A boundary approach to rigidity of $q$-Araki--Woods factors

In this paper, we study structural and rigidity properties of $q$-Araki–Woods factors arising from strongly continuous orthogonal representations of $\mathbb{R}$. We first introduce an AO-type boundary property that is formulated in terms of length projections on the $q$-Fock spaces, and prove that all $q$-Araki–Woods factors satisfy this condition. We next characterize uniform convergence of the associated malleable deformations. We then use these two ingredients as a substitute for weak containment in the coarse bimodule and deduce several rigidity theorems in the framework of Popa’s deformation/rigidity theory. As a main consequence, we prove that every $q$-Araki–Woods factor is stably strongly solid. We also prove a normalizer rigidity theorem for inclusions arising from direct sum representations. Finally, we compute Connes’ $\tau$-invariant, and construct a continuum of pairwise non-isomorphic type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ $q$-Araki–Woods factors with the same $\tau$-invariant.