Between reduced powers and ultrapowers

Abstract

One of our results is a transfer principle between ultrapowers and reduced powers associated with the Fréchet ideal. Although motivated by the Elliott classification programme, this result applies to any axiomatizable category. We also show that there exists a nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$ such that for every countable (or separable metric) structure $B$ in a countable language the quotient map from the reduced power associated with the Fréchet ideal onto an ultrapower has a right inverse. While the transfer principle is proved without appealing to additional set-theoretic axioms, the conclusion of the latter theorem relies on the Continuum Hypothesis and it is independent of the standard axioms of set theory. We also prove that in the category of $C^$-algebras, tensoring with the $C^$-algebra of all continuous functions on the Cantor space preserves elementary equivalence. As a side note, neither the Jiang–Su algebra $\mathcal{Z}$ nor any UHF algebra share this property.