On perturbations of Hilbert spaces and probability algebras with a generic automorphism

We prove that IHSA, the theory of infinite dimensional Hilbert spaces equipped with a generic automorphism, is א0-stable up to perturbation of the automorphism, and admits prime models up to perturbation over any set. Similarly, APrA, the theory of atomless probability algebras equipped with a generic automorphism is א0-stable up to perturbation. However, not allowing perturbation it is not even superstable.