Asymptotic Cyclic Cohomology

Michael Puschnigg (auth.)9783540619864, 3-540-61986-0

The aim of cyclic cohomology theories is the approximation of K-theory by cohomology theories defined by natural chain complexes. The basic example is the approximation of topological K-theory by de Rham cohomology via the classical Chern character. A cyclic cohomology theory for operator algebras is developed in the book, based on Connes’ work on noncommutative geometry. Asymptotic cyclic cohomology faithfully reflects the basic properties and features of operator K-theory. It thus becomes a natural target for a Chern character. The central result of the book is a general Grothendieck-Riemann-Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. Besides this, the book contains numerous examples and calculations of asymptotic cyclic cohomology groups.

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Table of contents :
The asymptotic homotopy category….Pages 1-18
Algebraic de Rham complexes….Pages 19-26
Cyclic cohomology….Pages 27-39
Homotopy properties of X-complexes….Pages 40-58
The analytic X-complex….Pages 59-96
The asymptotic X-complex….Pages 97-117
Asymptotic cyclic cohomology of dense subalgebras….Pages 118-126
Products….Pages 127-157
Exact sequences….Pages 158-181
KK-theory and asymptotic cohomology….Pages 182-201
Examples….Pages 202-231