A type I approximation of the crossed product
Ronak M Soni
Abstract
A bstract I show that an analog of the crossed product construction that takes type ๐ผ๐ผ๐ผ 1 algebras to type ๐ผ๐ผ algebras exists also in the type ๐ผ case. This is particularly natural when the local algebra is a non-trivial direct sum of type ๐ผ factors. Concretely, I rewrite the usual type ๐ผ trace in a different way and renormalise it. This new renormalised trace stays well-defined even when each factor is taken to be type ๐ผ๐ผ๐ผ. I am able to recover both type ๐ผ๐ผ โ as well as type ๐ผ๐ผ 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.