Weak* density of the quantum Killing Lie–derived subalgebra in local von Neumann algebras

Prove that, for any region O in the indexing class K(M, g) of open relatively compact causally convex globally hyperbolic subsets of the Lorentzian manifold (M, g), the subalgebra M°(O, QuantLie) generated by the quantum Killing Lie derivatives oz (with Z ranging over all Killing vector fields on (M, g)) is weak* dense in the local von Neumann algebra M(O). Here QuantLie = {oz : Z a Killing vector field} and M°(O, QuantLie) denotes the subalgebra associated to these derivations within M(O).

Background

The paper develops an AQFT framework on curved spacetimes using von Neumann algebras, including a construction of local von Neumann algebras M(O) for regions O in a globally hyperbolic spacetime (M, g). For Killing vector fields Z, the authors define quantum Lie derivatives oz as (locally) densely defined *-derivations generated by the lifted local flows acting on these algebras.

Having established existence and closability criteria for these derivations, the authors propose that the subalgebra formed from elements governed by all such oz (i.e., associated to the set QuantLie) should be weak* dense in M(O). If true, this density would underpin a noncommutative differential geometric framework for QFT on curved spacetime, in the spirit of du Bois–Violette, by ensuring sufficient smoothness under the derivations to recover differential structures at the algebraic level.