A Type $I$ Approximation of the Crossed Product

TL;DR

The paper introduces a novel type I approximation of the crossed product construction, enabling the transition from type III to type II algebras, with applications in holographic quantum error correction.

Contribution

It presents a new way to approximate crossed products in type I algebras, extending the construction to include type III factors through a redefined and renormalized trace.

Findings

Reformulation of type I trace to include type III factors

Recovery of type II_{ ext{infinity}} and II_1 algebras via constraints

Application to holographic quantum error-correcting codes

Abstract

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.