# Modular Structure and Inclusions of Twisted Araki-Woods Algebras

December 05, 2022

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.

Keywords:

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