An application of free transport to mixed $q$-Gaussian algebras

TL;DR

This paper demonstrates that mixed $q$-Gaussian algebras are isomorphic to free group von Neumann algebras using free transport methods, under conditions of small maximum $|q_{ij}|$ values.

Contribution

It extends free transport techniques to mixed $q$-Gaussian algebras, establishing their isomorphism to free group algebras for small $q_{ij}$.

Findings

Mixed $q$-Gaussian algebras are isomorphic to free group von Neumann algebras when $|q_{ij}|$ are sufficiently small.

The proof generalizes Dabrowski's estimates to the mixed $q$-case.

The approach relies on free monotone transport theorem of Guionnet and Shlyakhtenko.

Abstract

We consider the mixed $q$-Gaussian algebras introduced by Speicher which are generated by the variables $X_i=l_i+l_i^*,i=1,\ldots,N$, where $l_i^* l_j-q_{ij}l_j l_i^*=\delta_{i,j}$ and $-1<q_{ij}=q_{ji}<1$. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed $q$-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra $L(\mathbb{F}_N)$, provided that $\max_{i,j}|q_{ij}|$ is small enough. The proof relies on some estimates which are generalizations of Dabrowski's results for the special case $q_{ij}\equiv q$.