A dichotomy for the number of ultrapowers

Details A dichotomy for the number of ultrapowers A dichotomy for the number of ultrapowers

2009-12-02

arxiv.org/abs/0912.0406v1

Mathematics

Logic

Scientific paper

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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