∑math20

Operator algebras · Result 285

Counterexamples to Baum–Connes and Kadison–Kaplansky

3 papers

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 3

First page of A torsion-free counterexample to reduced Baum–Connes injectivity

A torsion-free counterexample to reduced Baum–Connes injectivity

OpenAI · 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.

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Cite
@misc{OAI:A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026,
  author = {{OpenAI}},
  title = {{A torsion-free counterexample to reduced Baum--Connes injectivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf}{OAI:A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026}},
  year = {2026}
}
First page of A torsion-free counterexample to the Kadison–Kaplansky projection conjecture

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

OpenAI · 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.

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Cite
@misc{OAI:A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A torsion-free counterexample to the Kadison--Kaplansky projection conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026/paper.pdf}{OAI:A-Torsion-Free-Counterexample-to-the-Kadison-Kaplansky-Projection-Conjecture-September-23-2026}},
  year = {2026}
}
First page of An irrational-trace counterexample to reduced Baum–Connes

An irrational-trace counterexample to reduced Baum–Connes

OpenAI · 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.

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Cite
@misc{OAI:An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026,
  author = {{OpenAI}},
  title = {{An irrational-trace counterexample to reduced Baum--Connes}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026/paper.pdf}{OAI:An-Irrational-Trace-Counterexample-to-Reduced-Baum-Connes-September-23-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.