Kaplansky's quasitrace problem: is every 2-quasitrace on a unital C*-algebra a trace?

A 1-quasitrace on a unital $C^*$ -algebra $A$ is a function $\tau:A\to\mathbb C$ with $\tau(x^*x)=\tau(xx^*)\ge0$ , $\tau(h+ik)=\tau(h)+i\tau(k)$ for self-adjoint $h,k$ , and linear on every abelian $C^*$ -subalgebra; it is a 2-quasitrace if it extends to such a function on $M_2(A)$ . Blackadar and Handelman showed 2-quasitraces correspond to dimension functions, and the question whether every one is linear is equivalent to Kaplansky's 1951 question whether every $AW^*$ -factor of type II $_1$ is a von Neumann algebra. Haagerup proved it for exact algebras, and Milhoj and Rordam showed it is equivalent to asking whether the minimal tensor product of two unital simple stably finite $C^*$ -algebras is stably finite. Is every normalized 2-quasitrace on every unital $C^*$ -algebra a tracial state?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Operator algebras; traces and quasitraces on C*-algebras
Posed by: Irving Kaplansky (1951, the type II1 AW*-factor question, per Gow's account); quasitrace form via Blackadar and Handelman (J. Funct. Anal. 1982); tensor form Milhoj and Rordam (2023, Question 2.5)
Year posed: 1951
Years open: 75y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 55 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: there are a separable unital $C^*$ -algebra $A=M_{48}(B)$ with a normalized 2-quasitrace and positive contractions $a,b$ with $\tau(a+b)-\tau(a)-\tau(b)\ge1/144$ for every normalized 2-quasitrace $\tau$ , so none is a trace. $B$ is generated by 24 shifts between vector bundles with $\sum x_i^*x_i=1$ and $\sum x_ix_i^*\le\frac23$ , Chern-class obstructions ruling out proper infiniteness in every matrix size. Hence a type II $_1$ $AW^*$ -factor that is not a von Neumann algebra exists. Corollary: two unital simple stably finite algebras, one $C_r^*(\mathbb F_2)$ , with properly infinite minimal tensor product (answering Milhoj-Rordam Question 2.5), and a separable stably finite algebra with no tracial state. Not shown: a simple or exact example (exact is impossible by Haagerup), or an explicit II $_1$ factor.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, using on average about three hours of ChatGPT Pro thinking compute per result, with outputs grouped into families and manuscripts. The README's two exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. Lean formalizations of the main theorem and of the stable-finiteness corollary are in the release's lean/ library (lean/docs/294.md). The release does not say how much human review happened before publication.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the TeX source against the posed problem, and both Lean statements. formalization.yaml lists ComparatorChallenges/KaplanskyQuasitrace.json, declaration OAI.Kaplansky.kaplansky_quasitrace_counterexample in OAI/Analysis/Kaplansky/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. Its statement, read here, defines 1-quasitraces with the three axioms above (linearity on closed commutative nonunital star-subalgebras), 2-quasitraces by extension to 2x2 matrices, and asserts a separable C*-algebra with a normalized 2-quasitrace and positive contractions a, b such that every normalized 2-quasitrace has Re(tau(a+b) - tau(a) - tau(b)) >= 1/144. That is the headline: no normalized 2-quasitrace there is additive. A second challenge, ComparatorChallenges/KaplanskyStableFiniteness.json (solution module OAI.Analysis.Kaplansky.StableFiniteness exists at the pinned commit), is not in the formalization catalogue; read here, it states Corollary 1.2 (i) to (iii) with the tensor product modelled concretely on l2(F2, H). Neither was rebuilt here. On paper, the AW*-factor reading and Corollary 1.2 (i) rest on the equivalences in Milhoj-Rordam and Gow's 2026 preprint; the constructed algebra is necessarily nonexact.

Kaplansky's quasitrace problem: is every 2-quasitrace on a unital C*-algebra a trace?

What was actually shown

What the AI did

Verification

Sources

Discussion