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 $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ has trivial bicentralizer if it is a type $\rm III_1$ factor. In particular, we obtain that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady's work, we derive that $Γ_q(H_{\mathbf{R}},U_t)^{\prime\prime}$ is a full factor provided that the weakly mixing part of $(H_{\mathbf{R}}, U_t)$ is nonzero.

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