Family 285Operator algebrasNo LeanNot yet digested
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 in this family · 3
A torsion-free counterexample to reduced Baum–Connes injectivity
Sep 23, 2026·paper.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.
A torsion-free counterexample to the Kadison–Kaplansky projection conjecture
Sep 23, 2026·paper.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.
An irrational-trace counterexample to reduced Baum–Connes
Sep 23, 2026·paper.pdf
We construct a finitely generated discrete group whose reduced group C-algebra contains a projection of irrational canonical trace. Its K-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.