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arXiv · 0912.0406

A dichotomy for the number of ultrapowers

Ilijas Farah·Saharon Shelah
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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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Ilijas Farah, Saharon Shelah. 2009-12-02. A dichotomy for the number of ultrapowers. https://arxiv.org/abs/0912.0406

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