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

### MSC:

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

### Citations:

Zbl 0784.19001; Zbl 0534.18009
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