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.

Manuscripts (3)

A torsion-free counterexample to reduced Baum–Connes injectivity

September 23, 2026 · source · cite as \cite{OAI:A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026}

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.

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

September 23, 2026 · source · cite as \cite{OAI:A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026}

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.

An irrational-trace counterexample to reduced Baum–Connes

September 23, 2026 · source · cite as \cite{OAI:An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026}

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.

Result 285 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.