Download e-book for iPad: Asymptotic Cyclic Cohomology by Michael Puschnigg

By Michael Puschnigg

ISBN-10: 3540619860

ISBN-13: 9783540619864

The goal of cyclic cohomology theories is the approximation of K-theory via cohomology theories outlined by means of usual 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 constructed within the e-book, in response to Connes' paintings on noncommutative geometry. Asymptotic cyclic cohomology faithfully displays the fundamental houses and lines of operator K-theory. It hence turns into a usual aim for a Chern personality. The imperative results of the publication is a basic Grothendieck-Riemann-Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. in addition to this, the ebook includes a number of examples and calculations of asymptotic cyclic cohomology groups.

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It can be written as nj nj b ,~ = y-~a~ | b{: ;4 II "? IIs(j)-> ~ II 0,s IIs(j)ll bL II}(j)} k=l k=l where f is a monotone increasing unbounded fimction and lira j--*~o II "r IIs(j) = 0; II ~J Ils(j)r 0; II as IIs(j)# 0; II ~ II} N o } , K " n U " = { ~ , j > _ X , } with relatively compact multiplicative closure.

The inverse of the rescaling map in a) yields an inclusion of coinplexes g, := ~P. dRA --% ker(1 - n,) 2 C X . ( R A ) In order to show b) it suffices to prove that the complementary subcomplexes ker(1 + K,,) and G ker(ns - ~) r in X . ( R A ) are contractible. The differentials on the subcomplex ker(1 + ~ ) are given by 51~:e,-(l+~) = - n b s + d o n l ~ 2n fl[ker(l+a~) = bs o n ~ ~ A contracting nullhomotopy is provided by h = Gd on f W ' --~fGd o n ~ 2n+l (The notations are those of [CQ]). The differentials on k e r ( ~ - ~); ~ r +1 equal 5lk,~(~_ O = 0 o n ~ ~v /31k~,,(~,_r = b~ - (1 + ~ ) d o n ~ ~ A contracting nullhomotopy is provided in this case by h = G ( d - (1 + r The claim c) is clear because N d = B on ker(1 - ns) 2 by [CQ].

F(A) (A- l~)-1d)~) (#- x)-1dl~ +~/ f(A) ~ ,g(/~)(/~-A) -~dp (A-x)-ldA 1/; = 27ri , g(P)f(#) (it - x)-ld# = (fg)(x) as the second integral vanishes because A belongs to the unbounded component of (I;- F'. 17: If A, B are admissible Fr6chet algebras, then so is A | Proof: Let { 1I - ltk, k E tW } ( { I1 - lift, l e ~W}) be sequences of seminorms defining the topologies of A, B . We may suppose that II - Ilk' -> II - Ilk (11 - I1', -> II - II't) for k' > k (Z' _> l). 13 a) for nullsequences of elements of the algebraic tensor product A | B C A N,~ B .

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Asymptotic Cyclic Cohomology by Michael Puschnigg


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