A bivariant Chern-Connes character. (English) Zbl 0798.46051

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 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 \(k= (2\pi i)^{-1})\).


46L80 \(K\)-theory and operator algebras (including cyclic theory)
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