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An irrational-trace counterexample to reduced Baum–Connes
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Abstract
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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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.
Formalization
- franzhusch, Baum–Connes conjecture (formal-conjectures issue #2165) (google-deepmind/formal-conjectures, opened 2026-02-04). An open proposal to formalize the statement of the Baum–Connes conjecture, labelled "needs-prerequisites"; it predates and does not mention this paper. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Łukasz Grabowski, On Turing dynamical systems and the Atiyah problem (arXiv:1004.2030, Invent. Math. 198 (2014)). Shows that every non-negative real is an l²-Betti number, using lamplighter-type groups; this is relevant to the paper's survey of irrational kernel dimensions in group von Neumann algebras, which cites Grabowski's 2016 paper but not this one.
- Paolo Antonini, Sara Azzali, Georges Skandalis, The Baum–Connes conjecture localised at the unit element of a discrete group (arXiv:1807.05892, Compositio Mathematica). Builds an assembly map localised at the canonical trace; this may bear on the paper's use of the trace on the assembly image (Lück's theorem) to place [b_0] outside that image.
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) and notes that the three papers share one architecture, so a defect could affect the whole family. [family-level]
- OpenAI Math Digest, An irrational-trace counterexample to reduced Baum–Connes (online guide, undated; accessed 2026-10-21). The guide diagrams the proposed chain from code-defined local distributions and a global spectral gap to an irrational-trace projection and the assembly obstruction. [AI-generated per source]
- Open Mathematics Courses, A projection of irrational trace and the Baum–Connes conjecture (online course, undated; accessed 2026-10-21). This seven-lesson exposition, attributed to Claude Opus 5.5 and marked as not independently reviewed, covers graphical geometry, lamp-module codes, the spectral gap and the trace calculation. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 285
Independent sites; their content is not reviewed here.
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}
}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
- A torsion-free counterexample to the Kadison–Kaplansky projection conjecture · September 23, 2026