Ultraproduct methods for mixed $q$-Gaussian algebras

Marius Junge and Qiang Zeng

Vol. 331 (2024), No. 1, 99–147

DOI: 10.2140/pjm.2024.331.99

Abstract

We provide a unified ultraproduct approach for constructing Wick words in mixed $q$-Gaussian algebras which are generated by $s_j=a_j+a_j^{\ast }$ for $j=1,\dots <!--FUNCTION\; APPLICATION-->,N$, where $a_i{a}_j^{\ast }-q_{ij}{a}_j^{\ast }{a}_i={\delta }_{ij}$. Here we also allow equality in $-1\le q_{ij}=q_{ji}\le 1$. Using the ultraproduct method, we construct an approximate comultiplication 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 $L_p$ Poincaré inequalities with constants $C\sqrt{p}$.

Keywords

$q$-Gaussian algebras, Wick product, hypercontractivity, Riesz transform, Poincaré inequality, approximation property, strong solidity

Mathematical Subject Classification 2010

Primary: 46L36, 46L53

Secondary: 46N50, 81S05

Milestones

Received: 2 October 2016

Accepted: 24 August 2024

Published: 2 October 2024