A Conjugate System for Twisted Araki–Woods von Neumann Algebras of Finite Dimensional Spaces

We compute the conjugate system of twisted Araki–Woods von Neumann algebras LT(H) introduced in [ 7] for a compatible braided crossing symmetric twist T on a finite dimensional Hilbert space H with norm $ \|T\| <1$. This implies that those algebras have finite non-microstates free Fisher information and therefore are always factors of type IIIλ (0<λ ≤ 1) or II₁. Moreover, using the (nontracial) free monotone transport [ 19], we show that LT(H) is isomorphic to the free Araki–Woods algebra L₀(H) when $ \|T\|=q $ is small enough.