A dichotomy for the number of ultrapowers

We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ≤ 2^א0 associated with nonprincipal ultrafilters on ℕ. They are either all isomorphic, or else there are 2 ^2א0 many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II₁ factors, as well as their relative commutants and include several applications. We also show that the C*-algebra B(H) always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on ℕ.