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 C_r^*(\mathbb F_2). (Lean)

Manuscript

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

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-counterexample-to-Kaplanskys-quasitrace-conjecture-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).

Result 294 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.