Singularity of the generator subalgebra in mixed $q$-Gaussian algebras

We prove that for the mixed $q$-Gaussian algebra $Γ_{Q}(H_{\mathbb{R}})$ associated to a real Hilbert space $H_{\mathbb{R}}$ and a real symmetric matrix $Q=(q_{ij})$ with $\sup|q_{ij}|<1$, the generator subalgebra is singular.

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