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

Abstract

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 HR with dimHR≥2, a strongly continuous one parameter group of orthogonal transformations on HR, a direct sum decomposition HR=⊕iHR(i), and a real symmetric matrix (qij) with supi,j|qij| < 1. This is achieved by first proving the existence of conjugate variables for a finite number of generators of the algebras (when dimHR<∞), following the lines of Miyagawa–Speicher and Kumar–Skalski–Wasilewski. The conjugate variables belong to the factors in question and satisfy certain Lipschitz conditions.