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

Abstract

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 ${\Gamma }_q(H_{\mathbf{R}},U_t{)}^{\prime \prime }$ the $q$-deformed Araki-Woods algebra introduced by Shlyakhtenko and Hiai. In this paper, we prove that ${\Gamma }_q(H_{\mathbf{R}},U_t{)}^{\prime \prime }$ has trivial bicentralizer if it is a type $\mathrm{II}{\mathrm{I}}_1$ factor. In particular, we obtain that ${\Gamma }_q(H_{\mathbf{R}},U_t{)}^{\prime \prime }$ always admits a maximal abelian subalgebra that is the range of a faithful normal conditional expectation. Moreover, using Sniady's work, we derive that ${\Gamma }_q(H_{\mathbf{R}},U_t{)}^{\prime \prime }$ is a full factor provided that the weakly mixing part of $(H_{\mathbf{R}},U_t)$ is nonzero.

14 pages