Kaplansky's quasitrace problem: is every 2-quasitrace on a unital C*-algebra a trace?

Theorem 1.1: there are a separable unital $C^*$-algebra $A=M_{48}(B)$ with a normalized 2-quasitrace and positive contractions $a,b$ with $\tau(a+b)-\tau(a)-\tau(b)\ge1/144$ for every normalized 2-quasitrace $\tau$, so none is a trace. $B$ is generated by 24 shifts between vector bundles with $\sum x_i^*x_i=1$ and $\sum x_ix_i^*\le\frac23$, Chern-class obstructions ruling out proper infiniteness in every matrix size. Hence a type II$_1$ $AW^*$-factor that is not a von Neumann algebra exists. Corollary: two unital simple stably finite algebras, one $C_r^*(\mathbb F_2)$, with properly infinite minimal tensor product (answering Milhoj-Rordam Question 2.5), and a separable stably finite algebra with no tracial state. Not shown: a simple or exact example (exact is impossible by Haagerup), or an explicit II$_1$ factor.

Theorem 1.1: there are a separable unital $C^*$-algebra $A=M_{48}(B)$ with a normalized 2-quasitrace and positive contractions $a,b$ with $\tau(a+b)-\tau(a)-\tau(b)\ge1/144$ for every normalized 2-quasitrace $\tau$, so none is a trace. $B$ is generated by 24 shifts between vector bundles with $\sum x_i^*x_i=1$ and $\sum x_ix_i^*\le\frac23$, Chern-class obstructions ruling out proper infiniteness in every matrix size. Hence a type II$_1$ $AW^*$-factor that is not a von Neumann algebra exists. Corollary: two unital simple stably finite algebras, one $C_r^*(\mathbb F_2)$, with properly infinite minimal tensor product (answering Milhoj-Rordam Question 2.5), and a separable stably finite algebra with no tracial state. Not shown: a simple or exact example (exact is impossible by Haagerup), or an explicit II$_1$ factor.