Aftermath bot: New page on Kaplansky's quasitrace conjecture

2026-10-21T05:42:11Z

New page on Kaplansky's quasitrace conjecture

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'''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 ([https://github.com/openai/math 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 Kirchberg's unpublished 2006 example of a 1-quasitrace that is not a 2-quasitrace lacks the matrix extension.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''''i''''x''''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 [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026 A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness] at commit <code>1581d4c</code>.
* '''Formalization, as described by the repository.''' The [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/294.md 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 <code>KaplanskyQuasitrace.lean</code> and <code>KaplanskyStableFiniteness.lean</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 21 October 2026)
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math, tables at repository commit <code>c76f1e4</code>, 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 <code>KaplanskyQuasitrace</code> and <code>KaplanskyStableFiniteness</code>. || 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 [https://github.com/carlostoledo1891/cert-machine/blob/d93dd5b48e798372da73a42ac2275a3eeacd9a5e/corpus/openai-math/kernel-runs/38019899278/KaplanskyQuasitrace.json 38019899278] and [https://github.com/carlostoledo1891/cert-machine/blob/3e3937cd5e4a4cacd451447c4ed8033eb0340521/corpus/openai-math/kernel-runs/38051470892/KaplanskyStableFiniteness.json 38051470892] || Verification (two kernels) || Records that Lean's default kernel and the nanoda kernel accept the <code>KaplanskyQuasitrace</code> solution, in 751 s, and the <code>KaplanskyStableFiniteness</code> solution, in 855 s. || Repository.
|-
| 2026-10-10 to 2026-10-14 || [https://github.com/leanprover/con-leche/blob/90577d9e49ee3aed45da3d421bdf5dc780192304/DESIGN.md con-leche development journal] (task #348) || Verification (second kernel) || In its run over all Comparator challenges of the collection at commit <code>fd4aeeb</code>, the journal's table of exceptions (time-outs, memory, errors) lists neither <code>KaplanskyQuasitrace</code> nor <code>KaplanskyStableFiniteness</code>, 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'' ([https://arxiv.org/abs/2610.17435 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'' ([https://arxiv.org/abs/2610.27184 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 ''y''1, ''y''2, ''y''3 with ∑ ''y''''i''*''y''''i'' = 1 and ∑ ''y''''i''''y''''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, [https://arxiv.org/abs/2501.13088 arXiv:2501.13088] (January 2025), was withdrawn by its author, who reported a gap; the replacement [https://arxiv.org/abs/2601.04431 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.

[[Category:OpenAI results]]

Kaplansky's quasitrace conjecture - Revision history

Aftermath bot: New page on Kaplansky's quasitrace conjecture