A Note on Pseudofinite W*-Probability Spaces
Jananan Arulseelan
TL;DR
The paper develops the theory of pseudofinite $\mathrm{W}^*$-probability spaces, defined as algebras elementarily equivalent to Ocneanu ultraproducts of finite-dimensional von Neumann algebras with faithful normal states. It proves ultraproducts of matrix algebras are factors and that pseudofinite spaces inherit factoriality, then constructs Powers-style pseudofinite factors realizing $\mathrm{III}_{\lambda}$ (including $\mathrm{III}_1$) and shows these share universal theories with the corresponding Powers factors, leading to uncomputable universal theories. It also shows pseudofinite factors are full, generalizing the tracial case and implying hyperfinite type $\mathrm{III}$ factors (Powers) cannot be pseudfinite. The work blends operator-algebraic tools with continuous-logic methods, and outlines a program to study pseudofinite weighted von Neumann algebras as a further direction. Key outcomes include: (i) factoriality transfer from matrix ultraproducts, (ii) explicit $\mathrm{III}_{\lambda}$ realizations and uncomputable universal theories, (iii) fullness of pseudofinite factors and non-pseudofiniteness of hyperfinite type $\mathrm{III}$ factors, and (iv) a pathway to weighted generalizations. These results advance the model-theoretic understanding of operator algebras beyond the tracial setting and clarify which properties are preserved under ultraproducts and elementary equivalence in the $\mathrm{W}^*$-probability framework.
Abstract
We introduce pseudofinite W*-probability spaces. These are W*-probability spaces that are elementarily equivalent to Ocneanu ultraproducts of finite-dimensional von Neumann algebras equipped with arbitrary faithful normal states. We are particularly interested in the case where these finite-dimensional von Neumann algebras are full matrix algebras: the pseudofinite factors. We show that these are indeed factors. We see as a consequence that pseudofinite factors are never of type $\mathrm{III}_0$. Mimicking the construction of the Powers factors, we give explicit families of examples of matrix algebra ultraproducts that are $\mathrm{III}_λ$ factors for $λ\in (0,1]$. We show that these examples share their universal theories with the corresponding Powers factor and thus have uncomputable universal theories. Finally, we show that pseudofinite factors are full. This generalizes a theorem of Farah-Hart-Sherman which shows that pseudofinite tracial factors do not have property $Γ$. It has the consequence that hyperfinite factors of type $\mathrm{III}$ (the Powers factors) are never pseudofinite. Our proofs combine operator algebraic insights with routine continuous logic syntactic arguments: using Łos' theorem to prove that certain sentences which are true for all matrix algebras are inherited by their ultraproducts.