We study the additivity and Haag duality of the von Neumann algebra of a quantum field theory \({\mathcal{T}}_{\mathcal{F}}\) with 0-form (and the dual (d − 2)-form) (non)-invertible global symmetry \(\mathcal{F}\) . We analyze the symmetric (uncharged) sector von Neumann algebra of \({\mathcal{T}}_{\mathcal{F}}\) with the inclusion of bi-local and bi-twist operators in it. We establish the connection between the existence of these non-local operators in \({\mathcal{T}}_{\mathcal{F}}\) and certain properties of the Lagrangian algebra \(\mathcal{L}\) of the extended operators in the corresponding symmetry topological field theory (SymTFT). We prove that additivity or Haag duality of the symmetric sector von Neumann algebra is violated when \(\mathcal{L}\) satisfies specific criteria, thus generalizing the result of Shao, Sorce and Srivastava to arbitrary dimensions. We further demonstrate the SymTFT construction via concrete examples in two dimensions.
