On the isomorphism class of $q$-Gaussian W${}^{\ast }$-algebras for infinite variables

Let $M_q(H_{\mathbb{R}})$ be the $q$-Gaussian von Neumann algebra associated with a separable infinite dimensional real Hilbert space $H_{\mathbb{R}}$ where $-1<q<1$. We show that $M_q(H_{\mathbb{R}})\simeq ̸M_0(H_{\mathbb{R}})$ for $-1<q≠0<1$. The C${}^{\ast }$-algebraic counterpart of this result was obtained recently in [BCKW22]. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.