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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978-3-540-49579-6 (9783540495796)
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Schweitzer Klassifikation
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.
