AG Lie-Gruppen: R. C. da Silva, FAU: Twisted Araki-Woods Algebras: structure and inclusions

Twisted Araki-Woods Algebras: structure and inclusions \- Vortragender:
Ricardo Correa da Silva, FAU Erlangen – Einladender: Karl-Hermann Neeb

Abstract: 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 generalize the construction of the Bosonic and Fermionic Fock spaces
and provide a general framework of the Bose and Fermi second quantization, the
S-symmetric Fock spaces, and the full Fock spaces from free probability.
Under the assumption of compatibility between $T$ and $H$, we are going to
present the equivalence between $T$ satisfying a standard subspace version of
crossing symmetry, and the Yang-Baxter equation (braid

equation) and the Fock vacuum being cyclic and separating for
$\\mathcal{L}_T(H)$. Under the same assumptions, we also determine the
Tomita-Takesaki modular data for Araki-Woods algebra and the Fock vacuum.
Finally, the 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.
This is joint work with Gandalf Lechner (arXiv: 2212.02298).