  We axiomatize in (first order finitary) continuous logic for metric
structures $\sigma$-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
($\sigma$-finite in the first case) von Neumann algebras. We give various
axiomatizability results corresponding to recent results of Ando and Haagerup
including axiomatizability of $III_\lambda$ factors for $0<\lambda\leq 1$ 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.
