On ultraproduct approximations and property (T) factors
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911690741776384 |
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| author | Peterson, Jesse |
| author_facet | Peterson, Jesse |
| contents | We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II$_1$ factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras $L(SL_3(\mathbb Z))$ and $L \mathbb F_2$ are not elementarily equivalent, and we show that the group von Neumann algebra $L\mathbb F_2$ is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II$_1$ factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which we can take to be group von Neumann algebras or group-measure space constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16669 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On ultraproduct approximations and property (T) factors Peterson, Jesse Operator Algebras Group Theory We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II$_1$ factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras $L(SL_3(\mathbb Z))$ and $L \mathbb F_2$ are not elementarily equivalent, and we show that the group von Neumann algebra $L\mathbb F_2$ is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II$_1$ factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which we can take to be group von Neumann algebras or group-measure space constructions. |
| title | On ultraproduct approximations and property (T) factors |
| topic | Operator Algebras Group Theory |
| url | https://arxiv.org/abs/2605.16669 |