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 the reduced free-group algebra C*_r(F_2).