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A dichotomy for the number of ultrapowers. (English) Zbl 1254.03068

Summary: We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality \(\leq 2^{\aleph _{0}}\) associated with nonprincipal ultrafilters on \(\mathbb N\). 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 \(\mathcal B(H)\) always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on \(\mathbb N\).

MSC:

03C20 Ultraproducts and related constructions
46M07 Ultraproducts in functional analysis
46L05 General theory of \(C^*\)-algebras
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