By Michael Puschnigg

The goal of cyclic cohomology theories is the approximation of K-theory through cohomology theories outlined by way of normal chain complexes. the elemental instance is the approximation of topological K-theory by means of de Rham cohomology through the classical Chern personality. A cyclic cohomology conception for operator algebras is built within the ebook, according to Connes' paintings on noncommutative geometry. Asymptotic cyclic cohomology faithfully displays the fundamental homes and lines of operator K-theory. It hence turns into a typical objective for a Chern personality. The principal results of the publication is a normal Grothendieck-Riemann-Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. in addition to this, the ebook includes quite a few examples and calculations of asymptotic cyclic cohomology groups.

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Da 2n = aa~ o(aa~ 1 . . da 2n - dada~ '~ - co(a, a~ 1 . . ,i,~ ocoi~ Q . . O O n O c o i'~ t--- O,0 @ a~-@... ) ~ A -+ 0 It is still called the I-adic filtration of ~ R A . Under the identification above the degree of an elementary tensor is io + il + " " + in. da2k+ 1 Under this isomorphism the I-adic filtration of the R A bimodule [~IRA~ on the left corresponds to the degree (Hodge)-filtration on the right side. 9: The universal extensions of the functors X,, X~)a from the categories of algebras (Fr~chet algebras with smooth families of morphisms) to the based linear categories C(Coc) are given by A -+ X,(RA) A ~ X~a(RA ) Both of these complexes consist of filtered vector spaces (under the I-adic filtration).

D a 2~ under which the I-adic filtration on R A corresponds to the degree (Hodge)-filtration on ~ A (IA)-~ ~Z_ ~ ) t~2kA k--m The product on R A corresponds to the Fedosov product on f ~ A . If A is a Fr6chet algebra then R A becomes a locally convex topological vector space under this isomorphism by giving ~ A the topology of Chapter 2. Proofi Consider the subatgebra G[Q,w] C R A generated by the elements Q(o,), ~ ( a ,' a " ), a . ' a " E A This subalgebra in fact equals R A because R A is generated by the elements Q(a).

O Qo. = (01 o 00). becomes obvious once the two expressions are written down explicitely. [] 25 The component of degree zero of the map above is just the family of maps of Xcomplexes induced by a given family of algebra homomorphisms. The components of higher degree however contain higher homotopy information which will be necessary to obtain homotopy formulas comparing the cohomology classes of members of a family of cocycles for different parameter values. 8: There is a natural transformation ch: K, ~ h ( X , ( M ~ ( - ) ) ) on the category of adnfissible Fr~chet algebras from topological K-groups to the homology of the X-cornplex of stable matrices.