A bivariant Chern character for \(p\)-summable quasihomomorphisms
type: Item
instance of: scholarly article
title: A bivariant Chern character for \(p\)-summable quasihomomorphisms
zbMATH Open document ID: 0784.19001
DOI: 10.1007/BF00533587
author: Victor Nistor
published in: \(K\)-Theory
publication date: 1992-09-26T00:00:00Z
review text: Bivariant \(K\)-theory (\(KK\)-theory) has become by now the natural framework for generalized index theorems of Atiyah-Singer type. Similarly, the cohomological versions of such index theorems should find their natural framework in bivariant cyclic cohomology theory. The translation of index theory from the \(K\)-theoretic to the cohomological picture should be provided by a bivariant Chern character. One only can hope to construct such a character for a ``smooth'' version of bivariant \(K\)-theory (considering ``smooth'', holomorphically closed, dense subalgebras of \(C^\ast\)-algebras rather than \(C^\ast\)-algebras themselves is necessitated by the pathological behaviour of cyclic theories for \(C^\ast\)-algebras). The author achieves an important step towards a realization of this program: The construction of a natural homomorphism \[ ch_{\mathrm{biv}}: KK^\ast_{\mathrm{smooth}}(A,B)\to PHC^\ast(A^ \infty, B^ \infty) \] which yields the commutative diagram \[ \begin{tikzcd} K_i(A^\infty)\simeq K_i(A) \ar[r,"\otimes_A(x)"]\ar[d,"ch"'] & K_i(B)\simeq K_i(B^\infty)\ar[d,"ch"]\\ PHC^\ast(A^\infty) \ar[r,"ch_{\mathrm{biv}}(x)"'] & PHC^\ast(B^\infty)\end{tikzcd} \] The smooth bivariant \(K\)-groups \(KK^\ast_{smooth}(A,B)\) are described in terms of finitely (\(p\)-)summable quasihomomorphisms à la Cuntz. These are given by a pair of homomorphisms \[ (\varphi_ 0,\varphi_ 1): A^ \infty\to {\mathcal B}({\mathcal H})\otimes_ \pi B^ \infty \] of smooth subalgebras \(A^ \infty\subset A\), \(B^ \infty\subset B\) such that \(\varphi_ 0-\varphi_ 1\) maps \(A^ \infty\) to \(l^ p({\mathcal H})\otimes_ \pi B^ \infty\) with \(l^ p({\mathcal H})\), \(1\leq p<\infty\), the \(p\)th Schattenideal in \({\mathcal B}({\mathcal H})\). Equivalently, using the exact sequence \[ 0\to qA \to QA\simeq A * A@>id * id>> A\to 0 \] a \(p\)- summable quasihomomorphism is described by a homomorphism of pairs \[ \varphi= \varphi_ 0 *\varphi_ 1: (QA^ \infty,qA^ \infty)\to \bigl({\mathcal B}({\mathca…
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 54869
zbMATH Keywords: smooth bivariant \(K\)-theory
zbMATH Keywords: bivariant cyclic cohomology
zbMATH Keywords: bivariant excision free Chern character
zbMATH Keywords: smooth bivariant \(K\)-groups
zbMATH Keywords: \(p\)-summable quasihomomorphism
zbMATH Keywords: chain homotopy
zbMATH Keywords: cyclic bicomplex
reviewed by: Michael Puschnigg
MaRDI profile type: Publication
cites work: The index of elliptic operators. IV, V
cites work: Q3730469
cites work: Q3317281
cites work: Non-commutative differential geometry
cites work: Quasi homomorphismes, cohomologie cyclique et positivité. (Quasi homomorphisms, cyclic cohomology and positivity)
cites work: A new look at KK-theory
cites work: Q3470615
cites work: Q5660202
cites work: Q5588893
cites work: Cyclic homology, derivations, and the free loopspace
cites work: Bivariant cyclic theory
cites work: Equivariant KK-theory and the Novikov conjecture
cites work: Cyclic homology, comodules, and mixed complexes
cites work: Caractère de Chern bivariant. (Bivariant Chern character)
cites work: Cyclic homology and the Lie algebra homology of matrices
cites work: Group cohomology and the cyclic cohomology of crossed products
cites work: Cyclic cohomology and algebra extensions
cites work: Q3857325
cites work: Q3667036
cites work: Excision in cyclic homology and in rational algebraic K-theory
description: scientific article; zbMATH DE number 54869