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
CONTENT
The asymptotic homotopy category -- Algebraic de Rham complexes -- Cyclic cohomology -- Homotopy properties of X-complexes -- The analytic X-complex -- The asymptotic X-complex -- Asymptotic cyclic cohomology of dense subalgebras -- Products -- Exact sequences -- KK-theory and asymptotic cohomology -- Examples
