the invariant
• Classification of simple purely infinite C*-algebras I

• Preface
• Introduction
• The isomorphism theorem
• The range of the invariant
• Bibliography

• Preface/Acknowledgements
• The Kauffman-Lins recoupling theory
• Graphs and connections
• An extension of the recoupling model
• Relations to minimal models and subfactors
• Bibliography