Modular Structure and Inclusions of Twisted Araki-Woods Algebras

Abstract

In 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) depend on the twist operator T and a standard subspace H in the one-particle space. Under a compatibility assumption on T and H, it is proven that the Fock vacuum is cyclic and separating for mathcal {L}_{T}(H) if and only if T satisfies 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) 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 Ksubset H satisfies an L^2-nuclearity condition, mathcal {L}_{T}(K)subset mathcal {L}_{T}(H) has type III relative commutant for suitable twists T. Applications of these results to localization of observables in algebraic quantum field theory are discussed.