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Let $M_q(H_{\mathbb{R}})$ be the $q$-Gaussian von Neumann algebra associatedwith a separable infinite dimensional real Hilbert space $H_{\mathbb{R}}$ where$-1 < q < 1$. We show that $M_q(H_{\mathbb{R}}) \not \simeqM_0(H_{\mathbb{R}})$ for $-1 < q \not = 0 < 1$. The C$^\ast$-algebraiccounterpart of this result was obtained recently in [BCKW22]. Using ideas ofOzawa we show that this non-isomorphism result also holds on the level of vonNeumann algebras.

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