  An early result of algebraic quantum field theory is that the algebra of any
subregion in a QFT is a von Neumann factor of type III$_1$, in which entropy
cannot be well-defined because such algebras do not admit a trace or density
states. However, associated to the algebra is a modular group of automorphisms
characterizing the local dynamics of degrees of freedom in the region, and the
crossed product of the algebra with its modular group yields a type II$_\infty$
factor, in which traces and hence von Neumann entropy can be well-defined. In
this work, we generalize recent constructions of the crossed product algebra
for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs,
paving the way to the study of entanglement entropy without UV divergences. In
contrast to previous works, we emphasize that this construction is independent
of gravity. In this sense, the crossed product construction represents a
refinement of Haag's assignment of nets of observable algebras to spacetime
regions by providing a natural construction of a type II factor. We present
several concrete examples: a QFT in Rindler space, a CFT in an open ball of
Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the
holographic setting, we provide a novel argument for why the bulk dual must be
the entanglement wedge, and discuss the distinction arising from boundary
modular flow between causal and entanglement wedges for excited states and
disjoint regions.
