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

Proves Kaplansky's quasitrace conjecture false, and that stable finiteness can fail for tensor products.

1 paper September 23, 2026 Explained by AI

A trace is a way to measure the size of elements of an operator algebra that adds up correctly. A quasitrace keeps most of a trace's rules but only has to add up for elements that commute. Kaplansky's conjecture said that, with one extra matrix condition, every quasitrace is a trace. This paper builds an algebra where every such quasitrace fails to add up on one fixed pair, and Lean checks it. As a result, two well-behaved algebras can combine into one that is not well behaved.

First step
1951
Kaplansky, who introduced AW*-algebras and asked whether every type II₁ AW*-factor is a von Neumann algebra
Significance
Landmark
It refutes a decades-old conjecture tied to Kaplansky's AW*-factor question, with a Lean check.
Trust
Formal proof listed
Field
Operator algebras
1 paper

The explanation

Take 2-by-2 matrices and the normalized trace τ, half the sum of the diagonal entries. Let a be the matrix with rows (1, 0) and (0, 0). Let b have every entry 1/2. Then τ(a) = 1/2, τ(b) = 1/2, and τ(a + b) = 1/2 · (3/2 + 1/2) = 1. So τ(a + b) = τ(a) + τ(b), as it must for a trace. A trace also gives τ(xy) = τ(yx). For x with rows (0, 1) and (0, 0), the products x*x and xx* are the two diagonal projections, and both have trace 1/2.

A C*-algebra is a family of operators closed under sums, products, adjoints and limits. A trace measures the size of its elements, adds up, and gives τ(x*x) = τ(xx*). A quasitrace keeps positivity and that identity, but only has to add up on parts where everything commutes. Here a and b do not commute, so a quasitrace need not add them correctly. A 2-quasitrace must also extend to 2-by-2 matrices over the algebra. Kaplansky's conjecture said every normalized 2-quasitrace is a trace. Haagerup showed this holds for every exact algebra, a large well-behaved class. The general case stayed open.

The paper builds a separable algebra with normalized 2-quasitraces. For one fixed pair of positive elements, every such quasitrace overshoots: τ(a + b) − τ(a) − τ(b) is at least 1/144, about 0.0069. The algebra is generated by 24 elements. The products x*x of these generators add up to 1, while the other products xx* add up to at most 2/3 of 1. A trace would give both sums the same size, so it would give 1 ≤ 2/3. That is false, so the algebra has no trace. The obstacle was to still have quasitraces. By the Cuntz–Blackadar–Handelman theorem, it is enough that no matrix algebra over it is properly infinite. Properly infinite means it holds two copies of the identity side by side. The paper rules this out with vector bundles, families of vector spaces over a shape, and their Chern classes, which record how they twist.

The result also answers Kaplansky's question about type II₁ AW*-factors negatively. It also gives two unital simple stably finite algebras whose minimal tensor product is properly infinite. Stably finite means no matrix algebra over it has a one-sided isometry. One factor can be the reduced free-group algebra C*_r(F₂).

How sure is this? Lean, a computer proof checker, has formal statements for the nonadditive quasitraces with the 1/144 gap. It also covers the tensor-product and stable-finiteness consequences. So the paper proves these results, as far as the Lean statements match the paper. No specialist review is published.

What it doesn't show. This does not affect exact C*-algebras, where Haagerup showed every 2-quasitrace is a trace, and it does not make the main algebra simple.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Landmark-level claim with a Lean check of the main theorem and its consequences. It only concerns nonexact algebras, as Haagerup's theorem requires. No specialist review is published.

Words used

C*-algebra
A family of operators on a Hilbert space closed under sums, products, adjoints and limits.
Trace
A positive linear size function τ with τ(xy) = τ(yx) and value 1 on the identity.
Quasitrace
A size function that keeps positivity and τ(x*x) = τ(xx*) but only needs to add up on commuting parts.
2-quasitrace
A quasitrace that also extends to 2-by-2 matrices over the algebra.
Exact C*-algebra
A C*-algebra in a large, well-behaved class defined by how it interacts with tensor products.
Properly infinite
An algebra holding two isometries with orthogonal ranges, so it contains two side-by-side copies of its identity.
Stably finite
No matrix algebra over it contains an isometry that is not unitary.
Minimal tensor product
A standard way to combine two C*-algebras into one acting on a product space.

The original paper

Lean notes

All quoted lines (8)
  1. A quasitrace retains positivity and the identity τ(x∗x)=τ(xx∗)\tau(x^*x)=\tau(xx^*), but is required to be linear only on commutative subalgebras, together with compatibility with real and imaginary parts.
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text
  2. The exact case was settled by Haagerup: every 22-quasitrace on a unital exact C∗C^*-algebra is a trace.
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text
  3. There exist a separable unital complex C∗C^*-algebra AA, admitting a normalized 22-quasitrace, and positive contractions a,b∈Aa,b\in A such that every normalized 22-quasitrace τ\tau on AA satisfies τ(a+b)−τ(a)−τ(b)≥1144.\tau(a+b)-\tau(a)-\tau(b)\ge\frac1{144}.
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text
  4. There exist a separable unital complex C∗C^*-algebra AA, admitting a normalized 22-quasitrace, and positive contractions a,b∈Aa,b\in A such that every normalized 22-quasitrace τ\tau on AA satisfies τ(a+b)−τ(a)−τ(b)≥1144.\tau(a+b)-\tau(a)-\tau(b)\ge\frac1{144}.
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text
  5. We first build a unital algebra B=C∗(1,x1,…,x24)B=C^*(1,x_1,\ldots,x_{24}) such that ∑i=124xi∗xi=1,∑i=124xixi∗≤23 1,\sum_{i=1}^{24}x_i^*x_i=1,\qquad \sum_{i=1}^{24}x_ix_i^*\le\tfrac23\,1, while no Mm(B)M_m(B), m≥1m\ge1, contains two isometries with orthogonal ranges.
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text
  6. The Cuntz–Blackadar–Handelman existence theorem provides a normalized 22-quasitrace whenever no matrix algebra over the given unital algebra is properly infinite.
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text
  7. By the equivalence recalled above, it also gives a negative answer to Kaplansky's II1\mathrm{II}_1 AW∗AW^*-factor question.
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text
  8. As a consequence, we show that the minimal tensor product of two unital simple stably finite complex C∗C^*-algebras can be properly infinite, even when one factor is the reduced free-group algebra Cr∗(F2)C_r^*(\mathbb F_2).
    A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness Back to text

Other sources

  1. openai/math, Lean scope note for family 294 Thus none is additive on this pair.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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