We show that the convex set of factorizable quantum channels which factor through finite dimensional C*-algebras is non-closed in each dimension greater than 11, and that there exist factorizable quantum channels that require an ancilla of type II_1. The proof uses analysis of correlation matrices arising from projections, respectively, unitaries, in tracial von Neumann algebras. In recent work, we relate factorizable quantum channels to traces on a certain free product C*-algebra, via their Choi matrices. This new viewpoint leads to central questions in C*-algebra theory and to yet another formulation of the Connes Embedding Problem.