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${}^{\ast }$-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${}^{\ast }$-algebras can be properly infinite, even when one factor is the reduced free-group algebra $C_r^{\ast }({\mathbb{F}}_2)$.