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- W1574361912 abstract "We provide a unified 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 $-1le q_{ij}=q_{ji}le 1$. This approach relies on Speicher's central limit theorem and the ultraproduct of von Neumann algebras. We also use the unified argument to 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 $Csqrt{p}$. Finally we prove that the mixed $q$-Gaussian algebra is weakly amenable and strongly solid in the sense of Ozawa and Popa. Our approach is mainly combinatorial and probabilistic. The results in this paper can be regarded as generalizations of previous results due to Speicher, Biane, Lust-Piquard, Avsec, et al." @default.
- W1574361912 created "2016-06-24" @default.
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- W1574361912 date "2015-05-28" @default.
- W1574361912 modified "2023-09-27" @default.
- W1574361912 title "Mixed $q$-Gaussian algebras" @default.
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