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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.06455 |
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| _version_ | 1866911444232044544 |
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| author | Arulseelan, Jananan |
| author_facet | Arulseelan, Jananan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06455 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Note on Pseudofinite W*-Probability Spaces Arulseelan, Jananan Operator Algebras Logic 46L10, 03C66 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. |
| title | A Note on Pseudofinite W*-Probability Spaces |
| topic | Operator Algebras Logic 46L10, 03C66 |
| url | https://arxiv.org/abs/2601.06455 |