Cyclic Homology in Non-Commutative Geometry

by Cuntz, Joachim; Skandalis, Georges; Tsgan, Boris

• ISBN13: 9783540404699
• ISBN10: 3540404694
• Format: Hardcover
• Copyright: 2004-03-01
• Publisher: Springer Verlag

Summary

This volume contains contributions by three authors and treats aspects of noncommutative geometry that are related to cyclic homology. The authors give rather complete accounts of cyclic theory from different and complementary points of view. The connections between topological (bivariant) K-theory and cyclic theory via generalized Chern-characters are discussed in detail. This includes an outline of a framework for bivariant K-theory on a category of locally convex algebras. On the other hand, cyclic theory is the natural setting for a variety of general index theorems. A survey of such index theorems (including the abstract index theorems of Connes-Moscovici and of Bressler-Nest-Tsygan) is given and the concepts and ideas involved in the proof of these theorems are explained.

Table of Contents

Cyclic Theory, Bivariant K-Theory and the Bivariant Chern-Connes Character

Cyclic Theory

Cyclic Theory for Locally Convex Algebras

Bivariant K-Theory

Infinite-Dimensional Cyclic Theories

Locally Convex Algebras

Standard Extensions

Noncommutative Geometry, the Transverse Signature Operator, and Hopf Algebras (after A. Connes and H. Moscovici)

Preliminaries

The Local Index Formula

The Diff-Invariant Signature Operator

The "Transverse" Hopf Algebra

Cyclic Homology

Introduction

Hochschild and Cyclic Homology of Algebras

The Cyclic Complex <$$$>C^{lambda}_{bullet}

Non-Commutative Differential Calculus

Cyclic Objects

Examples

Index Theorems

Riemann-Roch Theorem for D-Modules

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