Projections in reduced group C*-algebras with prescribed trace values in (0,1)

Building on OpenAI's preprint, we construct, for every real number $r\in(0,1)$, a finitely generated discrete group $G_r$ and a projection $e_r\in C_r^*(G_r)$ whose canonical trace is $r$. The construction encodes the geometry of expanders in the dynamics of a semidirect product of groups. For irrational $r$, the $K$-theory class $[e_r]$ is not in the image of the reduced Baum--Connes map. The groups $G_r$ have torsion and satisfy the maximal strong Novikov conjecture.