Generator masas in $q$-deformed Araki-Woods von Neumann algebras and factoriality

TL;DR

This paper investigates the structure of $q$-deformed Araki-Woods von Neumann algebras, proving factoriality under certain conditions and analyzing the properties of generator masas within these factors.

Contribution

It establishes factoriality of these algebras for non-ergodic orthogonal representations with dimension at least 2 and studies the mixing properties of generator masas.

Findings

Von Neumann algebras are factors when the representation is not ergodic and dimension ≥ 2.

The centralizer of the $q$-quasi free state has trivial relative commutant.

Generator masas are shown to be strongly mixing.

Abstract

To any strongly continuous orthogonal representation of $\R$ on a real Hilbert space $\CH_\R$, Hiai constructed $q$-deformed Araki-Woods von Neumann algebras for $-1< q< 1$, which are $W^{\ast}$-algebras arising from non tracial representations of the $q$-commutation relations, the latter yielding an interpolation between the Bosonic and Fermionic statistics. We prove that if the orthogonal representation is not ergodic then these von Neumann algebras are factors whenever $dim(\CH_\R)\geq 2$ and $q\in (-1,1)$. In such case, the centralizer of the $q$-quasi free state has trivial relative commutant. In the process, we study `generator masas' in these factors and establish that they are strongly mixing. The analysis is inspired by a previous work of \'{E}. Ricard on Bo$\overset{.}{\text{z}}$ejko-Speicher's factors \cite{ER} and measure-multiplicity invariant of masas introduced by K. Dykema, A. Sinclair and R. Smith in \cite{DSS06}.