On the factoriality of q-deformed Araki-Woods von Neumann algebras

TL;DR

This paper investigates the factoriality of q-deformed Araki-Woods von Neumann algebras, establishing conditions under which these algebras are factors depending on the deformation parameter and the dimension of the underlying real Hilbert space.

Contribution

It provides new results on when q-deformed Araki-Woods algebras are factors, extending known cases to include low-dimensional and small deformation parameter scenarios.

Findings

For all q in (-1,1), the algebras are factors if the real Hilbert space dimension is at least 3.

When the dimension is 2, the algebras are factors if the deformation parameter is small or trivial.

The results unify and extend previous knowledge on the factoriality of these algebras.

Abstract

The $q$-deformed Araki-Woods von Neumann algebras $\Gamma_q(\mathcal{H}_\mathbb{R}, U_t)^{\prime \prime}$ are factors for all $q\in (-1,1)$ whenever $dim(\mathcal{H}_\mathbb{R})\geq 3$. When $dim(\mathcal{H}_\mathbb{R})=2$ they are factors as well for all $q$ so long as the parameter defining $(U_t)$ is `small' or $1$ $($trivial$)$ as the case may be.