A Conjugate System for Twisted Araki-Woods von Neumann Algebras of finite dimensional spaces

Abstract: We compute the conjugate system of twisted Araki-Woods von Neumann algebras $\mathcal{L}<em>T(H)$ for a compatible braided crossing symmetric twist $T$ on a finite dimensional Hilbert space $\mathcal{H}$ with norm $ |T| &lt;1$. This implies that those algebras have finite non-microstates free Fisher information and therefore are always factors of type $\text{III}</em>\lambda$ ($0&lt;\lambda\leq 1$) or $\text{II}_1$. Moreover, using the nontracial free monotone transport, we show that $\mathcal{L}_T(H)$ is isomorphic to the free Araki-Woods algebra $\mathcal{L}_0(H)$ when $|T|=q$ is small enough.