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 $K\subset H$ 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.

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