  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.
