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

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.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Kaplansky's quasitrace problem: is every 2-quasitrace on a unital C*-algebra a trace? · Kaplansky's quasitrace problem (QT = T)

Confidence: Not scored

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

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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.