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

We consider the mixed $q$-Gaussian algebras introduced by Speicher which are generated by the variables $X_i=l_i+l_i^{\ast },i=1,\dots ,N$, 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 ${\max }_{i,j}\mathrm{\mid }{q}_{ij}\mathrm{\mid }$ is small enough. The proof relies on some estimates which are generalizations of Dabrowski’s results for the special case $q_{ij}\equiv q$.