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∈(0,1), a finitely generated discrete group Gᵣ and a projection eᵣ∈ Cᵣ^*(Gᵣ) 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 [eᵣ] is not in the image of the reduced Baum--Connes map. The groups Gᵣ have torsion and satisfy the maximal strong Novikov conjecture.