By Michael Puschnigg

The objective of cyclic cohomology theories is the approximation of K-theory through cohomology theories outlined through average chain complexes. the elemental instance is the approximation of topological K-theory by way of de Rham cohomology through the classical Chern personality. A cyclic cohomology idea for operator algebras is built within the publication, in response to Connes' paintings on noncommutative geometry. Asymptotic cyclic cohomology faithfully displays the elemental homes and contours of operator K-theory. It hence turns into a normal goal for a Chern personality. The primary results of the booklet is a common Grothendieck-Riemann-Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. along with this, the booklet comprises quite a few examples and calculations of asymptotic cyclic cohomology groups.

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One has $(U)@,~g(U') ~- g(U x U') as locally convex DG-modules. In the sequel we will be interested in the simplicial DG-module V. := g ( t g ~ ) . We denote the associated X-complexes by X~)a(A ) := X~)a,e(z~r)(A) X~)G(A, B) := X~)G,e(~T)(A, B) Under the identification g(h~)@~g(h~,~) _~ g ( ~ sition product yields maps x h~) -~ g ( N ~ ) the compo- X•G(A, B) | X~)G(B, C) -+ X]Da(A, C) XSG(A,B) | X~)a(B ) -+ X]DG(A) It is possible now to replace in our considerations single homomorphisms by whole families of algebra homomorphisms.

Then Xn = 1 -y,~ where (y~) is a nullsequence. Choose a m o n o t o n e increasing, mlbounded sequence of positive real numbers An > 1 such t h a t (Anyn) remains still a nullsequence. After deleting finitely many elements one m a y suppose t h a t {A,,yn; n E PC} is contained in a "small" ball. Therefore there exists a c o m p a c t set K C A containing the multiplicative closure of the sequence (A,~yn). Let II - II be a s e m i n o r m on A. Then supx, e g II x II< C for some C > 0. -, ,~ (nY,~) k=0 and thus {{~ - i {{_~ ~ A3~ {{( A .

The first condition asks that traces should be zero-cocycles and closed 1-traces should be l-cocyclcs in such a complex. *(A) := H o m ( ? ,(A)) contains bB(f~A). (In the notations of [CO]). 1: (Cuntz,Quillen) Let A be a graded unital algebra. The p e r i o d i c de R h a m c o m p l e x of A is given by ~PdR(A) := {~A/(b(Nd) + (Nd)b)fM, b + Nd} It is a filtered 28/2-graded chain complex. The Hodge filtration by the subcomplexes Fnf~P, dR generated by differential forms of degree at least n yields a sequence of quotient complexes [-~PdR , (A ) / F n~PdR , ( A ) that approximate successively the periodic de Rham complex itself.