A type I approximation of the crossed product

Abstract

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.