Reduced products of metric structures: a metric Feferman-Vaught theorem

TL;DR

This paper extends the Feferman-Vaught theorem to metric structures, showing reduced powers preserve elementary equivalence and exploring isomorphism and independence results for coronas of separable C*-algebras.

Contribution

It introduces a metric version of the Feferman-Vaught theorem and demonstrates independence results for coronas of certain C*-algebras.

Findings

Reduced powers of elementarily equivalent metric structures are elementarily equivalent.

Existence of non-commutative coronas with independence from ZFC.

Extension of classical logic results to metric structures.

Abstract

We extend the classical Feferman-Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum Hypothesis. We also prove the existence of two separable C*-algebras of the form $\bigoplus_i M_{k(i)}(\mathbb{C})$ such that the assertion that their coronas are isomorphic is independent from ZFC, which gives the first example of genuinely non-commutative coronas of separable C*-algebras with this property.