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.

arXiv comment16 pages, 3 figures

References in corpus (21)

• Black Hole Entropy is Noether Charge
• Black Hole Entropy in Canonical Quantum Gravity and Superstring Theory
• Entanglement entropy of black holes
• Relative entropy equals bulk relative entropy
• The Ryu-Takayanagi Formula from Quantum Error Correction
• Remarks on entanglement entropy for gauge fields
• A Quantum Focussing Conjecture
• An Algebra of Observables for de Sitter Space
• Decomposition of entanglement entropy in lattice gauge theory
• Renormalization of Black Hole Entropy and of the Gravitational Coupling Constant
• Gravity and the Crossed Product
• Large N algebras and generalized entropy
• Aspects of Entanglement Entropy for Gauge Theories
• A change of perspective: switching quantum reference frames via a perspective-neutral framework
• Holographic Renyi Entropy from Quantum Error Correction
• Generalized entropy for general subregions in quantum gravity
• Group Averaging for de Sitter free fields
• A proposal for 3d quantum gravity and its bulk factorization
• von Neumann algebras in JT gravity
• Entanglement branes and factorization in conformal field theory
• Algebra of operators in an AdS-Rindler wedge

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