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Continuous model theories for von Neumann algebras

We axiomatize in (first order finitary) 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 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 $III_λ$ factors for $0<λ\leq 1$ 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 $II_\infty$ for different ultrafilters.