1 · The essentials
The manuscript gives a counterexample to Kaplansky’s quasitrace conjecture, a question about special ways of assigning numerical values to elements of a mathematical system. The system it constructs allows assignments called normalized 2-quasitraces. Yet none of these assignments always respects addition: the value of a sum need not equal the sum of the values. In particular, the system has no tracial state, the more restrictive kind of assignment at issue here. The manuscript also shows that two systems satisfying a strong finiteness condition can lose that condition when combined in a specified way. Their combination can instead be properly infinite, even with one of the two systems fixed to the particular example named in the abstract. The main result has a Lean formalization, although the supplied description does not specify its coverage of individual claims.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a way of valuing bundles of goods that follows certain consistency rules but does not always price a combined bundle as the sum of its separate prices. The counterexample is like a setting where such valuation rules exist, but no rule satisfying the stronger requirements of a tracial state is available.
Why it matters
The construction separates the existence of normalized 2-quasitraces from the existence of a tracial state. Its tensor-product consequence also shows that stable finiteness of both factors does not, under the stated hypotheses, ensure stable finiteness of their minimal tensor product. The abstract makes this failure more specific by allowing one factor to be .
2 · Going further
The precise counterexample is a separable unital complex -algebra that admits normalized 2-quasitraces but has no tracial state. The repository summary further specifies that all its normalized 2-quasitraces are nonadditive. Thus the issue is not merely finding one badly behaved assignment: the algebra supports the weaker kind of assignment while excluding the stronger one. The source identifies this as a refutation of Kaplansky’s quasitrace conjecture.
As a consequence, the manuscript shows that there are two unital simple stably finite complex -algebras whose minimal tensor product is properly infinite. One factor can be the reduced free-group algebra . The contrast is between stable finiteness of each factor and proper infiniteness of their tensor product; the claim is an existence result, not a statement about every such pair.
The abstract describes the argument only at the level of constructing the quasitrace counterexample and obtaining the tensor-product statement as a consequence. It supplies no construction details or intermediate steps, so a more specific account of the mechanism is not available here. Only one manuscript is listed, and no companion-manuscript relationship is described. The main result has a Lean formalization, but the supplied material does not identify which individual assertions beyond that result are covered.
Useful notions
- Complex C*-algebras
- Tracial states and normalized 2-quasitraces
- Additivity
- Stable finiteness and proper infiniteness
- Minimal tensor products
What was known before / what is new
Before
The source identifies Kaplansky’s quasitrace conjecture as the question being refuted. It gives no account of earlier partial results, the history of the conjecture, or previously known tensor-product preservation results.
New
The manuscript constructs a separable unital complex -algebra with normalized 2-quasitraces but no tracial state; the repository summary states that all those quasitraces are nonadditive. It also shows that two unital simple stably finite complex -algebras can have a properly infinite minimal tensor product. One factor can be .
Limits and assumptions
The quasitrace counterexample is stated for a separable unital complex C*-algebra. The tensor-product consequence concerns two unital simple stably finite complex C*-algebras and specifically their minimal tensor product; it does not assert that every such pair has a properly infinite product. The main result has a Lean formalization, but the supplied description does not establish that every consequence or repository claim is included in its scope. No remaining open questions, construction details, or companion manuscripts are supplied.
Mini glossary
- Additivity —
- The requirement that the value assigned to a sum equal the sum of the assigned values.
- Normalized 2-quasitrace —
- The kind of numerical assignment admitted by the constructed algebra, all instances of which are nonadditive in this example.
- Tracial state —
- The stronger kind of numerical assignment that the constructed algebra does not admit.
- Minimal tensor product —
- The specified way of combining two C*-algebras used in the manuscript’s finiteness counterexample.
- Stable finiteness —
- The finiteness condition satisfied by each factor in the tensor-product example but not preserved by their combination.
- Proper infiniteness —
- The infiniteness property possessed by the minimal tensor product in the stated example.
Keywords : Kaplansky’s quasitrace conjecture · Nonadditive quasitraces · Tensor-product stable finiteness
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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 .
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
23/09/2026We 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 .