Ultraproduct methods for mixed $q$-Gaussian algebras

We provide a unified ultraproduct approach for constructing Wick words in mixed $q$-Gaussian algebras, which are generated by $s_j=a_j+a_j^*$, $j=1,\cdots,N$, where $a_ia^*_j - q_{ij}a^*_ja_i =\delta_{ij}$. Here we also allow equality in $-1\le q_{ij}=q_{ji}\le 1$. Using the ultraproduct method, we construct an approximate co-multiplication of the mixed $q$-Gaussian algebras. Based on this we prove that these algebras are weakly amenable and strongly solid in the sense of Ozawa and Popa. We also encode Speicher's central limit theorem in the unified ultraproduct method, and show that the Ornstein--Uhlenbeck semigroup is hypercontractive, the Riesz transform associated to the number operator is bounded, and the number operator satisfies the $L_p$ Poincar\'e inequalities with constants $C\sqrt{p}$.