A type I approximation of the crossed product

Abstract

I show that an analog of the crossed product construction that takes type \U0001d43c\U0001d43c\U0001d43c1 algebras to type \U0001d43c\U0001d43c algebras exists also in the type \U0001d43c case. This is particularly natural when the local algebra is a non-trivial direct sum of type \U0001d43c factors. Concretely, I rewrite the usual type \U0001d43c trace in a different way and renormalise it. This new renormalised trace stays well-defined even when each factor is taken to be type \U0001d43c\U0001d43c\U0001d43c. I am able to recover both type \U0001d43c\U0001d43c∞ as well as type \U0001d43c\U0001d43c1 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.