Ricardo Correa da Silva
Twisted Araki–Woods Algebras: structure and inclusions
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 $T$-twisted Araki-Woods algebras, which are interesting in Algebraic Quantum Field Theory. These algebras encode localization properties in the standard subspace 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 above, we also determine the Tomita-Takesaki modular data for Araki-Woods algebra and the Fock vacuum, and study the inclusions $\mathcal{L}_T(K)\subset \mathcal{L}_T(H)$ of such algebras and their relative commutants for standard subspaces $K\subset H$.
This talk is based on a joint work with Gandalf Lechner (arXiv: 2212.02298).
