Asymptotic Cyclic Cohomology
Puschnigg, Michael.
Asymptotic Cyclic Cohomology [electronic resource] / by Michael Puschnigg. - XXIV, 244 p. online resource. - Lecture Notes in Mathematics, 1642 0075-8434 ; . - Lecture Notes in Mathematics, 1642 .
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.
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.
9783540495796
10.1007/BFb0094458 doi
Algebra.
Algebraic topology.
K-theory.
Operator theory.
Category Theory, Homological Algebra.
Algebraic Topology.
K-Theory.
Operator Theory.
QA169
512.6
Asymptotic Cyclic Cohomology [electronic resource] / by Michael Puschnigg. - XXIV, 244 p. online resource. - Lecture Notes in Mathematics, 1642 0075-8434 ; . - Lecture Notes in Mathematics, 1642 .
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.
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.
9783540495796
10.1007/BFb0094458 doi
Algebra.
Algebraic topology.
K-theory.
Operator theory.
Category Theory, Homological Algebra.
Algebraic Topology.
K-Theory.
Operator Theory.
QA169
512.6