Modular Structure and Inclusions of Twisted Araki-Woods Algebras

Abstract

AbstractIn the general setting of twisted second quantization (including Bose/Fermi second quantization,S-symmetric Fock spaces, and full Fock spaces from free probability as special cases), von Neumann algebras on twisted Fock spaces are analyzed. These twisted Araki-Woods algebras$$\mathcal {L}_{T}(H)$$LT(H)depend on the twist operatorTand a standard subspaceHin the one-particle space. Under a compatibility assumption onTandH, it is proven that the Fock vacuum is cyclic and separating for$$\mathcal {L}_{T}(H)$$LT(H)if and only ifTsatisfies a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, the Tomita-Takesaki modular data are explicitly determined. Inclusions$$\mathcal {L}_{T}(K)\subset \mathcal {L}_{T}(H)$$LT(K)⊂LT(H)of twisted Araki-Woods algebras are analyzed in two cases: If the inclusion is half-sided modular and the twist satisfies a norm bound, it is shown to be singular. If the inclusion of underlying standard subspaces$$K\subset H$$K⊂Hsatisfies an$$L^2$$L2-nuclearity condition,$$\mathcal {L}_{T}(K)\subset \mathcal {L}_{T}(H)$$LT(K)⊂LT(H)has type III relative commutant for suitable twistsT. Applications of these results to localization of observables in algebraic quantum field theory are discussed.