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Asymptotic Cyclic Cohomology by Michael Puschnigg

By Michael Puschnigg

The target of cyclic cohomology theories is the approximation of K-theory by means of cohomology theories outlined via common chain complexes. the elemental instance is the approximation of topological K-theory through de Rham cohomology through the classical Chern personality. A cyclic cohomology thought for operator algebras is built within the ebook, in accordance with Connes' paintings on noncommutative geometry. Asymptotic cyclic cohomology faithfully displays the fundamental houses and lines of operator K-theory. It therefore turns into a traditional objective for a Chern personality. The primary results of the ebook is a common Grothendieck-Riemann-Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. in addition to this, the publication comprises various examples and calculations of asymptotic cyclic cohomology groups.

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We claim t h a t U' : = 1 - U consists o f i n v e r t i b l e elements. I f x E U then also y := Ax E U for some A > 1. T h e n z := 1 - x E U' is invertible with inverse z -1 = ~ A-ny n n:O where the sum converges because the multiplicative closure {yn; rt E ffV} of y is relatively c o m p a c t and thus bounded for any seminorm on A. For the continuity let (xn) E A, limn--+oo x,~ = 1 be a sequence of elements of A converging to 1. 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.

D a O . . d a 2i+1 = = ( -bs(~(",-~-~)) + ~ 2(;'z-i-i) d q- t~"2(n-i-1)+1 d ) ( a ~ 1 . d a 2n) to 37 S u m m i n g up yields E-b 6 = n,s2 " d + (~2(n-i-1)] st's 2(n--i--I). ~ . 11: We recall from [CQ] t h a t the Karoubi operator ~8:= ~s(a~ 1 - db~ - b~d : 9 A ~ f~A ... da n) = ( _ l ) n - : ( _ l ) ( l a ~ aOdal ... dan-1 satisfies the identity (my- 1)(~;; +: - 1) = 0 0 1 : a n A The K a r o u b i o p e r a t o r commutes with the differentials/~, 6 of X, (RA) so t h a t X . ( R A ) splits under the generalized eigenspace decomposition X .

If G : C ~ i9 is any contravariant functor the maps T r a n s f ( F , Gj) ~ T r a n s f ( F o R , G) c(o) x o F(~r) r o X are bijeetions inverse to each other. Therefore tile above extensions are universal. [] The algebra R A depends only on the underlying vector space of A (and the a u g m e n t a t i o n m a p A --+ ~). R A however admits a canonical filtration which enables one to recover the algebra structure of A. 5: Tile conmmtative diagram of flmctors Alg (aa-~dic nlt)~ Filtered Alg Jj. RB.

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