Result 294, Operator algebras

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

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 Cr∗(F2)C_r^*(\mathbb F_2).

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Can a generalized notion of size exist even when no fully additive one does? This manuscript claims such an example for operator algebras, with a consequence: combining two finite algebras can produce an infinite one.

What changes?

The unreviewed manuscript reports a counterexample to Kaplansky's quasitrace conjecture: a separable unital complex C*-algebra, an algebra of bounded operators with an identity and a countable dense subset. It admits normalized 2-quasitraces, generalized size measurements assigning the identity size one, additive on commuting elements and compatible with two-by-two matrix extensions. Yet it has no tracial state, the fully additive version of such a measurement. Thus every normalized 2-quasitrace on this example is nonadditive, not merely one specially chosen quasitrace.

What does that help mathematicians do?

The reported consequence rules out preservation of stable finiteness under the minimal tensor product. Two unital simple stably finite complex C*-algebras can have a properly infinite product, even when one factor is the reduced free-group algebra on two generators. Simple means having no nontrivial closed two-sided ideals. Stable finiteness excludes proper equivalent copies of the identity inside itself at every matrix size; proper infiniteness allows two disjoint equivalent copies inside it. The construction therefore crosses a sharp structural boundary.

Are there practical applications?

Its immediate value is foundational. The minimal tensor product is a standard way to combine operator algebras, and this example shows that finiteness of both factors alone cannot ensure finiteness of their combination. Researchers seeking such preservation results must impose additional hypotheses. Likewise, quasitraces cannot universally substitute for additive traces.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

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

September 23, 2026 16 pages Main result formalized in Lean

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 Cr∗(F2)C_r^*(\mathbb F_2).

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/294.md.

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

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

Scope

Kaplansky's quasitrace conjecture predicts that every 22-quasitrace on a C∗C^*-algebra is a trace. The formalization constructs a separable C∗C^*-algebra with normalized 22-quasitraces and fixed positive contractions a,ba,b for which every such quasitrace satisfies Re(τ(a+b)−τ(a)−τ(b))≥1/144\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∗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 22-quasitraces are lost. These are the stable-finiteness consequences named in the paper's title.

Comparator links

Result Comparator statement
Nonadditive normalized 22-quasitraces KaplanskyQuasitrace.lean
Tensor-product failure of stable finiteness and quasitrace consequences KaplanskyStableFiniteness.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.