Conjugate variables approach to mixed $q$-Araki-Woods algebras: Factoriality and non-injectivity

We establish factoriality and non-injectivity in full generality for the mixed $q$-Araki-Woods von Neumann algebra associated to a separable real Hilbert space ${\mathsf{H}}_{\mathbf{R}}$ with $\dim {\mathsf{H}}_{\mathbf{R}}\ge 2$, a strongly continuous one parameter group of orthogonal transformations on ${\mathsf{H}}_{\mathbb{R}}$, a direct sum decomposition ${\mathsf{H}}_{\mathbf{R}}={\oplus }_i{\mathsf{H}}_{\mathbb{R}}^{(i)}$, and a real symmetric matrix $(q_{ij})$ with $q={\sup }_{i,j}\mathrm{\mid }{q}_{ij}\mathrm{\mid }<1$. This is achieved by first proving the existence of conjugate variables for a finite number of generators of the algebras, following the lines of Miyagawa-Speicher and Kumar-Skalski-Wasilewski. The conjugate variables belong to the factors in question and satisfy certain Lipschitz condition.