I introduce a preprint, arXiv:gr-qc/9509057, by R. Wald. The paper argues that, in curved spacetime quantum field theory, the choice of a Hilbert-space construction is not fundamental. One should take the algebra of observables as the primary object and define states on it. In this way, the theory can accommodate all states arising in unitarily inequivalent Hilbert-space realizations on an equal footing.
Recently, there has been discussion of research connecting AQFT with non-commutative spaces (or spacetimes). For example, in arXiv:2503.24068, they establishes a quantum energy inequality for a quantum field theory in a non-commutative spacetime.