Asymptotic Cyclic Cohomology

1. Front Matter

   Pages i-xxiii

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2. The asymptotic homotopy category

   • Michael Puschnigg

   Pages 1-18

3. Algebraic de Rham complexes

   • Michael Puschnigg

   Pages 19-26

4. Cyclic cohomology

   • Michael Puschnigg

   Pages 27-39

5. Homotopy properties of X-complexes

   • Michael Puschnigg

   Pages 40-58

6. The analytic X-complex

   • Michael Puschnigg

   Pages 59-96

7. The asymptotic X-complex

   • Michael Puschnigg

   Pages 97-117

8. Asymptotic cyclic cohomology of dense subalgebras

   • Michael Puschnigg

   Pages 118-126

9. Products

   • Michael Puschnigg

   Pages 127-157

10. Exact sequences

    • Michael Puschnigg

    Pages 158-181

11. KK-theory and asymptotic cohomology

    • Michael Puschnigg

    Pages 182-201

12. Examples

    • Michael Puschnigg

    Pages 202-231

13. Back Matter

    Pages 232-238

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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.

• Cohomology
• De Rham cohomology
• Homotopy
• K-theory
• cohomology group
• cohomology theory
• homology

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