A type I approximation of the crossed product
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Abstract I show that an analog of the crossed product construction that takes type 𝐼𝐼𝐼1 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∞ as well as type 𝐼𝐼1 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.
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Acknowledgements: I thank participants in the type 𝐼𝐼 algebra study week at DAMTP, University of Cambridge, — Amr Ahmadain, Goncalo Araujo-Regado, Alex Frenkel, Rifath Khan, Prahar Mitra, Krishnendu Ray, Ayngaran Thavanesan, Bilyana Tomova and Manus Visser —, especially those who sat through my explanation of this work at a very early stage. I thank Wissam Chemissany, Matthew Heydeman, David K. Kolchmeyer, Shiraz Minwalla, Onkar Parrikar, Antony Speranza, Gabriel Wong and the audience of a talk at Tata Institute of Fundamental Research (TIFR) for discussions. I thank Chris Akers and Annie Wei for collaboration on a related project [22] that led to this work and comments on previus versions; and also David Tong for asking about the possibility of approximating the crossed product. Finally, I thank an anonymous reviewer of my talk proposal to the “It from Qubit 2023” conference for pointing out an important mistake in my presentation. This work has been partially supported by STFC consolidated grant ST/T000694/1. I am supported by the Isaac Newton Trust grant “Quantum Cosmology and Emergent Time” and the (United States) Air Force Office of Scientific Research (AFOSR) grant “Tensor Networks and Holographic Spacetime”.
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1029-8479
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