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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.12481 |
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| _version_ | 1866917572939612160 |
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| author | Soni, Ronak M |
| author_facet | Soni, Ronak M |
| contents | I show that an analog of the crossed product construction that takes type $III_{1}$ algebras to type $II$ algebras exists also in the type $I$ case. This is particularly natural when the local algebra is a non-trivial direct sum of type $I$ factors. Concretely, I rewrite the usual type $I$ trace in a different way and renormalise it. This new renormalised trace stays well-defined even when each factor is taken to be type $III$. I am able to recover both type $II_{\infty}$ as well as type $II_{1}$ algebras by imposing different constraints on the central operator in the code. An example of this structure appears in holographic quantum error-correcting codes; the central operator is then the area operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12481 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Type $I$ Approximation of the Crossed Product Soni, Ronak M High Energy Physics - Theory Quantum Physics I show that an analog of the crossed product construction that takes type $III_{1}$ algebras to type $II$ algebras exists also in the type $I$ case. This is particularly natural when the local algebra is a non-trivial direct sum of type $I$ factors. Concretely, I rewrite the usual type $I$ trace in a different way and renormalise it. This new renormalised trace stays well-defined even when each factor is taken to be type $III$. I am able to recover both type $II_{\infty}$ as well as type $II_{1}$ algebras by imposing different constraints on the central operator in the code. An example of this structure appears in holographic quantum error-correcting codes; the central operator is then the area operator. |
| title | A Type $I$ Approximation of the Crossed Product |
| topic | High Energy Physics - Theory Quantum Physics |
| url | https://arxiv.org/abs/2307.12481 |