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A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
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Abstract
We refute Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra that admits normalized 2-quasitraces but has no tracial state. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C∗-algebras can be properly infinite, even when one factor is the reduced free-group algebra $C_r^*(\mathbb F_2)$.
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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.
Follow-up works
- Jananan Arulseelan and James E. Hanson, Counterexamples to the Monotone Completeness Conjecture (arXiv:2610.27184, 19 Oct 2026). Modifies this paper's vector-bundle construction (called "the quasitrace manuscript") to get uniform failure of traciality, which together with model-theoretic reductions gives a counterexample to the monotone completeness conjecture. The abstract says the authors fully reproduce the quasitrace disproof with modified arguments, e.g. rank stratification and parametric transversality in place of the incidence-variety estimate.
- Jananan Arulseelan, Model theory of quasitracial AW*-algebras and the absoluteness of monotone completeness (arXiv:2610.17435, 12 Oct 2026). Develops a model theory of quasitracial AW*-algebras and shows that Kaplansky's II$_1$ AW*-factor problem is ZFC-equivalent to a $\Pi^0_2$ arithmetic sentence; a note added to the introduction says the OpenAI manuscript's proposed counterexample may settle one of its motivating questions.
Formalization
- OpenAI math release, Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness: Lean scope note (lean/docs/294.md, 2026-10-06). Describes the release's Lean formalization of the fixed-pair nonadditivity bound $\mathrm{Re}(\tau(a+b)-\tau(a)-\tau(b))\ge1/144$ and of the stable-finiteness and tensor-product consequences, with comparator statements
KaplanskyQuasitrace.leanandKaplanskyStableFiniteness.lean. [family-level]
Related prior literature mentioned by others
- Uffe Haagerup, Quasitraces on exact C*-algebras are traces (arXiv:1403.7653, 2014-03-29). The Latent Seminar exposition invokes this theorem to explain why the manuscript's quasitrace counterexample must be nonexact.
Possibly relevant literature (found by the agent, not checked)
- Narutaka Ozawa, The hyperfinite II_1 factor is not quasidiagonal (arXiv:2609.08892, 8 Sep 2026). Gives an MF C*-algebra whose spatial tensor product with the hyperfinite II$_1$ factor contains a proper isometry, so stable finiteness is not preserved under tensor products; this is related to Corollary 1.2. Its abstract says the result was obtained using OpenAI's ChatGPT Pro 6.0. [AI-generated per source]
- Henning Olai Milhøj, Quasitraces, Tracial States, and Kaplansky's Conjecture (PhD thesis, University of Copenhagen, defended 5 Jul 2022). Defines numerical invariants measuring the distance from having quasitraces or tracial states; background to the bounded-length strategy of Milhøj–Rørdam (Remark 4.7) that the paper adopts.
- Samuel Evington, Shanshan Hua, Christopher Schafhauser, Apurva Seth, Aaron Tikuisis and Stuart White, Regular simple nuclear C*-algebras (arXiv:2610.09390, 2026-10-07). Its K-stability result for simple pure C*-algebras assumes that quasitraces are traces, a hypothesis relevant to the manuscript's claimed failure of quasitrace additivity in general.
Digests and expositions
- Latent Seminar, Quasitraces: a uniform additivity defect and tensor finiteness (2026-10-07). Explains the scaling, Chern-class obstruction and telescoping argument, distinguishes the algebras in the three consequences, and states that it is not an independent verification of the complete proofs. [family-level]
- Claude Opus 5.5 / Open Mathematics Courses, Kaplansky's quasitrace problem (October 2026). Four lessons on the construction and additivity defect, including a claimed sharper constant, with named imported hypotheses and a statement that independent review is incomplete. [AI-generated per source]
- Gradient Damage, Kaplansky's Quasitrace Conjecture, Naimark's Problem and 7 More in Operator Algebras (2026-10-15). Video with a short illustrated introduction to the quasitrace claim at 1:04, distinguishing its animations from proofs.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 294
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026,
author = {{OpenAI}},
title = {{A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/294.md. The Lean statements have not been checked by people against the paper's claims.
Family 294: Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor $C_r^*(\mathbb F_2)$.
This is the only paper in this family.