The Chern-Connes character is not rationally injective

We show that the Chern-Connes character from Kasparov’s bivariant K-theory to bivariant local cyclic cohomology is not always rationally injective. Counterexamples are provided by the reduced group $C^*$-algebras of word-hyperbolic groups with Kazhdan’s property (T). The proof makes essential use of Skandalis’ work on K-nuclearity and of Lafforgue’s recent demonstration of the Baum-Connes conjecture with coefficients for word-hyperbolic groups.

💡 Research Summary

The paper investigates the injectivity properties of the Chern‑Connes character, a natural transformation from Kasparov’s bivariant K‑theory (KK) to bivariant local cyclic cohomology (HC_loc). While the classical Chern character provides a rational isomorphism between topological K‑theory and rational cohomology, its non‑commutative analogue, the Chern‑Connes character ch_biv : KK(A,B) → HC_loc(A,B), is known to be multiplicative, natural, and compatible with Kasparov products. For C*‑algebras that are KK‑equivalent to commutative ones, ch_biv is rationally injective; however, the general situation remained open.

The author produces counterexamples by exploiting two deep results. First, Skandalis showed that for a word‑hyperbolic group Γ with Kazhdan’s property (T), the reduced group C*‑algebra C*_rΓ is not KK‑equivalent to any nuclear (hence not to any commutative) C*‑algebra. In Kasparov’s framework there exists a distinguished “Gamma‑element” γ ∈ KK_Γ(ℂ,ℂ) which is idempotent under the Kasparov product. Descending γ to the reduced algebra yields an element jr(γ) ∈ KK(C*_rΓ, C*_rΓ). Skandalis proved that jr(γ) ≠ 1 in KK(C*_rΓ, C*_rΓ)⊗ℚ, i.e. it is a non‑trivial rational class in KK‑theory.

The second ingredient is Lafforgue’s proof of the Baum‑Connes conjecture with coefficients for all word‑hyperbolic groups. Lafforgue constructs a family of Fredholm representations (E_{s,t}) of the pair (ℓℂ, Γ) over the interval