Free Araki-Woods factors and Connes' bicentralizer problem

We show that for any free Araki-Woods factor $\mathcal{M}=\mathrm{\Gamma }(H_{\mathbb{R}},U_t)\mathrm{"}$ of type $\mathrm{I}\mathrm{I}{\mathrm{I}}_1$, the bicentralizer of the free quasi-free state ${\phi }_U$ is trivial. Using Haagerup’s Theorem, it follows that there always exists a faithful normal state $\psi$ on $\mathcal{M}$ such that $({\mathcal{M}}^{\psi }{)}^{\mathrm{\prime }}\cap \mathcal{M}=\text{C}$.