A Type $I$ Approximation of the Crossed Product

Abstract

I show that an analog of the crossed product construction that takes type $II{I}_1$ 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 $I{I}_{\infty }$ as well as type $I{I}_1$ 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.

16 pages, 3 figures

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Cited by in corpus (6)

• Generalized Black Hole Entropy is von Neumann Entropy
• Dynamical Edge Modes and Entanglement in Maxwell Theory
• Holographic Tensor Networks with Bulk Gauge Symmetries
• Gravitational entropy is observer-dependent
• Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
• On Infinite Tensor Networks, Complementary Recovery and Type II Factors