A dichotomy for the number of ultrapowers

Abstract

We prove a strong dichotomy for the number of ultrapowers of a given countable model associated with nonprincipal ultrafilters on N. They are either all isomorphic, or else there are $2^{2^{{\aleph }_0}}$ many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II${}_1$ 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.

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