Counterexamples to Baum–Connes and Kadison–Kaplansky

What OpenAI claims

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.

Papers (3)

1. A torsion-free counterexample to reduced Baum–Connes injectivity

   23 September 2026 · Preprint, not peer reviewed · PDF

   We disprove rational injectivity of the coefficient-free reduced Baum–Connes assembly map for torsion-free groups. Specifically, we construct a finitely generated torsion-free discrete group whose degree-zero assembly map has a kernel class of infinite order.

2. A torsion-free counterexample to the Kadison–Kaplansky projection conjecture

   23 September 2026 · Preprint, not peer reviewed · PDF

   We construct a finitely generated torsion-free discrete group whose reduced group C∗-algebra contains a projection other than zero and the identity. This disproves the Kadison–Kaplansky projection conjecture.

3. An irrational-trace counterexample to reduced Baum–Connes

   23 September 2026 · Preprint, not peer reviewed · PDF

   We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.