TITLE : The trace conjecture - A counterexample

Following Baum and Connes well known Isomorphism conjecture is their Trace conjecture:

If $\Gamma$ is a discrete group with torsion and $C^{*}\Gamma$ the reduced C*- algebra of $\Gamma$, the trace map $tr : K_{0}(C^{*}\Gamma ) \longrightarrow \ {\bf R}$ maps $K_{0}(C^{*}\Gamma )$ onto the additive subgroup of $\mathbb{Q}$ generated by all rational numbers of the form $\frac{1}{n}$, where $n$ is the order of a finite subgroup of $\Gamma$.

We construct a counterexample to this conjecture using asphericalization techniques developed by Davis-Januszkiewicz.