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 LT(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 LT(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 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⊂H satisfies an L2-nuclearity condition, LT(K)⊂LT(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.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- September 2023
- DOI:
- 10.1007/s00220-023-04773-y
- arXiv:
- arXiv:2212.02298
- Bibcode:
- 2023CMaPh.402.2339C
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematical Physics
- E-Print:
- Commun. Math. Phys. (2023)