Kaplansky's quasitrace conjecture
From OpenAI Aftermath
Kaplansky's quasitrace conjecture asserts that every normalized 2-quasitrace on a unital C*-algebra is a trace, that is, that a functional which is positive, satisfies τ(x*x) = τ(xx*), is linear on commutative subalgebras and extends to 2 × 2 matrices must be linear. It is equivalent to Kaplansky's question whether every AW*-factor of type II1 is a von Neumann algebra. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a counterexample, with consequences for the stable finiteness of tensor products, and a Lean formalization of the main statements. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Kaplansky introduced AW*-algebras in 1951 to characterize W*-algebras by intrinsic algebraic conditions. Every type I AW*-factor is a W*-algebra and type III counterexamples exist; the type II1 case remained open. Its equivalence with the quasitrace problem is recorded by Milhøj and Rørdam and in recent work of Gow, which adds further equivalent formulations.
- Cuntz developed dimension functions for simple algebras, Blackadar and Handelman related 2-quasitraces to lower semicontinuous dimension functions, and Blackadar and Rørdam developed the extension of states underlying the general existence criterion: a unital algebra has a normalized 2-quasitrace whenever no matrix algebra over it is properly infinite.
- Haagerup proved that every 2-quasitrace on a unital exact C*-algebra is a trace (first draft 1991, published 2014). Kirchberg's unpublished 2006 example of a 1-quasitrace that is not a 2-quasitrace lacks the matrix extension.
- Milhøj and Rørdam asked whether the minimal tensor product of two unital simple stably finite C*-algebras must be stably finite, showed that the question is equivalent to the quasitrace problem, and proposed a route to a counterexample.
The original result
- Claim. Theorem 1.1 of A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness states that there are a separable unital complex C*-algebra A admitting a normalized 2-quasitrace and positive contractions a, b in A such that every normalized 2-quasitrace τ on A satisfies τ(a + b) − τ(a) − τ(b) ≥ 1/144. Corollary 1.2 states that there are two unital simple stably finite C*-algebras whose minimal tensor product is properly infinite, one of which can be the reduced group C*-algebra of the free group on two generators; a separable algebra with a normalized 2-quasitrace whose tensor product with that algebra admits none; and a separable unital stably finite C*-algebra with no tracial state.
- Method, as described: the algebra is generated by 24 elements with ∑ xi*xi = 1 and ∑ xixi* ≤ (2/3)·1, which rules out tracial states by Haagerup's criterion. The generators are shifts between finite-rank vector bundles over compact bases, with one test for each matrix size and polynomial degree; a Chern-class obstruction forbids the trivial subbundle that a witness of proper infiniteness of a matrix algebra would produce, so the Cuntz–Blackadar–Handelman theorem supplies a normalized 2-quasitrace.
- Sources. The manuscript A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness at commit
1581d4c. - Formalization, as described by the repository. The scope note states that the formalization constructs a separable C*-algebra with normalized 2-quasitraces and fixed positive contractions a, b for which every such quasitrace satisfies Re(τ(a + b) − τ(a) − τ(b)) ≥ 1/144, and that it also gives the stable-finiteness consequences named in the title. The Comparator statements are
KaplanskyQuasitrace.leanandKaplanskyStableFiniteness.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 21 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statement) | Rates the family "full": both parts of the summary are stated. Lists among the definitions to check the hand-written quasitraces, with a 2-quasitrace defined as one that extends to the 2 × 2 matrices over the algebra, the construction of the reduced free-group C*-algebra, and proper infiniteness; records sandboxed Comparator passes for KaplanskyQuasitrace and KaplanskyStableFiniteness. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K: runs 38019899278 and 38051470892 | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the KaplanskyQuasitrace solution, in 751 s, and the KaplanskyStableFiniteness solution, in 855 s. |
Repository. |
| 2026-10-10 to 2026-10-14 | con-leche development journal (task #348) | Verification (second kernel) | In its run over all Comparator challenges of the collection at commit fd4aeeb, the journal's table of exceptions (time-outs, memory, errors) lists neither KaplanskyQuasitrace nor KaplanskyStableFiniteness, so both were accepted by con-leche and by Lean's official kernel. |
Development journal. |
| 2026-10-12 | J. Arulseelan, Model theory of quasitracial AW*-algebras and the absoluteness of monotone completeness (arXiv:2610.17435) | Remark (related framework) | Develops a model theory for AW*-algebras with a faithful normalized quasitrace and shows that Kaplansky's type II1 question and the finite case of the monotone completeness conjecture are each equivalent in ZFC to a Π02 sentence of arithmetic. A closing paragraph of the introduction notes that OpenAI has since released a manuscript proposing a counterexample to the quasitrace conjecture, and that joint work with Hanson using this paper's machinery has produced a proposed counterexample to the monotone completeness conjecture. | arXiv preprint. The AI disclosure states that errors in a previous version were found by GPT 6 Astra. |
| 2026-10-21 (listed) | J. Arulseelan, J. E. Hanson, Counterexamples to the Monotone Completeness Conjecture (arXiv:2610.27184) | Extension (monotone completeness); independent reproduction | Modifies the manuscript's bundle construction to obtain a finite AW*-factor with canonical quasitrace τ and elements y1, y2, y3 with ∑ yi*yi = 1 and ∑ yiyi* ≤ (1/2)·1, and deduces a type II1 AW*-factor, which can be taken separable in the quasitrace metric, that is not monotone sequentially complete. The authors state that they fully reproduce the disproof of the quasitrace conjecture with a modified construction, using rank stratification and parametric transversality in place of the manuscript's incidence and covering estimates, and that the properly infinite case of the monotone completeness conjecture remains open. | arXiv preprint. The AI disclosure states that the proof idea was the authors', that GPT 6 Astra produced a preliminary write-up which the authors substantially reworked and corrected, and that the bundle construction adapts one from the OpenAI manuscript. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- An earlier preprint claiming a positive answer to Kaplansky's question, arXiv:2501.13088 (January 2025), was withdrawn by its author, who reported a gap; the replacement arXiv:2601.04431 establishes equivalent formulations of the problem rather than an answer.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
