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 $\dim H_{\mathbb {R}}\geq 2$. 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 $\dim H_{\mathbb {R}}\geq 2$ and $U_{t}$ admits an invariant eigenvector.