Browse › Operator algebras › 285 Counterexamples to Baum–Connes and Kadison–Kaplansky
A torsion-free counterexample to the Kadison–Kaplansky projection conjecture
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- Michael Puschnigg, The Kadison–Kaplansky conjecture for word-hyperbolic groups (Inventiones Mathematicae, 2002; listed on the author's page). Proves the projection conjecture for torsion-free hyperbolic groups by local cyclic homology, independently of the Mineyev–Yu route; this is relevant to the paper's "Context" paragraph on known cases, which cites only the Baum–Connes route.
- Paolo Antonini, Sara Azzali, Georges Skandalis, The Baum–Connes conjecture localised at the unit element of a discrete group (arXiv:1807.05892, Compositio Mathematica). Localises assembly at the group trace; this may bear on the paper's claim that the small trace of the matrix projection [p_0] places it outside the reduced assembly image for G_0.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, math.HO, 2026-10-12). Section 9.4.2 treats family 285, centred on the injectivity paper, mentions this companion's trace-(0,1/2) projection, and notes that the three papers share one architecture, so a defect could affect the whole family. [family-level]
- OpenAI Math Digest, A torsion-free counterexample to the Kadison–Kaplansky projection conjecture (online guide, undated; accessed 2026-10-21). Diagrams and source references organize the claimed small-trace spectral-band construction and the separate passage from a matrix projection to a scalar projection in a free product. [AI-generated per source]
- Open Mathematics Courses, The Kadison–Kaplansky conjecture and scalar compression (online course lesson, October 2026). This lesson, attributed to Claude Opus 5.5 and self-checked by the writing AI, expands the free-product compression argument and links to the remaining construction. [AI-generated per source]
News and blogs
- Anoushka Nagle and Haley Rodriguez, Dartmouth mathematicians weigh artificial intelligence’s role in the future of their work (The Dartmouth, 2026-10-20). The article discusses the three counterexample manuscripts through an interview with Erik van Erp, who reports stopping after the introductions because he found the papers difficult to read; this is not a proof assessment. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 285
Independent sites; their content is not reviewed here.
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}
}Family 285: 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.
Other papers in this family:
- A torsion-free counterexample to reduced Baum–Connes injectivity · September 23, 2026
- An irrational-trace counterexample to reduced Baum–Connes · September 23, 2026