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- Title
Modular Structure and Inclusions of Twisted Araki-Woods Algebras.
- Authors
Correa da Silva, Ricardo; Lechner, Gandalf
- 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 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 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 L T (K) ⊂ 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 K ⊂ H satisfies an L 2 -nuclearity condition, L T (K) ⊂ 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.
- Subjects
VON Neumann, John, 1903-1957; BOSE Corp.; MODULAR construction; ALGEBRAIC field theory; VON Neumann algebras; FOCK spaces; QUANTUM field theory; DIFFERENTIAL inclusions; TRIANGULAR norms
- Publication
Communications in Mathematical Physics, 2023, Vol 402, Issue 3, p2339
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-023-04773-y