  Recent work has highlighted the importance of crossed products in correctly
elucidating the operator algebraic approach to quantum field theories. In the
gravitational context, the crossed product simultaneously promotes von Neumann
algebras associated with subregions in diffeomorphism covariant quantum field
theories from type III to type II, and provides the necessary ingredients to
gravitationally dress operators, thereby enforcing the constraints of the
theory. In this note we enhance the crossed product construction to the context
of general gauge theories with arbitrary combinations of internal and spacetime
local symmetries. This is done by leveraging the correspondence between the
crossed product and the extended phase space. We then undertake a detailed
study of constraint quantization from the perspective of a generic crossed
product algebra. We study and compare four distinct approaches to constraint
quantization from this point of view: refined algebraic quantization, BRST
quantization, path integral quantization, and the commutation theorem for
crossed products. Far from simply reproducing existing analyses, the operator
algebraic viewpoint sheds new light on old problems by reformulating the
dressing of operators in terms of conditional expectations and other closely
related projection maps. We conclude by applying our approach to the constraint
quantization of three distinct gauge theories including a discussion of gravity
on null hypersurfaces.
