An irrational-trace counterexample to reduced Baum–Connes

We construct a finitely generated discrete group whose reduced group C${}^{\ast }$-algebra contains a projection of irrational canonical trace. Its K${}_0$-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.