Between reduced powers and ultrapowers

We prove that there exists a nonprincipal ultrafilter $\mathcal U$ on $\mathbb N$ such that for every countable (or separable) structure $B$ in a countable language the quotient map from the reduced product associated with the Fr\'echet filter onto the ultrapower has a right inverse. The proof uses the Continuum Hypothesis. We characterize the ultrafilters $\mathcal U$ with this property, and show that consistently with ZFC such ultrafilters need not exist. We also prove a similar ZFC result sufficiently strong to obtain all concrete applications of the existence of a right inverse to the quotient map. Among applications, we prove a transfer theorem, answering a question of Schafhauser and Tikuisis, motivated by the Elliott classification programme. We also show that, in the category of C*-algebras, tensoring with the C*-algebra of all continuous functions on the Cantor space preserves elementarity. We also prove that tensoring with the Jiang--Su algebra or a UHF algebra does not preserve elementarity in general.