Nuclear counterexamples to Naimark's problem in ZFC with prescribed trace simplices

For every nonempty metrizable Choquet simplex K, we construct, in ZFC, a simple unital infinite-dimensional nuclear complex C*-algebra A whose nonzero irreducible representations are all unitarily equivalent and whose tracial state space is affinely isomorphic to K. We extend Tanaka's shell construction to a separable simple unital base admitting a projection excision sequence. Nuclearity of the base implies nuclearity, and hence exactness, of the shell algebra. When the excision projections have values tending to zero under every base trace, each such trace has a unique extension among all states of the shell algebra, and the extension is tracial. Classical realization by approximately finite-dimensional algebras then gives the prescribed simplex, including the case of a nonclosed extreme boundary. GitHub repository