A Note on Pseudofinite W*-Probability Spaces

TL;DR

This paper introduces pseudofinite W*-probability spaces, explores their properties, constructs examples of type III factors, and shows these factors are full, extending previous results and connecting operator algebras with continuous logic.

Contribution

It defines pseudofinite W*-probability spaces, constructs explicit examples of type III factors, and proves these factors are full, extending the understanding of their model-theoretic properties.

Findings

Pseudofinite factors are always factors.

Explicit examples of type III$_ ext{lambda}$ factors are constructed.

Pseudofinite factors are full and do not have property $ ext{Gamma}$.

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 $\mathrm{III}_\lambda$ factors for $\lambda \in (0,1]$. 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 $\Gamma$. 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 \L os' theorem to prove that certain sentences which are true for all matrix algebras are inherited by their ultraproducts.