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We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ≤ 20 associated with nonprincipal ultrafilters on ℕ. They are either all isomorphic, or else there are 220 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 CAF001-algebra always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on ℕ.

AMSC: Primary 03C20, Secondary 46M07


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