Counterexamples to Baum–Connes and Kadison–Kaplansky

Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C^∗-algebra, disproving the Kadison–Kaplansky conjecture.