The Universal Theory of Locally Universal Tracial von Neumann Algebras is not Computable
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908935427981312 |
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| author | Arulseelan, Jananan Manzoor, Aareyan |
| author_facet | Arulseelan, Jananan Manzoor, Aareyan |
| contents | Building on Lin's breakthrough MIP$^{co}$ = coRE and an encoding of non-local games as universal sentences in the language of tracial von Neumann algebras, we show that locally universal tracial von Neumann algebras have undecidable universal theories. This implies that no such algebra admits a computable presentation. Our results also provide, for the first time, explicit examples of separable II$_1$ factors without computable presentations, and in fact yield a broad family of them, including McDuff factors, factors without property Gamma, and property (T) factors. We also obtain analogous results for locally universal semifinite von Neumann algebras and tracial C*-algebras. The latter provides strong evidence for a negative solution to the Kirchberg Embedding Problem. We discuss how these are obstructions to approximation properties in the class of tracial and semifinite von Neumann algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21709 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Universal Theory of Locally Universal Tracial von Neumann Algebras is not Computable Arulseelan, Jananan Manzoor, Aareyan Operator Algebras Logic Quantum Physics 46L10, 03C98 Building on Lin's breakthrough MIP$^{co}$ = coRE and an encoding of non-local games as universal sentences in the language of tracial von Neumann algebras, we show that locally universal tracial von Neumann algebras have undecidable universal theories. This implies that no such algebra admits a computable presentation. Our results also provide, for the first time, explicit examples of separable II$_1$ factors without computable presentations, and in fact yield a broad family of them, including McDuff factors, factors without property Gamma, and property (T) factors. We also obtain analogous results for locally universal semifinite von Neumann algebras and tracial C*-algebras. The latter provides strong evidence for a negative solution to the Kirchberg Embedding Problem. We discuss how these are obstructions to approximation properties in the class of tracial and semifinite von Neumann algebras. |
| title | The Universal Theory of Locally Universal Tracial von Neumann Algebras is not Computable |
| topic | Operator Algebras Logic Quantum Physics 46L10, 03C98 |
| url | https://arxiv.org/abs/2508.21709 |