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

TL;DR

This paper computes the conjugate system for twisted Araki-Woods von Neumann algebras with finite-dimensional spaces, showing they have finite free Fisher information and are isomorphic to free Araki-Woods algebras under certain conditions.

Contribution

It introduces a conjugate system for twisted Araki-Woods algebras and establishes their classification as factors of specific types, also demonstrating isomorphism to free Araki-Woods algebras in the small twist norm regime.

Findings

Algebras have finite non-microstates free Fisher information.

They are factors of type III_λ or II_1.

Isomorphism to free Araki-Woods algebra for small twist norm.

Abstract

We compute the conjugate system of twisted Araki-Woods von Neumann algebras $ \mathcal{L}_T(H) $ for a compatible braided crossing symmetric twist $T$ on a finite dimensional Hilbert space $ \mathcal{H} $ with norm $ \|T\| <1$. This implies that those algebras have finite non-microstates free Fisher information and therefore are always factors of type $\text{III}_\lambda$ ($0<\lambda\leq 1$) 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.