This book describes a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, it details other approaches to bivariant K-theories for operator algebras.

Topological K-theory is one of the most important invariants for noncommutative algebras. Bott periodicity, homotopy invariance, and various long exact sequences distinguish it from algebraic K-theory. This book describes a bivariant K-theory for bornological algebras, which provides a vast generalization of topological K-theory. In addition, it details other approaches to bivariant K-theories for operator algebras. The book studies a number of applications, including K-theory of crossed products, the Baum-Connes assembly map, twisted K-theory with some of its applications, and some variants of the Atiyah-Singer Index Theorem.

First available book on bivariant topological K-theory for algebras other than C*-algebras with complete proofs Several new results about bivariant K-theory for general algebras in this book did not even appear in journal publications before The exposition is accessible to graduate students Modern ideas from category theory are used to study bivariant K-theory Includes supplementary material: sn.pub/extras

Autorentext

Joachim Cuntz is a full Professor at the Westfälische Wilhelms-Universität in Münster, Germany. Siegfried Echterhoff is a full Professor at the Westfälische Wilhelms-Universität in Münster, Germany. Xin Li is a Senior Lecturer in Pure Mathematics at Queen Mary University of London, United Kingdom. Guoliang Yu is Powell Chair in Mathematics and Professor at the Texas A&M University, USA.

Inhalt
The elementary algebra of K-theory.- Functional calculus and topological K-theory.- Homotopy invariance of stabilised algebraic K-theory.- Bott periodicity.- The K-theory of crossed products.- Towards bivariant K-theory: how to classify extensions.- Bivariant K-theory for bornological algebras.- A survey of bivariant K-theories.- Algebras of continuous trace, twisted K-theory.- Crossed products by ? and Connes' Thom Isomorphism.- Applications to physics.- Some connections with index theory.- Localisation of triangulated categories.