Turbulence, Representations, and Trace-preserving Actions

We establish criteria for turbulence in certain spaces of C∗-algebra representations and apply this to the problem of nonclassifiability by countable structures for group actions on a standard atomless probability space (X,μ) and on the hyperfinite II1 factor R. We also prove that the conjugacy action on the space of free actions of a countably infinite amenable group on R is turbulent, and that the conjugacy action on the space of ergodic measure-preserving flows on (X,μ) is generically turbulent.