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Math-2823 · mathematics · Analysis

Kaplansky quasitrace additivity conjecture

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is every normalized two-quasitrace on every unital complex C*-algebra additive on the positive cone and therefore a tracial state, including separable algebras without exactness or nuclearity?

Short English statement

Is every normalized two-quasitrace on every unital complex C*-algebra additive on the positive cone and therefore a tracial state, including separable algebras without exactness or nuclearity?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2823-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2823

Manuscript 1 (family 294, 2026-09-23): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed counterexample to that assertion in full.

Kaplansky quasitrace additivity conjecture

OpenAI's claim

A separable unital algebra is claimed to have normalized 2-quasitraces but no traces; consequently, minimal tensoring can turn two simple unital stably finite C*-algebras into a properly infinite algebra. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims a complex $C^*$-algebra $A$ that is unital and separable and supports a normalized $2$-quasitrace and fixed positive contractions $a,b\in A$ such that every normalized $2$-quasitrace satisfies \[ \tau(a+b)-\tau(a)-\tau(b)\ge1/144. \] Normalized means $\tau(1)=1$; the corner-preserving extension to $M_2(A)$ takes value two on its identity. The positive gap is uniform over all such quasitraces and prevents any of them from being additive tracial states.

Outline of the argument

Manuscript 1 (family 294, 2026-09-23): Twenty-four bundle-based generators have unit column sum and a strictly smaller row sum, excluding tracial states. Continuous operator scaling preserves this imbalance while nonzero Chern classes obstruct every finite matrix test for proper infiniteness. A bounded-product construction retains all tests simultaneously; established quasitrace and tensor-product implications give the claimed consequences, including a reduced-free-group tensor factor.

Formal verification

For family 294, the scope page documents the following selections. Selections exhibit positive contractions whose normalized quasitraces have a fixed additivity defect, as well as simple stably finite tensor factors with properly infinite spatial product and the corresponding trace failures. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: KaplanskyQuasitrace.lean, KaplanskyStableFiniteness.lean. Declarations: OAI.Kaplansky.kaplansky_quasitrace_counterexample, OAI.KaplanskyConsequences.stable_finiteness_bundle.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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