Result 285, Operator algebras

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.

Disproof or counterexample

The bigger picture

Why it matters

Can a group's topology fully account for the operator algebra built from its symmetries? These manuscripts claim counterexamples to that expected correspondence, including for torsion-free groups, where no nonidentity element returns to the identity after finitely many repetitions.

What changes?

The reduced group C*-algebra packages a group's regular action as operators; assembly maps topological data to algebraic K-theory. One manuscript constructs a finitely generated torsion-free discrete group with an infinite-order kernel class in degree zero, defeating rational injectivity of coefficient-free reduced assembly. Another constructs a finitely generated torsion-free discrete group with a projection other than zero or the identity. A third constructs a finitely generated discrete group with an irrational-trace projection whose K0-class is outside assembly's image.

What does that help mathematicians do?

The infinite-order kernel class means information loss persists even after passing to rational K-theory. Separately, the irrational-trace projection supplies a specific K-theory class that assembly cannot reach. A projection keeps one subspace and sends its orthogonal complement to zero; its existence in the torsion-free example would refute the Kadison–Kaplansky prediction. These are distinct obstructions, not claims that one group exhibits every failure.

Are there practical applications?

The immediate value is foundational: these claimed examples identify limits on using group topology to determine operator-algebra invariants. They would require researchers to justify additional assumptions before relying on assembly as a complete description, or on torsion-freeness to exclude nontrivial projections. This is guidance for mathematical theory, not a demonstrated computational application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

A torsion-free counterexample to reduced Baum–Connes injectivity

September 23, 2026 85 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

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

September 23, 2026 76 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

An irrational-trace counterexample to reduced Baum–Connes

September 23, 2026 23 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}