A bivariant Chern-Connes character
type: Item
instance of: scholarly article
title: A bivariant Chern-Connes character
zbMATH Open document ID: 0798.46051
DOI: 10.2307/2946556
author: Victor Nistor
published in: Annals of Mathematics. Second Series
publication date: 1994-10-23T00:00:00Z
review text: The study begun in the author's paper ``A bivariant Chern character for \(p\)-summable quasihomomorphisms'' [\(K\)-theory 5, No. 3, 193-211 (1991; Zbl 0784.19001)] where a character is defined with values in the Jones- Kassel bivariant cyclic group is continued here and a new character with values in Connes' bivariant cyclic group is defined. Denoting by \({\mathcal E}_ p^ i (A,B)\) the set of \(p\)-summable cycles of Kasparov's theory, by \(\Lambda\) the Connes' cyclic category and by \(A^ \#\) the cyclic vector space associated to any unital algebra \(A\) in \textit{A. Connes'} paper ``Cohomology cyclique et foncteurs \(\text{Ext}^ n\)'' [C. R. Acad. Sci., Paris, Sér. I 296, 953-958 (1983; Zbl 0534.18009)], this character is given by a map \[ \text{ch}_ i^{2n+i}: {\mathcal E}_ p^ i(A,B)\to \text{Ext}_ \Lambda^{2n+i} (A^ \#, B^ \#) \] defined for \((2-i)n\geq p-1\) which is additive, vanishes on degenerates, is compatible with the periodicity morphism \(S\), is homotopy invariant, functorial in both variables, compatible with the external Kasparov product and, for \(B=\mathbb{C}\) coincides with Connes' character. For this aim some new operations on Connes' bivariant cyclic groups, which were required by the necessity of extending the definition of \(S\) to this new setting are defined and the natural transformation from Connes' theory to Jones-Kassel bivariant cyclic groups is studied. For some low-dimensional cases of the character some explicit formulae are proved. Finally, related to the compatibility with \(K\)-theory, the following index theorem, the main result of the paper, is given: If \(A\) and \(B\) are smooth subalgebras of \(C^*\)-algebras and \(\varphi\) is a cyclic cocycle on \(B\), then the following result holds true: \[ \varphi_ * (e\otimes_ A x)= k(\varphi\circ \text{ch}_ i^{2n+i}(x))_ *(e) \] (\(k=1\) unless \(i=1\) and \(j=0\), in which case \(k= (2\pi i)^{-1})\).
Mathematics Subject Classification ID: 46L80
zbMATH DE Number: 510248
zbMATH Keywords: character with values in the Jones-Kassel bivariant cyclic group
zbMATH Keywords: Kasparov's theory
zbMATH Keywords: Connes' cyclic category
zbMATH Keywords: periodicity morphism
zbMATH Keywords: homotopy invariant
zbMATH Keywords: Kasparov product
zbMATH Keywords: index theorem
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946556
OpenAlex ID: W2329029502
description: scientific article; zbMATH DE number 510248