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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 II 1  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.

Turbulence, representations, and trace‐preserving actions