Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
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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).
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MANUSCRIPT 01
A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
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…
Full source abstract
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).
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