Full solution of the factoriality question for $q$-Araki-Woods von Neumann algebras via conjugate variables

TL;DR

This paper proves that $q$-Araki-Woods von Neumann algebras are factors with specific properties, extending previous methods to a broader setting and determining their type, non-injectivity, and solidity.

Contribution

It provides a full solution to the factoriality question for $q$-Araki-Woods algebras using conjugate variables, generalizing prior results beyond the tracial case.

Findings

Proved factoriality for $q$-Araki-Woods algebras with at least two generators.

Established non-injectivity and determined the type of these factors.

Showed that the factors are solid and full when finitely generated.

Abstract

We establish factoriality of $q$-Araki-Woods von Neumann algebras (with the number of generators at least two) in full generality, exploiting the approach via conjugate variables developed recently in the tracial case by Akihiro Miyagawa and Roland Speicher, and abstract results of Brent Nelson. We also establish non-injectivity and determine the type of the factors in question. The factors are solid and full when the number of generators is finite.