Continuous model theories for von Neumann algebras

Abstract

We axiomatize in continuous logic for metric structures σ-finite W⁎-probability spaces and preduals of von Neumann algebras jointly with a weak-* dense C⁎-algebra of its dual. This corresponds respectively to the Ocneanu ultrapower and the Groh ultrapower of (σ-finite in the first case) von Neumann algebras. We give various axiomatizability results corresponding to recent results of Ando and Haagerup including axiomatizability of IIIλ factors for 0<λ≤1 fixed and their preduals. We also strengthen the concrete Groh theory to an axiomatization result for preduals of von Neumann algebras in the language of tracial matrix-ordered operator spaces, a natural language for preduals of dual operator systems. We give an application to the isomorphism of ultrapowers of factors of type III and II∞ for different ultrafilters.