Speaker: **Ricardo Correa da Silva (Friedrich-Alexander-Universität Erlangen-Nürnberg)
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Title: Structure and Inclusions of Twisted Araki-Woods Algebras
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Time/Date: 4:45-6:15pm, Wednesday, March 8, 2023.
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Room: 128 Math. Sci. Building (It is also on Zoom. The Zoom link is the same as before.
If you don't have the one, please ask Kawahigashi.)
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Abstract: Similar to the construction of Bosonic and Fermionic Fock spaces, we will introduce the family $\mathcal{L}_T(H)$ of von Neumann algebras with respect to the standard subspace $H$ and the twist $T\in B(\mathcal{H} \otimes \mathcal{H})$ known as Araki-Woods algebras. These algebras provide a general framework of Bose and Fermi second quantization, S-symmetric Fock spaces, and full Fock spaces from free probability.
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Under a compatibility assumption on $T$ and $H$, we are going to present the equivalence between the Fock vacuum being cyclic and separating for $\mathcal{L}_T(H)$, and $T$ satisfying a standard subspace version of crossing symmetry and the Yang-Baxter equation (braid equation). In this case, we also determine the Tomita-Takesaki modular data. Finally, inclusions $\mathcal{L}_T(K)\subset \mathcal{L}_T(K)$ of such algebras and their relative commutants, for standard subspaces $K\subset H$, will be discussed as well as applications in QFT.
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