Remarks on factoriality and $q$-deformations

Abstract

We prove that the mixed $q$-Gaussian algebra ${\Gamma }_Q(H_{\mathbb{R}})$ associated to a real Hilbert space $H_{\mathbb{R}}$ and a real symmetric matrix $Q=(q_{ij})$ with $\sup |q_{ij}|<1$, is a factor as soon as $\dim H_{\mathbb{R}}\ge 2$. We also discuss the factoriality of $q$-deformed Araki-Woods algebras, in particular showing that the $q$-deformed Araki-Woods algebra ${\Gamma }_q(H_{\mathbb{R}},U_t)$ given by a real Hilbert space $H_{\mathbb{R}}$ and a strongly continuous group $U_t$ is a factor when $\dim H_{\mathbb{R}}\ge 2$ and $U_t$ admits an invariant eigenvector.

9 pages; v2 corrects a few minor points. The paper will appear in the Proceedings of the AMS