On the local structure of the representation of a local gauge group

We discuss the local structure of the net O 7→ M(O)′′ of von Neumann algebras generated by a representation of a local gauge group C∞ c (M,G). Our discussion is independent of the singularity of spectral measures, which has been discussed by many authors since the pioneering work of Gelfand-Graev-Veršic. We show that, for type (S) operators UA,b, second quantized operators with some twists, the commutativity only with those U(ψ) is sufficient for the triviality of them, where ψ belongs to an arbitrary (small) neighborhood of constant function 1. Some properties of 1-cocycles for the representation V : ψ 7→ Adψ are also discussed.