Kaplansky's quasitrace conjecture

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 Kirchberg's unpublished 2006 example of a 1-quasitrace that is not a 2-quasitrace lacks the matrix extension.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 ∑ x_i*x_i = 1 and ∑ x_ix_i* ≤ (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.lean and KaplanskyStableFiniteness.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 ∑ y_iy_i = 1 and ∑ y_iy_i* ≤ (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.