Connes' bicentralizer problem for q-deformed Araki-Woods algebras

Let $(H_{\mathbf{R}},U_t)$ be any strongly continuous orthogonal representation of $\mathbf{R}$ on a real (separable) Hilbert space $H_{\mathbf{R}}$. For any $q\in (-1,1)$, we denote by ${\mathrm{\Gamma }}_q(H_{\mathbf{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that ${\mathrm{\Gamma }}_q(H_{\mathbf{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ has trivial bicentralizer if it is a type $\mathrm{I}\mathrm{I}{\mathrm{I}}_1$ factor. In particular, we obtain that ${\mathrm{\Gamma }}_q(H_{\mathbf{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady’s work, we derive that ${\mathrm{\Gamma }}_q(H_{\mathbf{R}},U_t{)}^{\mathrm{\prime }\mathrm{\prime }}$ is a full factor provided that the weakly mixing part of $(H_{\mathbf{R}},U_t)$ is nonzero.