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Also, (P ⊥ A, b + B) is contractible and A retracts on P A for the differential (b+B). Let c : an A → A be the bornological vector space isomorphism c(a0 da1 . . dan ) = (−)[n/2] [n/2]! a0 da1 . . dan ∀n ∈ N . (11) Bivariant Chern Character for Families of Spectral Triples 53 Then c maps isomorphically P an A onto P A, and under this correspondence, the boundaries ( d, b) and b + B coincide: c−1 (b + B)c = ( d, b) on P an A. It follows that the X-complex X(T A) is homotopy equivalent to the (b + B)-complex of entire chains A.
In the context of C ∗ -algebras, such objects correspond to the unbounded version of Kasparov’s bivariant K-theory . In this picture, an element of the group KK(A, B) is represented by a triple (E, ρ, D), where E is an Hilbert B-module. D should be viewed as a family of Dirac operators over B, acting by unbounded endomorphisms on E, and ρ is a representation of A as bounded endomorphisms of E commuting with D modulo bounded endomorphisms. In the particular case B = C, this description just reduces to spectral triples over A.
In Sects. 2 and 3 we recall the basic definitions and properties of bornological spaces and entire cyclic cohomology. In Sect. 4 we present our construction of the Goodwillie equivalence [γ ] ∈ H E0 (A, T A). The semigroup of unbounded bimodules ∗ (A, B) is introduced in Sect. 5. Sections 6 and 7 are devoted 48 D. Perrot to the fundamental construction of the element χ ∈ H E∗ (T A, B). Finally, we end the paper with an application of our Chern character to the non-bivariant cases, namely ordinary K-theory and K-homology in Sect.