II_1 factors with non-isomorphic ultrapowers

Abstract

We prove that there exist uncountably many separable II${}_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II${}_1$ factors originally introduced by McDuff \cite{MD69a,MD69b} are such examples. This entails the existence of a continuum of non-elementarily equivalent II${}_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.

16 pages

References in corpus (3)

• Model theory of operator algebras III: Elementary equivalence and II_1 factors
• Logic and operator algebras
• Games and elementary equivalence of $\mathrm{I}{\mathrm{I}}_1$ factors

Cited by in corpus (5)

• Asymptotic freeness in tracial ultraproducts
• Sequential commutation in tracial von Neumann algebras
• Elementary equivalence and disintegration of tracial von Neumann algebras
• Explicit sentences distinguishing McDuff's II${}_1$ factors
• Weight theory for ultraproducts