  We identify a direct correspondence between the crossed product construction
which plays a crucial role in the theory of Type III von Neumann algebras, and
the extended phase space construction which restores the integrability of
non-zero charges generated by gauge symmetries in the presence of spatial
substructures. This correspondence provides a blue-print for resolving
singularities which are encountered in the computation of entanglement entropy
for subregions in quantum field theories. The extended phase space encodes
quantities that would be regarded as `pure gauge' from the perspective of the
full theory, but are nevertheless necessary for gluing together, in a path
integral sense, physics in different subregions. These quantities are required
in order to maintain gauge covariance under such gluings. The crossed product
provides a consistent method for incorporating these necessary degrees of
freedom into the operator algebra associated with a given subregion. In this
way, the extended phase space completes the subregion algebra and subsequently
allows for the assignment of a meaningful, finite entropy to states therein.
