2 problems

Let $(A,\varphi)$ and $(B,\psi)$ be nontrivial $C^*$-probability spaces inducing faithful GNS representations. Form their reduced free product $(A,\varphi)*(B,\psi)$. Choda–Dykema…

Non-converse conjecture. There should be no converse to the implication that if $A$ is algebraically purely infinite, then $A$ is purely infinite in the analytic sense.