Structural and non-isomorphism results for $q$-Araki-Woods factors

It is proved that the $q$-Araki-Woods factor ${\mathrm{\Gamma }}_q({\text{sH}}_{\mathbb{R}},U{)}^{\mathrm{\prime }\mathrm{\prime }}$ associated with a strongly continuous orthogonal representation $U:\mathbb{R}\to \text{cO}({\text{sH}}_{\mathbb{R}})$ is strongly solid for all $q\in (-1,1)$ if the representation $U$ is almost periodic. We also show that the $q$-Araki-Woods factor ${\mathrm{\Gamma }}_q({\text{sH}}_{\mathbb{R}},U{)}^{\mathrm{\prime }\mathrm{\prime }}$ is not isomorphic to any free Araki-Woods factor for any $q\in (-1,1)\setminus \{0\}$ if the representation $U$ has nontrivial weakly mixing part or infinite dimensional almost periodic part with bounded spectrum.