Würzburg Seminar on Quantum Field Theory and Gravity

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.