On the factoriality of q-deformed Araki-Woods von Neumann algebras

Abstract: The $q$-deformed Araki-Woods von Neumann algebras $\Gamma_q(\mathcal{H}\mathbb{R}, U_t)^{\prime \prime}$ are factors for all $q\in (-1,1)$ whenever $dim(\mathcal{H}\mathbb{R})\geq 3$. When $dim(\mathcal{H}_\mathbb{R})=2$ they are factors as well for all $q$ so long as the parameter defining $(U_t)$ is `small' or $1$ $($trivial$)$ as the case may be.