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Operator algebras · Result 294

Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness

1 paperLean

Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C∗-algebras can have a properly infinite minimal tensor product, with one factor \(C_r^*(\mathbb F_2)\).

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First page of A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness

A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness

OpenAI · September 23, 2026

We refute Kaplansky's quasitrace conjecture by constructing a separable unital complex C∗-algebra that admits normalized 2-quasitraces but has no tracial state. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C∗-algebras can be properly infinite, even when one factor is the reduced free-group algebra \(C_r^*(\mathbb F_2)\).

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@misc{OAI:A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Kaplansky's quasitrace conjecture predicts that every \(2\)-quasitrace on a \(C^*\)-algebra is a trace. The formalization constructs a separable \(C^*\)-algebra with normalized \(2\)-quasitraces and fixed positive contractions \(a,b\) for which every such quasitrace satisfies \(\mathrm{Re}(\tau(a+b)-\tau(a)-\tau(b))\ge1/144\). Thus none is additive on this pair.

The formalization also gives simple stably finite \(C^*\)-algebras whose spatial tensor product is properly infinite, a separable stably finite algebra with no tracial state, and a tensor-product example in which normalized \(2\)-quasitraces are lost. These are the stable-finiteness consequences named in the paper's title.

Comparator links

ResultComparator statement
Nonadditive normalized \(2\)-quasitracesKaplanskyQuasitrace.lean
Tensor-product failure of stable finiteness and quasitrace consequencesKaplanskyStableFiniteness.lean

Scope note on GitHub →