  If quantum gravity respects the principles of quantum mechanics, suitably
generalized, it may be that a more viable approach to the theory is through
identifying the relevant quantum structures rather than by quantizing classical
spacetime. This viewpoint is supported by difficulties of such quantization,
and by the apparent lack of a fundamental role for locality. In finite or
discrete quantum systems, important structure is provided by tensor
factorizations of the Hilbert space. However, even in local quantum field
theory properties of the generic type III von Neumann algebras and of long
range gauge fields indicate that factorization of the Hilbert space is
problematic. Instead it is better to focus on the structure of the algebra of
observables, and in particular on its subalgebras corresponding to regions.
This paper suggests that study of analogous algebraic structure in gravity
gives an important perspective on the nature of the quantum theory. Significant
departures from the subalgebra structure of local quantum field theory are
found, working in the correspondence limit of long-distances/low-energies.
Particularly, there are obstacles to identifying commuting algebras of
localized operators. In addition to suggesting important properties of the
algebraic structure, this and related observations pose challenges to proposals
of a fundamental role for entanglement.
