A bivariant Chern-Connes character and the higher \(\Gamma\)-index theorem
Let \((H,D)\) be an unbounded Fredholm module over the \(C^*\)-algebra \(C\) which is \(\Theta\)-summable over the dense subalgebra \({\mathcal C}\). Then \((H,D)\) possesses a Chern-Connes character in entire cyclic cohomology \[ ch(H,D) \in HC^*_\varepsilon ({\mathcal C}) \] which is given by a canonical cocycle, the Jaffe-Lesniewski-Osterwalder JLO-cocycle. In the present paper this is generalised to a bivariant setting as follows. If \(B\) is another \(C^*\)-algebra, then \((H \otimes B,D \otimes 1)\) defines an unbounded Kasparov bimodule over \(A: =B \otimes_{\min} C\). The author constructs then a bivariant JLO-cocycle which defines a bivariant Chern-Connes character \[ ch_{biv} (H\otimes {\mathcal B}, D\otimes 1): HC^\varepsilon_* ({\mathcal A}) \to H_*^{dR} ({\mathcal B}) \] from the entire cyclic homology of a suitable dense subalgebra \({\mathcal A}\subset A\) to the noncommutative de Rham homology of a subalgebra \({\mathcal B} \subset B\). Let \(e\in M_k ({\mathcal A})\) be an idempotent such that the operator \[ D_e: =e(D \otimes 1)e \in \text{End} \bigl(e (H\otimes {\mathcal B}) \bigr) \] is \({\mathcal B}\)-Fredholm. Then the bivariant Chern-Connes character is compatible with the index pairing in the sense that \[ ch^{dR} \bigl( \text{Ind} (D_e)\bigr) =ch_{biv} (H\otimes {\mathcal B}) \bigl(ch(e) \bigr) \] in \(H^{dR}_*({\mathcal B})\). This follows from a noncommutative version of the MacKean-Singer formula which is also used to establish a new proof of the \(\Gamma\)-index theorem of Connes and Moscovici. Finally, the author studies transgression formulas for the bivariant JLO-cocycle to define higher \(\eta\)-cochains for the Dirac operator on infinite coverings of spin-manifolds with boundary.
scientific article; zbMATH DE number 992026