The Chern-Connes character is not rationally injective

TL;DR

This paper demonstrates that the Chern-Connes character from Kasparov's K-theory to local cyclic cohomology is not always rationally injective, using counterexamples from hyperbolic groups with property (T).

Contribution

It provides the first known counterexamples to the rational injectivity of the Chern-Connes character, leveraging recent advances in K-theory and the Baum-Connes conjecture.

Findings

Counterexamples from hyperbolic groups with property (T)

The Chern-Connes character is not always rationally injective

Utilizes recent results on K-nuclearity and Baum-Connes conjecture

Abstract

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.